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Medical_Lectures
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Thyroid_Hormones_and_Thyroid_Function_Tests.txt
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[Music] today I will be discussing thyroid function tests the learning objectives will be first to be able to describe the general structure and function of the hormones involved in thyroid gland regulation second to be able to diagram the relationship between the thyroid hormones next to use thyroid function tests to diagnose the four General categories of thyroidal illness and last to list non-thyroidal conditions which can affect tfts tfts can be a confusing topic for a number of reasons their interpretation superficially seems like it should be incredibly easy and for the majority of patients it is however a significant minority of tfts don't conform to what logic suggests for reasons I'll discuss also the biosynthesis of thyroid hormone is unusually complex but as most of the synthetic steps are of minimal clinical relevance I'll be covering only the most important aspects finally even the terminology surrounding tfts is confusing while the term thyroid hormone generally refers to one of two specific and nearly identical compounds there are nevertheless a total of 4 hormones involved in the process I'll briefly review each the first is thyrotropin releasing hormone or TR TR is a tripeptide amide which essentially means it's composed of three amino acids it's formed in the hypothalamus and travels to the anterior pituitary via the hypothalamic hypothal portal system the primary effect of TR is to stimulate release in the pituitary of the next hormone in the pathway TSH another effect of TR which won't be discussed further in this particular video is to stimulate secretion of prolactin now TSH known forly as thyroid stimulating hormone as well as thyrotropin is a glycoprotein it has a number of different specific effects all of which are focused on increasing the physiologic actions of thyroid hormone these are to increase the release of preformed thyroid hormone to increase the rate of thyroid hormone formation and to increase the size and number of thyroid cells which produce that hormone next let me talk about the true thyroid hormones that is those hormones which are actually synthesized and released by the thyroid gland they are almost universally called T3 and T4 though formally are triiodothyronine and thyroxine as mentioned a minute ago the biosynthesis of T3 and T4 is very complicated but requires tyrosine and iodine the tyrosine used for thyroid hormone is actually stored as a glycoprotein called thyroglobulin each molecule of which has about 70 molecules of tyrosine and the thyroid hormone is actually synthesized while still attached to the larger thyro globulin and remain so until it is secreted the thyroid gland is relatively unique among endocrine organs in its ability to store large amounts of pre-formed hormone this can become relevant in diseases such as Hashimoto thyroiditis and autoimmune mediated destruction of the thyroid gland which can transiently result in hyper thyroidism due to abrupt release of hormone in normal physiologic conditions in response to TSH the thyroid secretes predominantly T4 along with a small amount of T3 the ratio of T4 to T3 is about 90% to 10% after they're released over 99% of the thyroid hormones are bound to plasma proteins in the circulation the predominant one is called thyroxine binding globulin or tbg some is also bound to albumin as well as a protein called transthyretin so named because it's responsible for the transport of thyroxine and retinol only the tiny free Unbound form of thyroid hormone is physiologically active the effect of protein binding is clinically important because changes in the concentration of thyroxine binding globulin can affect the concentration of Unbound hormone for example in steroid use and in curosis less of thyroxin binding globulin is produced resulting in a higher proportion of free hormones the opposite effect occurs during pregnancy in which the liver produces higher than normal amounts of thyroxin binding globulin now despite the fact that the thyroid secretes much more T4 than T3 the biological activity of T3 is much greater than T4 the peripheral tissues convert some of the relatively inactive T4 to the relatively active T3 as well as some to the completely inactive reverse T3 also known as rt3 in fact some recommend considering T4 more of an inactive pro hormone and T3 really the only proper thyroid hormone what do T3 and T4 do they have the greatest diversity of actions of any hormone their primary effects are first to increase the basil metabolic rate which results in increased heat generation and oxygen consumption second they sort of rev up metabolism specifically increasing gluconeogenesis glycolysis glucose absorption from the GI tract lipolysis and protein turnover next they stimulate bone maturation and growth and their last major effect under normal physiologic conditions is to increase cardiac output by increasing both the heart rate and contractility you may have noticed that many of the physiological effects of the thyroid hormones are similar to those from the activation of the sympathetic nervous system and there may be interactions between the two which are not yet understood explaining why beta blockers are a commonly employed a treatment for some manifestations of hyperthyroidism I'll now move on to discuss how thyroid hormones are regulated which will partly be a visual representation of what was just covered it begins in the brain at the hypothalamus where TR is released into the hypophysial portal system to be delivered directly to the anterior pituitary where it stimulates TSH release and TSH released into the systemic circulation travels to the thyroid where it stimulates among other things the formation of predominantly T4 from thyroid globuline and iodine T4 travels in the bloodstream largely bound to thyroxine binding globulin and when it reaches certain peripheral tissues it's converted to T3 by an enzyme more accurately a collection of very similar enzymes called deiodinases which as their name implies can add or remove an iodine Atom from a molecule there actually a number of different names for these enzymes in medical textbooks and in the literature one of the more common alternate names is five Prime iodase now not all of the T4 gets converted to T3 as some also gets converted into the molecule reverse T3 which as I said before is inactive both active T3 and reverse T3 only differ in which of the four iodine atoms were removed by the enzyme normally there's about a one to1 ratio of T3 to reverse T3 that it's produced however pregnancy fasting hepatic and renal failure and beta blockers all result in preferential conversion to reverse T3 thus decreasing the amount of active hormone there are a number of important negative feedback mechanisms in this pathway first the thyroid hormones themselves exert negative feedback on the hypothalamus and pituitary to reduce secretion of TR and TSH respectively so for example if TSH starts to get too high this leads to higher levels of T3 and T4 which leads to Greater inhibition of TSH thus the system can keep TSH and thyroid hormones within a specific window in addition to hormonal negative feedback there are a few others as well for example both physiologic and emotional stress can inhibit TR and TSH which is a likely contributor to the U thyroid 6 syndrome to be discussed in a few minutes in addition exposure to coold temperature appears to inhibit TR though the clinical significance of this in humans appears to be low the final point to bring up with this diagram concerns the two possible effects of an acute iodine load on the actions of the thyroid the first possible effect is called the wolf chof effect which is a reduction in thyroid hormone levels seen after an acute iodine load which presumably evolved as a mean to prevent hyperthyroidism in response to being suddenly provided a high amount of substrate this effect explains both the hypothyroidism seen by some patients after starting the iodine containing anti- rythmic medication amiodarone as well as the use of highd do iodine after nuclear emergencies as it will prevent uptake of radioactive iodine from the atmosphere the second possible effect of an acute iodine load is called the yode bastal effect which is stimulation of the thyroid gland's production of hormone unlike the wolf chof effect which can be seen in patients with completely normal thyroids the yode bastal effect is only observed in patients with pre-existing thyroid pathology a typical scenario occurs when the patient with a goer and hypothyroidism from chronic iodine deficiency moves to an area where iodine is abundant in the diet it can also be seen in patients with a multi-nodular goiter or Graves disease who has started on am oone if you feel confused by these two seemingly contradictory effects don't worry you're not alone luckily with the exception of amoon use these effects come up pretty uncommonly in routine clinical practice okay well now I'm finally going to get to discuss what you've probably been waiting for the actual thyroid function tests or tfts tfts are complicated due to the variety of substances that can be measured and the variety of specific assays that different Labs can use the only three tests which are commonly checked are TSH free T4 and to a lesser extent free T3 we typically only order T3 when we are specifically concerned about hyper thyroidism because it will be more sensitive based on our current understanding of thyroid disease and current lab technology TSH free T4 plus or minus fre T3 are more or less the only choices within the domain of thyroid function tests that you will ever need to order or interpret unless you become an endoc chronologist in which case there are a few more that may come up from time to time some of these other tests which are uh sometimes commercially available for clinical purposes and sometimes used only for research uh include the following total T4 and T3 and something called the T3 resin uptake together these tests could be used to estimate free T4 and free T3 via a calculation called the free T4 T3 index luckily with the relatively recent wide availability of free T4 and T3 tests these others are no longer necessary another rarely ordered test is the reverse T3 it was once invogue as a means to distinguish true thyroid disease from TFT abnormalities caused by non-thyroidal illness although physiologically this use seems to make sense its usefulness has not panned out and it's now phenomenally rare for a conventional doctor to check one having said that reverse T3 has become very popular among doctors and other providers working more on the fringes of medicine where they often Market it as the critical hormone your own doctor won't tell you about as if endocrinologists somehow make more money by intentionally not diagnosing thyroid disease next is TR I honest honestly don't know if the fact it's so rarely ordered is because it's not available not validated for clinical use generally not helpful or all three there's also thyroxine binding globulin which I've never seen ordered on a patient and last is thyroid globulin which actually has a very important though uncommon role in following differentiated thyroid cancer since thyroid globulin is only produced by thyroid folicular cells having detectable levels in the serum is indicative of residual thyroid tissue being present thus if a patient has had the thyroid gland removed entirely as part of the treatment of thyroid cancer but still has detectable thyroid globulin it is suggestive of recurrent or metastatic disease so if TSH freet T4 and freet T3 are the only tests which are typically used to diagnose thyroid pathology and free T4 and T3 typically Trend together it should be pretty easy to interpret tfts well here's a basic summary the most common scenario is for TSH to be high and thyroid hormones to be low that occurs in primary hypothyroidism meaning there is something wrong with the thyroid gland itself if the gland is diseased and not producing the hormone there's less negative feedback on the hypothalamus and pituitary leading to the secondary increase in TSH a less common combination is a low sh and high hormones as seen with primary hyper thyroidism including the situation of excess exogenous thyroid replacement much much rare combinations include low TSH and low free T4 and T3 which is referred to as Central hypothyroidism which the problem may be in the pituitary or theoretically the hypothalamus and last is something called secondary hyperthyroidism in which there is a TSH producing tumor somewhere so interpretation of tfts doesn't seem so bad does it four combinations four General Diagnostic categories unfortunately in a significant minority of patients it can be substantially more challenging on account of the effect of non-thyroidal conditions on tfts tfts can be affected by non-thyroidal conditions for a variety of reasons there could be transient acquired pituitary dysfunction and critical illness an alteration the level of thyroid bonding globulin an increase in circulating free fatty acids which displace thyroid hormone from thyroxine binding globulin decreased peripheral conversion of T4 to T3 and an altered ratio of T3 to rt3 the net consequence of these potential effects may be some combination of low TSH either low or high free T4 low free T3 and high rt3 the abnormal tfts due to non-thyroidal illness is often referred to as either U thyroid SI syndrome or sick U thyroid syndrome common causes of euthyroid six syndrome include pregnancy any clinical illness liver disease renal disease malnutrition and various medications let's relook at how to interpret tfts keeping in mind you thyroid 6 syndrome as well as a few other diagnoses and this time let's have a column and row for a normal TSH and normal thyroid hormones first the obvious now the most common scenario other than everything being normal is still high TSH and low T4 plusus T3 which is consistent with just primary hypothyroidism there isn't really anything else that commonly causes that P pattern in the event that TSH is low and T4 and T3 are high that's still consistent with primary hyperthyroidism but it's also consistent with u thyroid 6 syndrome particularly if the free T4 is high but freet T3 is low if everything is low it can be due to either Central hypothyroidism as before as well as you thyroid 6 syndrome if everything is high it can still be due to secondary hyperthyroidism from a TSH produc ucing tumor it can also be due to a very rare inherited disorder of thyroid hormone transport or hormone metabolism now if the thyroid hormones are normal but TSH is not we have a category of thyroid disease called subclinical hyper thyroidism or subclinical hypothyroidism I'm not a huge fan of the terms since usually subclinical means without symptoms but in this case the diagnosis is essentially based just on lab tests I suppose the patient is presumably not symptomatic if thyroid hormone is in the normal range since that's the final mediator of thyroid function but the term still feels a little misused to me in addition a low TSH with normal T4 and T3 can also be from U thyroid 6 syndrome a high TSH with normal T4 and T3 can occur during the recovery phase of you thyroid 6 syndrome a normal TSH with low T4 and T3 can be from you thyroid 6 syndrome and a normal TSH with high T4 and T3 is sometimes seen in acute psychiatric illness and various drugs most notably amiodarone can cause almost any pattern of TFT abnormalities so as you can see interpretation of tfts can become far more complicated if a patient's results don't fall neatly into one of the four classic patterns previously shown when it comes to ordering tfts the selection of tests is very straightforward remember unless you are testing for something highly unusual only worry about TSH freet T4 and free T3 if you have a low suspicion for a thyroid disease and the patient is acutely ill consider deferring tfts until the patient's better it can be very difficult to sort out you thyroid 6 Syndrome from True thyroid pathology in acutely ill patients if the patient is not acutely ill check the TSH by itself first if it's normal consider yourself done with no further testing if it's high check a fre T4 to distinguish hypothyroidism from subclinical hypothyroidism if it's low check a fre T4 and freet T3 if the freet T3 is elevated it's consistent with hyperthyroidism if it's normal the patient likely has subclinical hyperthyroidism if there is high suspicion for hypothyroidism check a TSH and free T4 and if there is a high suspicion for hyper thyroidism check a TSH free T4 and free T3 you could ask if we always check a T3 when hyperthyroidism is suspected why in the world do we bother with a T4 at all that's a good question I don't really have a good answer for it I don't think I've ever seen a T3 ordered without a T4 but I'm not positive a T4 is actually always necessary I'm going to end the video with a discussion of three more tests which aren't technically thyroid function tests though they are logically included with this topic these are the anti-thyroid antibodies the first two of these antibodies are anti-thyroglobulin antibody an anti-thyroid peroxidase antibody thyroid peroxidase is an enzy enzyme involved in the synthesis of thyroid hormone at least one of these is seen in almost all cases of Hashimoto's thyroiditis the most common ideology of hypothyroidism the last category of antibody is anti-tsh receptor antibody of which there are three subtypes stimulating blocking or neutral Graves disease a common cause of hyperthyroidism is due to stimulating anti-tsh receptor anti body while various anti-tsh receptor antibodies can be seen in different stages of Hashimoto measurement of these antibodies is not always needed for making the associated diagnosis if the history and exam strongly favor a particular diagnosis already however if the history or exam is inconsistent with a common ideology of either hypo or hyperthyroidism or if there is concern about confounding U thyroid 6 syndrome in a critically own patient with symptoms or signs consistent with thyroid disease checking antibodies can be helpful that concludes this video on thyroid function tests and their interpretation if you found it interesting or useful please remember to like and share it
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Medical_Lectures
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Introduction_A_Gut_Reaction_to_Obesity_The_Impact_of_Diet_the_Microbiome_and_the_Environment.txt
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okay good morning and thank you all for joining us my name is rachel mandlebaum and i am the director of speakers for the 2015 ucla healthcare symposium and a second year medical student at ucla we are very proud to present to you the 19th annual ucla healthcare symposium a gut reaction to obesity the impact of diet the microbiome and the environment the symposium is an annual event organized by medical students at the david geffen school of medicine for students of all areas as well as ucla faculty staff and community members today's topic focuses on one of the most pressing medical financial and public health crises of our time the obesity epidemic our expert speakers will discuss obesity from unique but intricately interrelated perspectives all crucial to curbing this epidemic we will first start on a microscopic level quite literally exploring the microbes with whom we share our bodies and the newest research on their impact on diet obesity and other diseases from there we will shift to a more macroscopic view of obesity and explore the public health realities food industries incentives and marketing strategies and innovative solutions for the future we will delve into the fundament the fundamental questions of how the obesity epidemic reached this order of magnitude and most importantly where do we go from here we have a very exciting and enriching program in store and we encourage you all to actively participate in today's events before we get started i want to go over a few housekeeping items with you all first of all we have had a change in agenda our first speaker had a family emergency and had to cancel this morning so we switched around the agenda a little bit so our first speaker will be dr meyer this morning and we'll have a little more time for coffee break and everything should run a little more comfortably other housekeeping items please feel free to use the restroom at any time the restrooms are located outside the main entrance doors to the left in terms of raffle you all probably received a raffle ticket so those are going to be entered to receive either gift cards from places like whole foods sprouts we also have a company called aura aquaponics that has donated a really great system where it grows a plant and a fish lives underneath the fish creates fertilizer for the plant and the plant filters the water it's great so enter to win one of those opportunities to get extra raffle enterings if you hashtag on facebook twitter instagram hashtag ucla hcs standing for ucla healthcare symposium hashtag healthycampus and for undergraduates if you're interested in entering to win a mcat class or an mcat set of books please also add hashtag mcat and you'll get an extra entrance into the raffle in terms of asking questions today we'll have one q a a change to your agenda we'll have one q a session instead of two right before the panel discussion so please write down your questions clearly on the cards on your tables and hand them to either our student directors our medical student directors or medical student volunteers who will be circling the room and then selected questions will be answered by dr dr schlesser our mcm panel moderator at the q a i would now like to introduce dr clarence braddock our faculty advisor for this event to deliver some welcome remarks to all dr braddock is a phi beta kappa graduate of stanford university and an aoa graduate of the university of chicago pritzker school of medicine he trained in internal medicine at the u.s naval hospital in oakland followed by service at the naval hospitals in naples italy and in oakland dr braddock has been on faculty at ucsf the university of washington stanford and we are honored that he has chosen to recently bring his wealth of experience and leadership to ucla at ucla dr braddock serves as our vice dean of education and he oversees all aspects of medical education including undergraduate graduate and postgraduate medical programs dr braddock is a national leader in medical education curriculum development and innovation with influential work in physician patient communication and informed decision making my fellow student directors and i have felt extremely privileged to have had the opportunity to work closely with dr braddock while planning this event thank you all again for joining us this morning for today's symposium and please join me now in welcoming dr braddock well good morning good okay i want to make sure everyone's there it's just so exciting to see all of you here today so many people coming for what's going to be just a fabulous day of conversation learning and and also i hope interaction i want to start with with a couple of uh thanks i want to thank rachel and her colleagues uh mustafa jennifer elise and becky you guys have done a phenomenal amount of work i i don't know if folks in the audience recognize that these medical students while taking classes and studying and they have a little board exam and a little bit that they're not trying to think about today they've done a phenomenal amount of work planning the program selecting the topic selecting our fabulous speakers and i think they've done just a terrific job so thank you i also want to thank our speakers for taking time out on a saturday morning to come and and present your thoughts reflections and expertise and share them with us and we're deeply grateful for uh for your contributions and to thank all of you for coming um i think the food was pretty good but it is a saturday morning so uh but thank you for coming how many just out of curiosity how many students do we have here great faculty fabulous community members staff terrific just a terrific turnout we couldn't be more pleased and again this is a the 19th of these so it's been a great tradition the david geffen school of medicine and a wonderful way for our students to really embody what we talk about as our goal which is to create world leaders in health and science and that connection between health and science is apropos today science informs how we think about health and disease but we're trying now to broaden our thinking in medicine and at the health system here to think about not just treating disease and not just prevention but how do we actually promote health of our communities and this this symposium really embodies that that step towards a new way of framing what we in the ucla health system can do to foster the health in our community so i'm now going to introduce my colleague and friend dr wendy schlesser who will be our emcee today dr schlesser is a clinical professor and associate vice provost for the healthy campus initiative the healthy campus initiative was envisioned and supported by jane and terry semel and as the name implies it's a broad set of activities that dr schlesser directs over the campus including food nutrition physical activity and and more dr schlesser in her own right is a national figure in nutrition and health both promoting healthy behaviors and promoting nutritional change both at this campus and beyond so with that i'm delighted to welcome to the stage dr wendy schlesser well thank you dr braddock rebecca and the incredible medical student team that put this together i just am in awe they can now uh handle their weddings and everything else no problem okay i'd also like to have a special shout out thank you to julie quan who's the medical librarian who is responsible for helping put out these beautiful images the medicinal herbs on your table comes from our rare book collection and we got wind of it from the healthy campus initiative when we were planting these medicinal herbs just south of the ronald reagan hospital and it's open to the public there are amazing books uh these are original lithographs and watercolors so thank you so much julie and all the incredible talent um that she brings to the healthy camps initiative in this initiative i'd also like to uh thank the doctors who are walking the talk here the ones who organize this the food uh is incredible and these are our future doctors and i'm so proud of the fact that they are really embracing this kind of health and wellness and i have hope i have hope so today i have the great privilege to introduce all these incredible speakers and the first speaker is dr meyer and he'll discuss the microbiome neural impacts of obesity and he's a professor in the department of medicine physiology and psychiatry and biobehavioral sciences at ucla he's also executive director of the oppenheimer family center for neurobiology of stress and he's the co-director of the cure digestive disease research center at ucla he's the director of the nih funded center for neurovisceral sciences and women's health now he has a career-long interest in the role of the mind-brain body interactions in health and chronic disease his research efforts during the past few years have focused on several new areas of the brain gut interactions in particular on the role of the gut microbiota in influencing brain structure and function and associated behavior and on the role of food addiction and obesity he started his career in germany and after moving to the us he became a gastroenterologist and focused his work in basic translational and clinical aspects of brain gut interactions he's published over 300 peer-reviewed articles 90 chapters and reviews and co-edited four books and organized several interdisciplinary symposia in the area of visceral pain and mind-body interactions he served on the editorial boards of the leading journals and digestive diseases including gastroenterology gut digestion and the american journal of physiology and he served as a reviewer for a wide range of medical and neuroscience journals so please welcome dr meyer the brain gut microbiome access in obesity i just have a gut feeling this is gonna talk will keep you on the edge of your seats now i wanna tell you there's another thing i'd like to say keep track of all of those little sayings gut feeling you whoever gets the right amount will get a little prize at the end thanks you
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Medical_Lectures
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18_Biochemistry_Signaling_I_Lecture_for_Kevin_Aherns_BB_450550.txt
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Kevin Ahern: Well, sorry about that. I have one last thing to finish up saying about carbohydrates, in general, and then we're going to turn our attention to a very interesting phenomenon known as signaling. When we get into signaling, we start beginning to see how controls of cell division and processes like that lead to, ultimately, important health considerations for cancer and other things. The one thing I want to mention today about carbohydrates that I didn't finish up with last time also has some significant health considerations, as well, and it actually has to do with viral receptors. The example I have for you is that of the flu virus. It's the time of the year where the flu is moving around and many people have not gotten vaccinated against it. For students that want to go into health professions, I think that's pretty outrageous if you haven't gotten inoculated against the flu. But, in any event, that aside, flu is an important health consideration. What you see on the screen is an interesting depiction about how a flu virus affects a blood celló"infects," not "affects"óinfects a blood cell. The flu virus is a virus that contains RNA, not DNA. If you look inside the virus right here, you see several, one, two, three, four, five, six, seven, eight, actually, in this caseófragments of RNA that consist of the entire coding information for the flu virus. One of the questions people commonly ask about the flu virus is, "How come there are so many different types?" and so forth, and it's partly because of mixing and matching of different strands of RNA that can occur when you start mixing infections from different organisms. So flu virus is a very important, as I say, health consideration, and also very, very, variable in terms of the different forms that it can come up with. Well, a common feature of many of the forms of the flu virus is what you see on the screen. They infect blood cells by attaching to an extracellular component. You can see this extracellular component is actually here. The virus has, on its outside coat it has projections sticking out, and two of them are of interest to us. The first one is hemagglutinin. Hemagglutinin, again, as its name implies, is what's responsible for the virus agglutinatingóthat is, attaching itselfóto a blood cell, "hem - " referring to the blood cell. So hemagglutinin is a protein that recognizes and binds to a specific carbohydrate residue on the surface of a red blood cell. So this protein hemagglutinin, you can see it's projecting all around this virus. It's just basically waiting to latch onto the appropriate carbohydrate residue on the surface of a blood cell. Once it has latched onto that specific carbohydrate, then the virus has to get its RNA into the blood cell. It turns out that, in order for this to happen, that there has to be an opening created in the blood cell for the entry of the viral RNA. The opening creation requires action of this enzyme known as neuraminidase. What neuraminidase does is it cleaves a residue, a modified carbohydrate residue known as neuraminic acid, and that cleavage is necessary we can see it depicted over here for the entry of the viral RNA. So the combination of the hemagglutinin binding to a specific carbohydrate residue and the neuraminidase cleaving a neuraminic acid containing residue on the surface of the red blood cell allows the viral RNA of the flu virus to enter the cell, infect the cell and cause many more copies of the virus to be made as a result of that. Interestingly, this neuraminidase is a target of anti-flu drugs. When you hear of the anti-flu drug known as Tamiflu, it works because it is a neuraminidase inhibitor. It inhibits the action of neuraminidase. If the neuraminidase can't cleave that residue, there's no entry, there's no way for the viral RNAs to enter the blood cell and the flu virus is pretty much left waiting out there. So that's one place where cellular carbohydrate residues on the surface obviously play important roles in human health. Student: [unintelligible] Kevin Ahern: Sorry? Student: Do inoculations actually work against the cell or just boost your immune system? Kevin Ahern: Inoculations always boost your immune system and they are targeted at recognizing specific proteins on the surface of flu viruses, as they are for any virus. There are many other strategies for viruses, as well, but Tamiflu is a very cool one. That's what I want to say about carbohydrates. I want to turn our attention now to talking about cellular signaling, and I think you'll find some interesting and important considerations of signaling for human health. Signaling is essential for multicellular life. When we have differentiated cells of an organism, it's important that those cells of an organism all pull their oars in conjunction with each other, and that is coordinated by the action of small molecules that move through the body that basically communicate. Those small molecules are known as hormones, and hormones are basically produced in one part of the body, by cells in one part of the body. They travel, usually through the bloodstream, and get to their target tissues, where they bind to specific receptors and cause, inside of the cells of those target tissues, a response. That response might be, "Let's activate a bunch of enzymes," "Let's inactivate a bunch of enzymes," "Let's tell the cell to divide," "Let's tell the cell not to divide." All kinds of possible responses can happen as a result of the binding of hormones to the cell surface receptors. So we're going to spend some time talking about those receptors, as well as about a few of the signaling pathways that are there. I will caution you as I go into this, that this lecture is the beginning of a series of "This goes to this, goes to this, goes to this, goes to this," and it is important for you to understand and know what those pathways are. So if I talk about them here, yes, you will be responsible for them. Before we get to the "This goes to this, goes to this, goes to this," let's take a look at three receptors that we'll be talking about over the next day and a half or so. The three receptors are: the beta-adrenergic receptor that, as we will see, is very, very important in, ultimately, in controlling levels of glucose in the body; the insulin receptor, which plays a very important role in also controlling levels of lucose in the body, but it works in opposite fashion to the beta-adrenergic receptor. The insulin receptor also is involved in many other processes. It's not only involved in blood glucose. And a third receptor, the EGF receptor, which stands for "epidermal growth factor receptor," which is intimately involved in helping cells to decide, "Do I divide or do I not divide?" That decision of dividing or not dividing is a very important one, as we can imagine. If the cells are continually getting signals telling them to divide and they shouldn't be dividing, we may have uncontrolled growth and, of course, the definition of an uncontrolled growth is a cancer. This very simple figure shows what happens in signaling in the body. I told you that the tissues in one part of the body communicate by making a molecule that they release, called a hormone. They release that into the bloodstream of the body. It goes and travels to the place where it encounters other cells that have a receptor specific for binding to that. So that's the reception part of the process. As we will see, the reception part of the process invlolves not only binding of the molecule that was released óthat is, the hormoneó but that binding induces some very important structural changes inside of the receptor protein, the receptor protein being located in the membrane of the target cells, and the changes in that structure of the receptor protein results in a process we call "transduction." So when you hear the term "signal transduction" what we're talking about is the communication of information outside the cell to mediate a response inside of the cell. So that's the phenomenon of transduction. Transduction, in turn, causes those responses to happen that I talked about earlier. It might be activating an enzyme. It might be inactivating an enzyme. It might be activating entire classes of enzymes, proteins, quite a wide variety of things that can happen. Then, it's very important that cells be able to turn that process off. Cells are not one-way machines, as it were. They turn something on, they need to have the ability to turn it off. Question? Student: Could you say the definition of "signal transduction" one more time? Kevin Ahern: So signal transduction is the phenomenon whereby information outside the cell is communicated inside the cell. An example is a hormone binding to a receptor. That receptor has some shape changes, as we will see, that will cause several things to happen inside the cell as a result of that. But the transduction is just that general phenomenon. Now, turning this process off is also important. I will spend more time talking about turning the processes on, but I will point out to you some places where turning them off is important considerations. Now I want to introduce a term to you, "second messengers," before I introduce the term to you, "first messengers." That's kind of odd, but I need to do that. What is a second messenger? A second messenger is a molecule inside of a cell, and it's a molecule inside of a cell that is made as part of that signal transduction process. So it's made as a result of that signal transduction process. Well, now you ask the question, "What's the first messenger?" In the scheme that I've been depicting for you, the first messenger is the hormone. The hormone is extracellular. The first messenger is extracellular. "Hormone," "first messenger," I will use those terms interchangeably. It never makes it, at least in the scheme we'll be talking about here, it never makes it into the cell. It binds to a protein receptor on the cell surface. That protein changes shape. That shape change causes some things to happen, and those things happen inside the cell, but the hormone does not make it into the cell. Second messengers, in general, are very small molecules. Second messengers are not proteins. Second messengers, one we'll talk about a lot is cyclic AMP. Next term we'll talk briefly about cyclic GMP, because cyclic GMP plays a very important role in our vision. Cyclic AMP is a much more generic second messenger. It occurs in a lot of cells and is used for a lot of signaling purposes. Calcium, as we will see today, or if I don't finish today then certainly on Wednesday, plays an important role in this signal transduction, that is, the signaling process inside of cells, and it happens, it is not something that is made, but it is something that is released from various stores that cells have inside of them. Inositol 1,4,5-triphosphate or as you're more likely to know it, IP3, is an important messenger that is made as a result of action on a bigger molecule that also makes diacylglycerol, and I'll show you that later. So the four messengers that we will be concerned with in the lectures here are shown on the screen. Let's think about those receptors. Receptors are important because receptors, we remember, have to basically bind to that first messenger. They have to change shape upon binding, and that change of shape has to, somehow, result in the production of a second messenger. We will see that, in most cases, that production of a second messenger does not occur immediately but, instead, occurs after several steps later. There are different classes of receptors. One of the more common classes that we have on our cells are called "7TM receptors" and, no, you don't need to memorize this table. But you can see that 7TM receptors play some very important roles in a wide variety of processes: neurotransmission, hormone secretion, smell, taste, vision, embryogenesis, control of blood pressure. All of these things are very, very important processes that are mediated by 7TMs. So 7TMs play very, very important roles in all of these processes. Well, why do we call them "7TMs"? The reason we call them 7TM is if we schematically examine them, first of all, we remember that they are located in the membranes of target cells. Schematically, they look like this. They project through the lipid bilayer. This is the membrane, the outer membrane of the cell. They cross. here's the start. This is the N-terminus and here is the C-terminus. Here's the end, over here. They cross the membrane seven times. So the "TM" part of it doesn't stand for "transcendental meditation." In fact, it stands for "transmembrane." "Seven transmembrane domain," that's what it's called. Yes, sir? Student: Is the amino or carboxyl end preferentially on the inside or outside of the cell? Kevin Ahern: Yes. It will generally be as you see it here. So this arrangement is common among many, many different receptors involved in these processes that I'll be describing to you. Now, this is a very simplistic way of depicting the way they look. A more realistic way of how they actually appear in three dimensions is something like what you see on the screen. There are some similarities. Here's one involved in rhodopsin. Rhodopsin is light sensitive, for example. Here's one that is a beta-adrenergic receptor that we'll be talking about, here. But we see some similarities in terms of the organization of the seven transmembrane domains. You'll notice in the middle of these 7TMs that there is a binding site for a molecule. That binding site for the molecule is, in fact, the first messenger. So the 7TM has a binding site for the first messenger to come and do its thing. The one I'll be talking about first is the beta-adrenergic receptor, as I said earlier, and the beta-adrenergic receptor is sensitive to, that means it binds to, the hormone epinephrine. Epinephrine is shown on the screen. No, you don't need to know the structure of it, but I will tell you that epinephrine is derived from tyrosine, and epinephrine is also known as "adrenaline." So epinephrine is the thing that we produce when we get scared, we get anxious or whatever. It's produce in other conditions, as well, but a big dump of epinephrine can have an enormous effect on our bodies, as we shall see a little bit today, but much more in the next couple of weeks. So epinephrine is a first messenger. It's a hormone produced by the adrenal glands. It is essential for the flight-or-fight response, basically. Now, what happens in this response? I'm going to show you the first half of that today and I'm going to allude to the second half of it, and then I'll show you more detail about the second half of it in about two weeks. Let's imagine that I am out going for a hike in the woods and I discover that there's a grizzly bear that is on my tail. So the grizzly bear starts to chase me and I realize that I'd better get my butt moving or I'm going to be in trouble. I get scared and my body produces epinephrine. Epinephrine goes and binds to target cells. Now, these target cells that I'll be describing to you here we can think of as muscle and/or liver cells, because these are both important for us to produce glucose, in the case of the liver, and get away, in the case of the muscle cells. So epinephrine is released into the bloodstream. It travels. It hits the receptor, and when it hits the receptor, as I've said previously, what happens is the receptor goes through a slight change of shape upon binding. So this guy has bound to epinephrine. It's the little yellow ball inside of there. That schlight. "schlight." That slight change of shape what did I have to drink before I came to class today, right? Some Schlitz. The slight change of shape causes the interaction of the 7TM that is, the beta-adrenergic receptoróit causes the interaction between it and a cellular protein known as a G protein to change. So this G protein is normally just sitting here, right next to the 7TM. It's sitting here right next to the 7TM. Binding of epinephrine changes the interaction between these two. You can see the result of this change is that this G protein which has three proteins in it, known as alpha, beta and gamma, changes from holding GDP to holding GTP. That's number one. Now, I will tell you that, first of all, that is a replacement reaction. That is, the receptor does not make GTP. It causes this guy to dump its GDP and pick up GTP. That action causes a change in the shape of the G protein. This whole complex is known as the G protein, by the way. It causes a change in the shape of this G protein, such that, when GTP is bound, the beta and the gamma subunits no longer bind to the alpha subunit. [student sneezes] Gesundheit! [student sneezes] Gesundheit, again. Why is that important? Well, it turns out that the beta and the gamma subunits, when they bind to the alpha, they cover up a region of the alpha that would otherwise bind to an enzyme. The enzyme is known as adenylate cyclase. So we can see this process happening. When this guy has GTP, it sheds its beta gamma subunits and now can interact with this enzyme known as adenylate cyclase again, the "-ase" telling us it's an enzyme. Adenylate cyclase, you can see, is also a membrane protein. But it's a membrane protein that is an enzyme. When the alpha subunit of the G protein binds to adenylate cyclase, we see that adenylate cyclase catalyzes the formation, of cyclic AMP from ATP. ATP is converted into cyclic AMP. Now we've seen several steps happening in this process: binding of the hormone, alteration of the interaction with the G protein, replacement of the GDP on the G protein with GTP, the interaction of the GTP with the alpha subunit of the 7TM with the adenylate cyclase, and now adenylate cyclase is activated to make cyclic AMP. Well, you remember from our discussion last week that protein kinase A is allosterically activated by cyclic AMP, and that's what's depicted on the screen here. So we see this being activated, and, if you recall what I said about protein kinase A, I said its name told you what it does: "kinase" means it puts phosphate onto, "protein" means it's putting phosphates onto proteins. So what the result of this entire action of the screen is, is that protein kinase A has now been converted from an inactive form to an active form. It will start putting phosphates onto serines and threonines of target proteins. What we will see in a couple of weeks is really interesting with that. I'm going to give you just sort of a preview of that right here. Putting phosphates onto target proteins is going to affect those proteins, and it generally has an effect of either turning them way on or turning them way off. A really good example, in our liver, for example, our liver has glycogen. I told you last week that glycogen was a storage carbohydrate for glucose. We store it so when we need glucose we can make it. We can release it from glycogen. We have glycogen in our liver because we have enzymes that make it. We also have enzymes in our liver that can break it down. Now, what protein kinase A does is it puts phosphates onto both the enzymes that make glycogen as well as the enzymes that break down glycogen. Why is that important? Well, it has opposite effects on them. Putting phosphates onto the enzymes that break down glycogen activates them. Putting phosphates onto enzymes that make glycogen inactivates them. That's kind of important. We don't want to be making glycogen as quickly as we're breaking it down at the same time. Student: Say that again? Kevin Ahern: Okay. So putting phosphates onto enzymes that break down glycogen, activates them, and putting phosphates onto enzymes that make glycogen inactivates them. I'll talk about that, as I said, in the next couple of weeks, so don't panic on that. I'm just giving you a broad view here. Let's think about what's happened with this hormone action. I got scared. My adrenal glands produced epinephrine. Epinephrine went out into the bloodstream. It bound to target receptors. Let's think about these for the moment in the liver. That caused a G protein to be activated, and putting a GTP in it is what activates it. That activation allows it to interact with adenylate cyclase. Adenylate cyclase makes cyclic AMP. Cyclic AMP activates protein kinase, and what is protein kinase doing in the liver? It's stimulating the breakdown of glycogen, and breakdown of glycogen gives me, ultimately, glucose. I got scared. My blood supply gets a giant infusion of glucose. When you hear the stories about people who see a baby under an automobile and they go out and they grab the automobile up and they pick it up because they got scared, those are real, because of the enormous dump of glucose that's happening as a result of this hormone action. So this pathway is pretty phenomenal. You look at this, wow, there's a lot of steps to this pathway. This process happens in seconds and it actually would happen faster if the hormone itself didn't have to travel through the bloodstream. So this is a pretty phenomenal process and it happens really rapidly. This process is the one I usually start with in talking about signaling because it gives us a good taste of what signaling pathways are like. We like to think about, well, one thing happens and then all of a sudden the cell responds. But, in fact, we saw several things that had to happen here, sequentially, in order for the signal to be communicated. There was our second messenger, right there. We had to go through all of this before we made this second messenger. As we will see, cyclic AMP has many effects inside of cells. One of them is activating protein kinase. There are other effects that it has, as well. It might be a good place for me to tell you a story about cyclic AMP. Cyclic AMP, when I say we turn this process on, we also like to have ways of turning this process off. Right? Well, let me show you one of them. One of the processes that we have to turn off is shown right here. Actually, I'll leave that there. Here's my activated G protein. I've got GTP in there. I want to and I need to not leave that thing in the active state because if I leave the G protein with its GTP, what's going to happen? It's going to stimulate the production of a lot of cyclic AMP, and the production of a lot of cyclic AMP is going to activate a lot of protein kinase, and that's going to activate a lot of glycogen breakdown enzymes, and, bang! all of a sudden, I burn up all my glycogen. I need to control that fairly readily. For G proteins, cells have a very interesting but a very odd way of controlling them. G proteins get their name from the fact that you find them carrying guanine nucleotides, either a GDP, where it's inactive, which is what we see over here, or a GTP, where it's active. How do we get the GDP from the GTP? Well, you can see right here that there's a hydrolysis reaction that occurs, and it turns out that G proteins are really bad enzymes. I'll repeat that. G proteins are really bad enzymes. Bad in what sense? They're terribly inefficient. What do they catalyze? They catalyze the breakdown of GTP. They catalyze the breakdown of the very thing that activates them. They bind to it and over the course of minutes, they'll say, "Okay, I'm going to break you down," and when they break it down, they basically turn themselves off. So G proteins are self-regulating. They turn themselves off over time. That keeps the cell from making too much activated enzyme for breaking down glycogen. That's very important. We don't want to be doing that. So G proteins are very inefficient enzymes. How about some other considerations? Well, what happens if I have a receptor that binds to epinephrine, but the epinephrine gets stuck? In the normal scheme of things, it just goes backwards this way, dissociates, the epinephrine is gone. The cell doesn't go on and do its thing. So it stops activating G proteins and the process stops because the G proteins, in turn, inactivate themselves. But what happens if that gets stuck in there? Well, one of the considerations if it gets stuck in there, is cells have yet another way of turning off the beta-adrenergic receptor. That's by action of this enzyme known as receptor kinase. What receptor kinase does is it puts phosphates onto that C-terminus of the G protein. I'm sorry. Not the G protein, the C-terminus of the 7TM. That's important because now those phosphorylated residues on the C-terminus are a target for binding by the enzyme known as arrestin. This now binds the 7TM and stops the 7TM from activating G proteins. So the cell has a way of turning off that signal if there's a problem with the receptor. Again, the fact that this machinery is built into the cell says something very important about the need to control signaling processes. If I don't control signaling processes, just like I don't control enzymes, I'm in deep doodoo. Well, that's good. That's all fine and dandy. Student: The arrestin is attracted to the phosphates? Kevin Ahern: Beta arrestin binds to the phosphates on the beta-adrenergic receptor. Well, that's fine and dandy, but there's one thing I haven't told you. I've shown you how we can knock out the epinephrine, I've shown you that the G protein turns itself on. What about this guy over here? Once I've made it, isn't it just going to sit there forever? If I have this cyclic AMP, isn't it just going to sit there? Even if I turn everything else off, isn't cyclic AMP going to be a problem? Well, it turns out, no. Cells have an enzyme floating around inside of them, ubiquitously, known as phosphodiesterase. What does phosphodiesterase do? Well, it breaks down cyclic AMP. That means that cyclic AMP, if we look at the cyclic AMP levels in the cell, we see that when signaling happens, they go up, but they fairly quickly come back down. The reason that they come back down is the phosphodiesterase starts catching up and starts breaking down that cyclic AMP. So now we've seen three things here that can help to shut down this signal when cells don't want to have it going all the time. I tell you this because it's very interesting that a critical player in this process is phosphodiesterase. Phosphodiesterase is a target for a very important drug. It's known as caffeine. Caffeine inhibits phosphodiesterase. Now, I'd like you to think about the buzz you get from drinking your coffee. The buzz is real, and, by the way, the buzz occurs at a couple of levels. I'm only describing one level to you. But one of the levels at which it occurs is you are inhibiting phosphodiesterase, therefore you have less breakdown of cyclic AMP. Less breakdown of cyclic AMP means more breakdown of glycogen. More breakdown of glycogen means more blood glucose. I just got a buzz! Kind of cool. Questions about that? I'll slow down. Shannon? Student: And that's why you crash, right? Kevin Ahern: That's why you crash? You will crash, yeah, and the other reason that you crash is people don't just drink coffee. They drink what I describe as chocolate milk syrup cream macchiato latte espresso. They've got all this sugar crap that's in there, that not only is their body dumping sugar out there, but they've got all this stuff that they've put into their system, so the blood glucose levels go, "Bo-ing!" As we will see, when the body sees blood glucose levels going "Bo-ing!" glucose is a poison, so the body acts to take it out of the bloodstream and there's your crash. I'll talk about that in a bit. Yes? Student: Without all the sugar and stuff you might add to your coffee, would just the caffeine be bad for a diabetic, then, because it can alter. Kevin Ahern: It's a very common question. She's asked if caffeine is bad for a diabetic. It's one of the most common questions I get and I don't know. I suspect for some people it could be a problem, but in general, diabetics have problems more with what they ingest than what their body is producing. Yes? Student: What about synthetic sugars? Would your body recognize those and crash? Kevin Ahern: What about synthetic, you mean like artificial sweeteners? Student: Yeah. Kevin Ahern: Does your body crash from artificial sweeteners? The idea of the artificial sweetener is that you stimulate the sense receptors that taste sweet and there's no calories that's contained in them. For a long time, it was felt that in fact, that there was very little response that happened to that. But there have been recent studies now that have suggested that artificial sweeteners, in fact, actually are inducing the production of insulin, which is what happens in the consumption of sugar. It's not as pronounced, but there is some production that's there, not because of the process that I'll describe to you, but probably more likely because of a learned response in your brain. Okay, so you learned something about how your body works, there. There's adenylate cyclase. It is not a 7TM, but you see it's bound in the membrane. Blah, blah. What else do I want to say here? There's words telling you what I've showed you on a figure there. That's what I want to say about the beta-adrenergic receptor. I've got some time, so I want to talk now about another 7TM system. I'm not going to talk about the first messenger, and that's not really important for our purposes. The process I want to talk about here is another 7TM system. It involves a receptor. It involves a G protein. The receptor that most commonly is associated with this is called "angiotensin," which is an important receptor for modulating blood pressure, so some of the things that I have to say here will have effects, ultimately, on blood pressure. The angiotensin receptor is involved in using a different kind of second messenger. The second messenger that it uses is called "PIP2." PIP2 is basically a compound that's found in the membrane of cells. You know that cells have a lipid bilayer. That lipid bilayer tends to have molecules that are long and nonpolar stuck into the layer, and then a polar portion that is projecting out of the layer. In this case, "out of" means on the inner portion of the membrane. What this signaling pathway does is it activates an enzyme known as phospholipase C. Phospholipase C acts by cleaving this guy in the membrane into two pieces. Both pieces are second messengers. So when the angiotensin receptor gets stimulated, it activates phospholipase C by action of a G protein. Phospholipase C then catalyzes the breakdown of PIP2 to DAG, D-A-G, and this long name which you can call IP3. So that's what up in this signaling pathway. Now, if we look at it at the level of the cell, this is what it looks like. Here's the cell membrane. You'll notice that we're not depicting, in this case, the angiotensin receptor out there, at all. But we have, in this membrane, we have some PIP2. When the angiotensin receptor activates the G protein, the G protein activates phospholipase C, and phospholipase C now cleaves PIP2 and makes two things. One is, it makes DAG, which remains in the membrane, and second, it makes IP3, which is water soluble, and IP3 leaves the membrane and travels into, in this case, a calcium storage reservoir. It's labeled as "ER" here, endoplasmic reticulum. Sometimes you'll see it labeled as "sarcoplasmic reticulum." They're both involved in sequestering calcium ions. So what's happened? Receptor has activated a G protein. G protein activates phospholipase C. Phospholipase C makes DAG and it makes IP3. IP3 travels to a receptoróthis is a second receptor, now in the endoplasmic reticulum and binds to it. When it binds to it, it causes the receptor to open up and let calcium out. That, then, increases the concentration of calcium in the cytoplasm and an increased concentration of calcium in the cytoplasm goes and binds to this protein called "protein kinase C." Protein kinase C requires two things to be active. It requires calcium and it requires DAG. The combination of these activate, now, a different protein kinase. This different protein kinase then is active in phosphorylating a variety of target proteins that mediate a cell's response. I always like to point out, when I talk about this, that though we don't really talk much about muscular contraction in this class, that you should know, ultimately from your basic biology classes, that calcium is a signal to initiate muscular contraction. Calcium is a signal to initiate muscular contraction. Angiotensin is favoring the forming of tension by stimulating the release of calcium in these target cells, and we could imagine how this tension in target blood vessels, for example, could affect blood pressure. So calcium is described in this system as a second messenger. I would describe it, actually, as a third messenger, and you can call it either one, as far as I'm concerned, for the exam. But I think you can make a case for it being a third messenger, because here it is produced as a result of action of a second messenger, IP3. IP3 is a second messenger. DAG is a second messenger. Calcium, if you call it a second or a third, it really doesn't matter, but it happens only after the action of a second messenger. Student: Couldn't you also think of IP3, then, as a first messenger? Kevin Ahern: No, it's not a first messenger because first messengers will always be outside the cell. So that is the phospholipase C system. Now calcium, it turns out, in the cell is a bit of a problem for cells. The reason it's a bit of a problem for cells, you saw that the cell had it sequestered in the endoplasmic reticulum. There's a reason it keeps most of the calcium in the endoplasmic reticulum. A, it allows it to release it and signal so that's kind of important, but even more importantly is the fact that calcium really likes to bind to DNA. Calcium binding to DNA can actually cause your chromosomes to precipitate. So you don't want that calcium concentration to be too high. So it's let out in little batches. One of the ways that cells keep the calcium concentration low, even when calcium is released, is by using calcium binding proteins to help communicate the signal that, "Hey, calcium's been released." Now, when we examine the structure of these calcium binding proteins in the cell, we discover they all have a common shape, and the common shape is that they're known as "EF hands." It's depicted here. You can see the finger in yellow. You can see the sort of fist of the hand down here below, and you can see that the calcium site is right here in where these fingers are curled around. This common feature is found in many calcium binding proteins. One of the most abundant calcium binding proteins that we have inside of cells is known as calmodulin, C-A-L-M-O-D-U-L-I-N. Calmodulin binds to calcium. It has EF hands, and the binding of calcium by calmodulin induces, not surprisingly, a big structural change in calmodulin. Here's calmodulin without calcium bound to it. Here's calmodulin with calcium bound to it. We see that the binding of calcium induces a big structural change and that big structural change allows calmodulin to interact with, in this case, something called "CaM kinase" that it couldn't interact with before. Why is that important? Well, if calcium is a signaling ion and calcium is a problem in concentration, what if I have a protein that gobbles up the calcium but still communicates the signal? That's what calmodulin is doing. It's saying, "Hey, calcium has been released. Do your thing." In this case, it's activating a kinase by binding to it, and that activation happens only because this protein has bound to calcium. So tricks that cells use to counter the effects of high concentrations of calcium arise because of this binding, in this case, of calcium by calmodulin. Let's see here. I'll start and I won't finish this. I'll start one last thing. The last thing I want to say very briefly about is insulin. When I introduced the hormones originally, I said to you that insulin is important because it really counteracts the effects of the beta-adrenergic receptor. The beta-adrenergic receptor, when it's activated, stimulates an increase in blood glucose, by the process I described to you. Blood glucose levels go very high. Well, glucose in our bloodstream is a poison. If there's one message I want you to take out of this class, that's it. It's a poison. In high levels, glucose is a problem. So our body has a defense against glucose being a poison. It's insulin. Insulin is also a hormone. Insulin is a first messenger. What insulin does is it stimulates target cells to take up glucose, thereby lowering the glucose concentration in the bloodstream. You're sitting there saying, "But you said it was a poison. Cells are taking up a poison." If cells did nothing with it, they would have a problem. But cells do things with glucose. They may burn it. They may store it in the form of glycogen. They may do other things to it, but the important thing is, the blood glucose levels are falling. It's glucose in the bloodstream that is the real problem. That's how we get kidney damage. That's why some diabetics have to have limbs amputated, for example. That's why they may go blind. Because they've got too much glucose in their bloodstream. Now, next time I will tell you how insulin does all of that. It's a pretty cool processóand remind you again about the poisonous nature of that compound. Captioning provided by Disability Access Services at Oregon State University
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Medical_Lectures
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Immunology_Lecture_MiniCourse_13_of_14_FailuresHost_Defense_Mechanisms.txt
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some have the short span of Life how does that happen well uh the first question again the first question is how how how do immune boosters differ from anti retroviral mean has immune treatment or immune response they are like there are these things that are being sold here mostly in South Africa they call them immune boosters every shop you go to immune boosters Bo yeah differ from anti retroviral okay so as far as the first question is concerned since I don't know what these immune Bo boosters consist of I can't give you a specific answer but what what I would tell you is that uh now that I think a lot of you have more of a context of the immune response you could appreciate that immune boosters may be a double-edged sword because what HIV preferentially infect are activated te- cells and macrofagos so if you have a resting te- cell it's a lot more difficult to infect it with HIV people who work in the laboratory know that if you take peripheral blood from a patient put it into tissue culture and add HIV it's very difficult to infect the only way that you could really infect those cells is that you need to give them uh in vitro a mitogen Incan 2 maybe anti CD3 and anti-cd 28 and then only then can get infection right people familiar with that well think in terms of a patient IM immune boosters may actually be again if they make the te- cells activated they may actually make things worse and in addition people who have an underlying infection may be more susceptible to HIV because their immune system is activated so again I just want to number one say that if it really does activate te- cells it may actually not be beneficial but the second point I would make is that uh immune I mean anti-retal viral therapy works there's absolutely no question about that I mean anybody who has taken care of patients before art was available knows what the natural course of the disease was within a period of time patients went uh deteriorated ultimately died and lost weight I mean the whole clinical Spectrum when you put patients on anti retal viral they can live relatively normal lives so that's so art definitely works okay so immune boosters I can't give you a specific answer on I mean clearly if and again it depends and that actually goes to your third question the third question is why does some people go have a more rapid deterioration than other individuals and again that's one of the reasons why a lot of the studies are going on here looking at a large group of patients that try to start answering those questions but one observation has been made clearly that HLA type plays a significant role so for example HLA b57 has been associated with individuals that are able to control HIV infection better than others well why should HLA play a role well now that you've learned some you know Immunology aot Immunology what would you postulate why would someone whose HLA b57 be better able to control HIV infection well okay let's think about it what do HLA what do what do MHC molecules do they present peptides and what defines what kind of peptides an HLA molecule can present the anchor motifs right so again I'm just throwing this out but it may very well be that some HLA types may have unique anchor motifs that uniquely allow them to express peptides from HIV that may not be as that may be from highly conserved regions that that are more difficult to mutate in that case if you mount a strong CTL response against those epitopes it may be able to control better I mean that's one possibility but again you know clearly HLA is variable and again we all know that everybody's immune system is different so there are some people that they're they get a cold they can't clear it they're always coughing and they're always getting one you know infection after another and other people are never sick a day in their life and part of that we say is cuz the qual the quality people's immune responses may be different and that translates into HIV and the other uh answer that people have proposed is it depends upon the strain of the virus and that's again another kind of semi controversy is it the virus or is it the immune system and when I was at when I was at uh uh at Aaron Diamond I think about 10 years ago to give a talk everybody there was wearing a button that said it's the virus stupid cuz like you know that's what they thought just it has nothing to do with the immune system it has just something to do with the virus and I'm sure that in in Bruce's lab people walking around with button saying the immune system stupid uh and clearly it's a combination of both but there's no question that there are some strains of HIV because of the fact that their uh long terminal repeat or their accessory proteins may work a little bit better are more virulent so that is another possible explanation of that and the second question is how does Narine prevent transmission again it's actually quite straightforward uh the lower the level of virus in the patient the less virus there is to be transmitted so if you treat somebody and you bring their viral loads down to non-detectable levels there's just not a lot of virus in the bloodstream so therefore either at Birth when the baby's exposed to maternal blood or potentially in utero you just don't have that same level of exposure to the high level of inoculum and the same thing is too in terms of transmission individuals that have high viral loads tend to be higher transmitters individuals that have much lower viral loads okay so I mean you know classic if if you would the experiments were done if you would take like you know a million a million uh salmonella and drink it you'll you can get very sick if you take like 10,000 or a th000 salmonella it doesn't bother you at all so clearly the amount of the inoculum plays a role and what the Narine does is it dramatically lowers the maternal level of HIV and therefore when the baby's exposed to the blood it just don't have that much virus there to infect the baby so does that mean that if you come into first contact with HIV you don't get infected no if you come if you can absolutely be infected with HIV the first time that you're exposed to it there's no if it's a high enough if it's a high enough inoculum you could be exposed to you can get it better so it depends upon the the viruses in the body well that's one variable I mean you know the numbers that people quote is that the transmission rate in in in terms of sexual encounter is between one and 200 and one in 400 so that sounds you know like good odds right but we all know that that that one in 200 could happen the first time or could happen the the 200th time uh so that it's all question of now if someone has 200 exposures then the Chan of them getting infected are pretty much much going be almost 100% just from a probability point of view but there's also variables in terms of mucosal tears in terms of blood blood exposure so there's a lot of variations and as you know things like circumcision is another variable that people have identify in terms of transmission okay okay fantastic so again for me the most exciting part of teaching is actually stimulating questions because when you stimulate questions you're really demonstrating that number one people have an understanding of the material and the second point is that people are thinking and triggering thinking is really the most important thing to do with students because as we get older uh older people really aren't as good at thinking so we need younger people to do a thinking for them question I have a ter CH hi exploit the normal rout maybe Lang to but there seems to be some evidence of it entering into the epithelial cells with transcytosis how do you explain that the way I would explain it is is that there multiple roots that the virus can take and uh and which one it chooses to take it's it's it's it's uh um maybe random um but at the end of the day act the virus ultimately has to come in contact with a cell that it could productively infect and to my knowledge there's not a lot of evidence that epithelial cells can be productively infected in a way that they then turn around and make a lot of virus so if you even if you infect an epithelial celf the virus sits there integrated but doesn't make any more virus then patient's not going to get infected so at the end of the day the virus has to infect a CD4 positive t- cell or maccrage in order now to trigger the the the cycle of expansion of virus and then expanding infection but there's no question there's more it's like a building okay most buildings have more than one entrance so which entrance that you choose to use it's a variable this is the in terms of being in vacular so there even seems to be some talk about Epal cell in the kidney for example that I'm aware of that might be latent res it's not enough evidence but there's talk about it right and and and in in the uh early 80s there was talk of HIV infecting colonic epithelial cells and there's actually data for that and probably does um and but again in terms of a latent infection you really don't need a lot of virus as long as it then is able to infect a te- cell where it could then take off so think about like a fire if you have a tiny spark but there's no material that could that's flamable nearby you could you know be making Spark s for your whole life and you're not going to start a fire but if you do Sparks right in front of very very flammable Tinder one spark folds on that Tinder and then poof the whole thing goes up and flames and if it's in a house the whole house can burn down so in terms of being a reservoir for latent infection you don't need to pump out a lot of virus as long as those virion can now turn around and infect a te- cell or a maage that can now dramatically expand it okay does that help okay so any more questions yeah I've always assumed that trauma played an important role in in in sexual transmission of HIV and because you needed some sort of break in the mucosa but it appear that that might not be the case because you mentioned that dritic cells do have the ability to so so how would you explain that the the the increased Transmission in homosexual Behavior as opposed to heterosexual Behavior yeah you know that's a question that a lot of people have debated for a long period of time there's no question that trauma facilitates the ability of HIV to to get into the uh bloodstream and I mean cause infection get get past the epithelial barrier and cause infection and in fact in animal models if you traumatize the um genital mucosa you actually have increased uh infection rate um and clearly uh there's also the issue of co-infection so in addition some individual may be infected with other sexually transmitted diseases which also compromises their mucosal Integrity so again that is is an addition so you clearly can get transmission across an intact mucosa but clearly if there are breaks in the mucosa or the mucosa is is compromised you probably increase the transmission rate which makes sense and would you say that STI increase that risk because you were more likely to have dritic cells with dritic cells come into the muc at a higher you mean again you you're you know on clearly postulating things that may very well be the case you have a lot more dendritic cells they're activated your tea cells that are present there are activated again as I mentioned to you as you know HIV preferentially infects activated te- cells so if you have a STD going on and you have a lot of te- cells that are activated in the lymph node now with the dendritic cell brings the virus to the lymph node that you probably have a higher chance of getting infected okay great okay so so now uh we'll this lecture is actually talking about failures and host defense mechanisms that are congenitally uh immune deficiencies that have a lot of the similarities to HIV but they're not caused by an infectious agent that caused by mutations and questions that I want to consider and very few of you here are clinicians so the way I like to structure this lecture is not like these are the immuno deficiencies that patients can get and you need to know about them but rather what insights into the working of the immune system can we learn from patients because a lot of these patients have mutations that affect immune cells immune molecules and then the question is what are the manifestations of that and a lot of things that we rationally think should happen because of what we know about Immunology from invitro studies turns out not necessarily to be true when we look at patients so the same way you do an experiment with mice you knock out the gene see what the phenotype is to a degree we could do the same thing in humans they have genetically knocked out genes because of a mutation and now we look at the patient see what the phenotype is and that teaches us what proteins and what signal what pathways are critical in having an appropriate immune response and then we could dissect exactly what stages these uh different uh mutations impact on it how how does uh this obviously is not as relevant to you but how does one list at a clinical history but it's also interesting just to know how do utilize our knowledge of the maturation of the immune system to understand what kind of pathogens can infect an individual and well how can ignorance of Immunology kill your patients so if people are interested in learning about that I'm more than happy to teach that but um this is a picture of David and if you recall from the uh first lecture that I gave which uh is only four days ago but uh may seem like a while uh is you know he had uh had severe Comin imuno deficiency and when he was born he was placed in a bubble and this actually illustrates one of the early bubbles that he was placed in and all the toys that he had for example everything had to be sterilized so clearly this is before Fisher Price because all they had back then were blocks but and then here you could see this is a a glove box so the same way when you're doing uh p uh pl3 TB work everything is done in a glove box that's the environment that he lived in because if he had no uh functioning Bells C cells and NK cells he was basically susceptible to infections with every pathogen so the way that he was being protected was by putting him in a sterile environment and therefore even though he had no functioning immune system he wasn't exposed to any pathogens and therefore he was able to do fine did he have did he have a sterile bow um so so he the question is did he have a sterile bowel and they actually did some stool smears and he probably had some commensals that weren't uh affecting him it's another interesting uh aspect that if you have mice that are nomic they're completely sterile a lot of them actually develop volvulus of their intestines because having no commensals actually is apparently not very good for it so he may have had low levels and they actually the original papers probably they did some stool smears but I don't recall that data but I I I I totally uh would say that he wasn't absolutely positively sterile now you also have to understand something else he had a perfectly functioning innate immune system his macrophages worked fine his polymorphonuclear lucites looked fine worked fine so in essence it wasn't like he had no immune system and as you learned from one of the lectures if an angel asks you what's immune limb would you want to give up right you'd rather give up your acquired immune as opposed to inate and he had a fully functioning inate and that may be sufficient to control small inoculums of kenils and as he grew up again he got larger and larger quarters but again he has another Glo box for example and again everything was sterile and this is what he looked like uh when he was becoming a teenager again everything was done through through gloves and again as I mentioned earlier unfortunately he passed away but the question at that time was why did he have an imuno deficiency and that time no one could really answer that question and in addition what can we do to treat him now as opposed to a patient with HIV where you could treat with drugs because you could basically prevent HIV replication and therefore now treat the underlying cause and in the case of A congenital disease that won't help the only way you could really treat this individual is by correcting their genetic defect well uh back then the way they tried to do that was by giving a bone marrow transplant by giving cells from someone that had a functioning immune system but unfortunately he succumbed to complications of the treatment but as I'll go on since then we have now have developed technology that actually allows us to treat this disease or this congenital disease now now if you ask the question what are the causes of imuno deficiency in the general population this is a very very helpful pie chart because over half of them are purely B cell imuno deficiencies and again based on what you know about Immunology it probably makes a lot of sense because imunoglobulin rearrangement is a very complicated process it involves a large number of proteins and there's a lot of places for things to go wrong uh fosic uh diseases about 14% And combination BNT cells are about 25% of the imuno deficiency so again the the numbers all suggest the most common are B cell amuno deficiencies if you want to evaluate a child that has con that has you suspect of congenital amuno deficiency the first uh question you want to ask is do they have increased frequency of infection now that makes a lot of sense because if you don't have a good immune system you can't you get a lot of infections but what's also important is the inability to clear infection so normally we get sick we we're sick but then we get better because we're able to clear we we amount an appropriate immune response and then we clear the infection if you lack the ability to clear the infection that suggest is something wrong with your immune system so you always have ear infections you're always coughing also what's a really red flag is infection with opportunistic pathogens and that many of you are familiar with with HIV it's the same concept someone who's infected with pyus for example it normally it doesn't normally infect individuals with a functional immune system that suggests that something is wrong because normally these pathogens are very easily controlled by the immune system if you are having infection it means that there's something wrong with your immune system that you can control these Ager presentation is actually very very important so if patients are are are fine until 6 months of age that indicates an antibody deficiency any can anybody tell me why that should be excuse me matern exactly maternal antibes because maternal antibes cross the placenta in utero and the halflife of IG is between a month or two for the first six months of Life the baby still has enough antibod to protect the baby by six months that antibody is gone and then the individual is susceptible to infection however if they are report being sick in the first few months of life that now suggests they have a cellular imuno deficiency something wrong with their tea cells because clearly maternal T cells do not cross the placenta okay so that's you know right there without even doing any lab tests you could actually get a tremendous amount of information and this is to illustrate the point that you made is this is in Udo and during uh during uh placental life maternal antibod specifically IG crosses the placenta as I had mentioned previously through active transport through FC mediated transport across the placenta and for the most part migm does not cross the placenta IG does not cross the placenta only IGG G does and in fact at Birth the baby's IGG levels are actually slightly higher than the mother's now like a minor point is is that if you have a premature baby so let's say you have a baby that's born at 28 weeks which is about um 6 months well you look here and you see that the levels of IGG that have crossed the placenta is much much much lower in fact that's why premature babies are at much higher risk for developing infections because they have not yet gotten that high level of antibody which probably comes in the last two uh to three months of gestation so that's something also that has clinical relevance this particular slide actually shows an example what's called transient hypog gam globin of the newborn for some reason some babies takes them a little bit longer to to to initiate their production of imunoglobulin and but again it's a benign uh condition just a matter of of waiting for their immunoglobulin ultimately to kick in and afterwards they do very very fine okay in addition as I mentioned before you want to look at what pathogens are infecting them so again based on what you now know if an individual is getting a lot of bacterial infections what do you think their defect is going to be antibodies because that plays the critical role in protecting from bacterial infections if they have a lot of viral infections or fungal infections what Lim of the immune system do think is compromised te- cells because that plays important role there so again by getting a good clinical history and taking what you know about Immunology you could actually U make a very good educated guess about what's going on this is basically a review for is showing that you have the PLO PLO hematopoetic stem cell it differentiates to the common lymphoid progenitor B cell T cell and K cell and on and then the myoid differentiation but the point of this slide is again just to tell you that clearly different genes play critical roles at different maturation Forks in this differentiation pathway and if the mutation is at that protein then you impact any of the downstream maturational processes so now is having a mutation that prevents maturation of the plop poent hematopoetic stem cell is that compatible with Life why do not well right you're not going to get PL and you're not going to get red blood cells so we don't see that clearly so for the most part we don't see a lot of myoid mutations uh in terms of differentiation but again if you have a mutation that affects the common lymphoid progenitor you won't have any B cells T cells and K cells if you have a mutation that affects t- cell maturation you won't have t- cells but you'll have normal numbers of B cells and you could have M if you have a mutation that impacts on suass imunoglobulin maturation you'll have normal cells and t- cells and K cells but you lack immunog Global in some classes okay so that it's pretty straightforward and again as you all know very well you could basically characterize lymphocytes based on unique surface markers they express and this is a very this is the way we we tend to we we would evaluate a child with suspected immuno deficiency ask what are the cells that this individual has and it's kind of like I guess uh in a soccer match you know if you want to know the team is losing and you want to know what's the problem with the team you have to look at the individual players and say what player is not playing well what player is missing so for example if you just happen to forget to have a goalie on your team right you could imagine that the other team is going to score a lot of goals so you basically analyze and say oh missing a goalie that's what the problem is clearly that is a uh a fixable problem so this way you know do do you have B cells do you have T cells do you have CD4 cells Etc Etc and again all of you are familiar with this basically uh using flow cetric analysis this is showing a two-color analysis and this is looking at B cells and here this is IGM versus igd and here you have a population of cells that is both IGM positive and igd positive and what what would you call this kind of cell a what a mature B cell because you know enough now to know that if this B cell expressed IGM so you'd have a very nice population here but it did not express igd what would you call that B cell you call immature B cell so again this is very nice way of of identifying both maturational state as well as the cell population so what David had was severe Comin Amino deficiency he had Marly decreased uh T cells and B cells interesting that some of them actually could have normal cells the cells just for some reason just aren't very functional he had markedly decreased immunoglobin level it presented pretty much after birth and these individuals can be infected with bacterial viral fungal pathogens if they don't have functioning B cells and t- cells they're basically wide open to the spectrum of infection theologies is of mutation and therapy at that point is bone transplant and I'll discuss a little bit later how gene therapy is being utilized to treat this now this is showing again the hematopoetic lineage but in the case of of David where where it was skid it was xlink which means that it affected males and did not affect females and therefore the postulate was is that there must be some mutation that is affecting A protein that plays a critical role in NK cell T cell and B cell maturation from the common lymphoid progenitor okay is that makes that's clear but what's interesting about David for example is that David otherwise was perfectly normal a lot of the amuno deficiencies affect proteins that not only are critical for the the immune system but they're also critical for other aspects of differentiation so they'll have other manifestations of the mutation but in this case it was purely impacting on the immune system which led to a question of What gene is only affecting the immune system and this uh turned out to be really very interesting observation because as you recall the Incan 2 receptor has an alpha chain a beta chain and a gamma chain and we but then when we they did Knockouts of Incan 2 and mice it turned out that mice had as you may recall had a relatively normal immune function if you knocked out the alpha chain and the beta chain of the Incan 2 receptor it turned out that they had a similar relatively normal immune system but then they knocked out the gamma chain of the Incan 2 receptor and guess what these mice had severe combinant amuno deficiency so intuitively it made no sense why if you knock out the cyto the two of the chains of the receptor you didn't have any sign ific effect but now when you knocked out the gamma chain now you had severe Comm imuno deficiency any suggestions why that should be isn't it that you don't need the out chain well it can be low2 simulation gam right but if you knocked out intran 2 you had no incin 2 at all on the body you also had minimal impact on the immune system CH is exactly that that don't don't you got to think a little bit out of the box so it turns out that whereas the alpha and beta chains are usually are specific for the cylin so for a 2 i 4 I 21 I 7 I 9 and l15 they have unique chains that specifically bind to that particular cyto kind it turns out that the gamma chain is is a basically allpurpose utility signal transduction recept ctor and in essence whereas These Chains give specificity for binding of the cyto the gamma chain is what actually does the signal transduction and in fact uh if you knock out the gamma chain now you not only don't have interlukin 2 you don't have I 4 I 21 I 7 is 9 and I 15 function and now it makes a lot of sense why now you have severe combine immune deficiency because each one of these cylin plays critical roles as shown here in different stages of T cell and B cell mat ation and now you've knocked out a whole large number of cyto kindes critical for BNT cell maturation and the gamma chain is a unique molecule to immune maturation and that explained why David's manifestations solely were his immune system because the gamma chain really doesn't play a role in other aspects of of physiology doesn't not important in the liver in the kidney etc etc okay is that clear now if you then go into a imuno deficiency clinic and ask the question what what fraction of these patients have mutations uh half of them have mutations of the common gamma chain uh a very small amount Incan 7 and I'll discuss in a minute Jack 3 deficiency and I'll also discuss in a minute Dennis and damin deficiency which also can cause immuno deficiencies now uh again severe Comm deficiency does the presentation uh but so David had knockout of his common gamma chain but there's also a population of patients that their idiology is a Denine deaminase deficiency and what's interesting about these patients is they can actually be relatively normal for a year or two after birth but then they start losing their B cells and te- cells so what that suggested was that the problem wasn't an underlying maturational problem but there was some toxin or toxic metabolite that was being generated as they were growing that was slowly killing off their B cells and T C cells and then it was identified the G neing was a Denine deaminase and in fact this is just a metabolic pathway a Denine in order to eliminate it It ultimately has to get uh deaminated and ultimately becomes uric acid where it gets secreted in the urine if you lack adenosine deaminase this is blocked and you basically are going to get buildup of adenosine and again you know this is in a biochemistry lecture so you don't have to go into detail but it turns out that the major uh the major problem whereas adenosine has alternate pathway for being metabolized by pporting and then using an enzyme called the Denine monophosphate deoxyadenosine lacks that and therefore you get buildup of deoxyadenosine well why should buildup of deoxyadenosine be toxic and it turns out that deoxyadenosine is as you all know is the source of datp and datp is a critical nucleus TI that you utilize for making DNA well in order to um and what makes it unique for lymphocytes is that lymphocytes once they phosphorate datp they have very very low levels of the enzyme five Prime nucleotidase to Def phosphorated and phosphorated datp can't leave the cell so you end up doing is you're taking all your deoxyadenosine your phosphorilated it it gets built up but you can't def phosphorate it so lymphocytes uniquely have buildup of this datp and that's why it's uniquely toxic to to lumoid lineage cells other cells are able to Def phosphorated def phosphorilated uh deoxy Denine could just diffuse out of the cell so it's kind of like it becomes this sync for datp well why is that toxic again the major one of the major reasons why it's toxic is that there's an enzyme called ribonucleotide reductase and ribonucleotide reductase is a critical enzyme for generating all these deoxynucleotides for making DNA however enzymes as you know have feedback loops that tell it when it it could stop because you have enough of the substrate and it turns out that ribonucleotide reductases even though it's involved in making dctp dttp and dgtp the only re feedback it utilizes is the levels of datp so if if ribo nucleotide reductases these high levels of datp it basically says I don't need to phosphorate dgtp dttp and dctp so now what's going to happen to your ability to make DNA you can't make it because you lack the building blocks and therefore that's and lymphocytes as you know have to proliferate well if you can't make the nucleotides for for DNA you can't proliferate and then it's ultimately toxic and that explains why it takes a while for these patients to get imuno deficiencies because they undergo proliferation and then the cells basically uh make too many mutations because they lack appr dntps okay another mechanism for severe combin immune efficiency is Jack stat defect if you recall Jack stats play a role in what kind of signal transduction cyto kind exactly so now the same way if you knock out the common gamma chain and you lose a panel of cyto kindes what happens if you particularly knock out a specific Jack stat that's critical for cyto required and again this is just review Jack stat basically cyto Jack comes together phosphorate stat and then the heterodimer gets into the nucleus and turns on genes and it turns out now if you look at uh the different cyto kindes the all these cyto kindes I 15 i 4 9 7 and two all use Jack 3 so now you can imagine if you knock out Jack 3 what's going to happen to the function of these receptors you're going to lose them and then you basically have inability of maturation that's dependent upon these multiple cyto and it turns out you get severe communo deficiency so now in in 1990 the first gene therapy was performed and it was performed with for a patient that had a Denine deaminase deficiency and again to do gene therapy you want to do it in a situation where we've identified the Gene and uh the level of the gene that needs to be expressed in order for normal cellular function you have you should have a lot of leeway cuz when the biggest problems with doing gene therapy is putting it in the way that the gene is appropriately expressed the way it is physiologically but if you have a gene that if it's if it's if it's uh two-fold threefold fivefold it's good enough then gene therapy is a lot easier to do while Denine deaminate deficiency it was toxic metabolite buildup so you just need a little Denine deaminase and that was able to resolve it that's one of the reasons why this was chosen and in fact in this case they actually did not treat hematopoetic stem cells they harvested peripheral blood from a patient they treated the peripheral te- cells and B cells and gave it back to the patient the reason for doing that is first of all it's a lot easier to do second of all you don't have to worry about introducing mutations into hematopoetic stem cell that can then go on and develop malignancy and also that Denine deaminase patients they had mature t- cells and B cells just if you let them go too long they would start uh deteriorating and these patients also can be treated with enzyme Replacements so they also can be treated by giving them pegol adenosine deaminase and so these patients had a reasonable level of B cells and t- cells and it turned out that of two patients treated uh peripheral blood Tel counts in in patient one increased and became normal and uh patient two showed an increase so it would looked very very promising that it was working in 2002 and 2003 a much more uh aggressive gene therapy approach was utilized basically these are children that had David had gamma chain deficiency severe Comm immune efficiency for that reason and these patients you had to treat their hematopoetic stem cells because you couldn't they really didn't have a lot of mature b cells in t-s to treat and in this case in this group in France nine of 11 children were successfully treated with hematopoetic stem therapy which is a mindboggling thing I mean you had a previously uh untreatable conest General imuno deficiency and now by gene therapy you could basically have a child that could leave the bubble and have a relatively normal life unfortunately uh four I think it may even be five now of these children develop leukemia due to insertional mutagenesis so the the length of the retroviral vector they utilized inserted adjacent to a enco gene and that caused malignant transformation so that really has cast a p upon the field of gene therapy in for using hematopoetic stem cells but an additional issue particularly for this kind of stem cell therapy is that you're giving a growth factor receptor and you could appreciate if you express too much of a growth factor receptor that's also going to make cells proliferate too much and the combination of M mutation plus a hyper proliferating cell is a recipe for potentially Iman transformation if you're doing gene therapy with a gene that is not associated with proliferation you may not have as high an incidence now if we move down the the the road to uh djor syndrome it's uh decreased t- cells normal B cells decreased IGG they and the patients present with hypocalcemia and seizures now you're probably thinking this is an imuno deficiency why should it be presenting with hypocalcemia and seizures we'll get to that in one second they because now the other point I want to make is that even though they have normal B cells if you lack t- cells what what impact is that going to have on your B cell function you're right you you won't make IG you won't make IGA only IG so even though you have functional B cells it's like having a a soccer player and you tie both hand well in soccer if you tie both hands behind the back it doesn't really matter uh just means they won't get a penalty but if you kind of like put a 50 lb weight on his on his or her leg it's going to compromise them so so even though the body is there they not as functional so therefore they have functional uh immuno deficiency based on their lack of IGG so they get bacterial infections as well and the ideology of it is a deletion in chromosome 22 q11 therefore it impacts on maturation Medi style maturation and they they the third and fourth brachial pouches do not develop normally and again the treatment potentially for this patient can be thymic transplant now why do these patients have hypo hypercalcemia and seizures and how could you make this diagnosis uh using an x-ray well the tissue that these patients lack is the thymus and you could actually do a lateral chest x-ray and this is of a of a of a baby and here here is the heart and right here in the anterior amyum is the thymus that's a perfectly normal uh child with a normal thymus if you look at a patient with djor syndrome and now you look at at they uh card do you see something in x-ray black means nothing's there so you see there's nothing here they have a heart and what do you think is lacking the thymus so and so you could basically make the diagnosis without even being an immunologist just doing a lateral chest x-ray well why do these patients have hyper calcemia is because in addition to the thymus the lack of the third and fourth brachial pches also means they lack a par thyroid gland and the parathyroid gland plays I'm sorry hypoc calcium the the hypoparathyroid gland plays a critical role in controlling calcium metabolism so they're presenting with seizures and hypoc calmia because they lack of parathyroid not because a problem with their thymus now I I said in the beginning how can it knowledge of Immunology prevent you from killing a patient and the answer is well this child has no functioning te- cells he's he's a day or two old he's you're doing all these blood tests on him him and then you finally uh uh nurse tells you the patient is anemic because you've taken so much blood and when a patient's anemic what do you do for how do you treat that you give a blood transfusion that makes a lot of sense right well if you give this patient a blood transfusion what potential problem can this child have again well but that you could choose what immunological problem could this child have well a transfusion reaction but like be more specific what unique transfus F usion reaction where a child that lacks te- cells have and what I'll say is is that in the red blood cell transfusion there are a few te- cells that are mixed in that will also be given to the patient graph versus host because these te- cells get inside the patient and they look around and they say whoa you know everything is foreign here you know we've been invaded we've the entire body has been taken over by foreign H molecules and now what they do is they now start proliferating out the Wazoo and they start attacking the recipient because they think they're doing a good job they're protecting themselves from from Invasion and the reality is they're completely messed up they're the Invaders uh so so therefore if you would give this child the blood transfusion this child would have horrible graph versus host disease eventually and you can't reverse it you can't take those cells back out so once you've made that mistake ultimately you could lead to the death of his child so therefore but what what how would you treat blood to prevent those lymphocytes from proliferating how do you treat cells to prevent them you radiate them for red blood cells irradiation is no problem because there's no nucleus in red blood cells but if you radiate the the blood now those few te- cells you Crosslink their DNA they get into the body they say oh we got to proliferate to fight this uh forign cells but they can't because you've basically again you've castrated them they can't replicate and therefore they can't do any harm so therefore if patient in first day or two of life has hypocalcemia and seizures do the lateral chest x-ray you see there's no thymus that tells you do not give this baby uh Red Cell transfusion unless you've radiated the blood products and that way you won't kill this child okay question how if a child had t- cells how would that stop the gra okay so if a child had t- cells how would that stop a graph versus host because all of us uh get trans well all us don't get trans but we frequently transfused patients and they don't get gra person's host well let's think about it you know you have a war and on one side are 100 soldiers and on the other side are a billion soldiers who's going to win can t- cells F T Cell um well the same way that the that the tea cells that come into the body see the body as foreign our tea cells would see the transfused te- cells as far and eliminate them so that's so routinely and again some patients do get trans Fusion reactions for for that and other reasons but at the end of the day if you have normal T cell function it's no big deal to eliminate that handful of te- cells only if you lack te- cells that would be a problem okay and this is just showing that this is they actually have an interesting looking faces because it's a whole midline maturational problem okay so now and this is just again reinforce the point that if you lack t- cells you actually compromise B cell function and again as I had mentioned helper T cells require a second signal provided by interaction of cd40 Lian expressed by helper T cells and cd40 that's present on the surface of B cells and now if someone has a mutation in their cd40 liand what kind of antibod deficiency do you think they would have they would lack IGG lack IGA and in fact I'll discuss that in a minute it's called hyper IGM syndrome but again lacking te- cells these patients would only make IGM for the most part okay now we get into excellent now we're getting to into imunoglobulin common one xlin a gabag globin emia have decreased B cells uh they lack antibody completely they can present after 6 months because of maternal antibody they infected with bacteria and in this case their defect is BTK which I mentioned briefly is a critical enzyme required for B cell maturation so if you lack this enzyme you specifically don't mature b cells and then you don't have any antibodies uh the treatment for these patients you just give them IV gammaglobulin so the same way that maternal antibod protects the baby first to 6 months if you give these children monthly gam globulin injections they actually do quite well but they need to get this continuous treatment for pretty much the rest of their lives but as opposed to a t- cell defect you can't transfuse te- cells to these patients so that makes imunoglobulin deficiency a little easier to treat now hyperm syndrome I just mentioned the second they present after 6 months um and and one possible ideology would be cd40 Lian deficiency can anyone think of another critical enzyme in class switching that could also cause you not to be able to make IGG or IGA what a a exactly uh Aid is critical for class switching if you don't have Aid you can't switch and in fact uh uh one cause is the absence of cd40 Li again but as you so nicely remembered the other one is lack of Aid you can't uh basically uh recruit all those repair enzymes to to cause class switching and again what I want to underline to you is that this really allows us to appreciate that these enzymes do what they we say they do and they're the only ones doing it because if you knock that out there's nothing to pick up the slack so again that's as I started the talk by saying that's why these patients are so instructive in teaching us how the immune system works because now I can tell you that you could take it to the bank that dd40 Lian is absolutely important for class switching I can tell you take it to the bank Aid is absolutely important class switching because if we have individuals that lack that they don't have the class switching okay okay this is just showing again the CD4 Li again okay now there's also subclass deficiencies and they basically have deficiencies of subclasses and the major uh ideology is unknown but there's been reports of just m if you have mutations in the switch regions of the imog globular molecule you could appreciate you're not going to be able to bring them together to Loop out the appropriate ones now IG deficiency they I mentioned yesterday decreased IGA bacterial and failure of isotope switching but the qu the question is that they really don't have dramatic deficiencies and now you know why because what's picking up the slack for these patients in they gut now what umal factor can also IGM because IGM could also bind to the to the has a j chain and it could also be transported across epithelial cells and could pick up some of this slack be lymphocytic have markedly decreased T cells and they have normal B cells and it turns out that as the name applies they're bare because they lack MHC Class 2 it's a mutation of the MHC Class 2 promoter and therefore now if you don't have MHC Class 2 what kind of te- cells won't be able to mature in the thymus because they won't be able to get positively selected CD CD4 so these patients really lack CD4 so it's again clinically like severe AIDS because they don't have CD4 cells and in addition if a maccrage is infected it doesn't have a class 2 molecule even if you did have some CD4 cells to present to so therefore these patients have very significant uh deficiencies and again they could be treated with B Mar trans plant and this is just a slide to to show you all the different genes in the context of T activated t- cells and B cells and again showing you how all of them are associated with amuno deficiencies uh variable so now the only other one I want to mention is tap deficiency so if you recall what does tap do I give you it's a pump what's it pumping okay allows the so MHC Class 2 is getting loaded in the vacu so it's not going to need to get pumped peptides so what's going to need to get peptides pumped to it what MHC molecule needs to get loaded with peptides in order to leave the endoplasmic particul MHC one and tap actually does the pumping so it turns out if you don't have tap you can't load the mhd CL one with peptides and then you can't express class 1 MHC now you don't express class 1 MHC what kind of cell T cells won't you see in the periphery C cd8 so these patients actually have deficiency in cd8 and they'll be they tend to get a lot of viral infections okay so job syndrome is actually uh the English are always um a lot more interesting than the Americans because and now I get Veronica's attention here I mean Victoria's attention so so the reason is that when Americans discover something they name it after themselves because you know Americans are relatively ego itical country you I'm sure you all know that by now uh uh in England though they tend to be a lot more poetic they have a classical education so if they see something interesting they try to relate the discovery to Something in the Arts and Sciences or literature so who is Joe people know who job was who's in the Bible and what what what dis disease or problem did job have he had skin boils so so these children presented with uh cold abscesses and the individuals looking at them said oh my gosh you know this kids look like job and then it was called job syndrome you know if it was discovered in America they would have named it after whoever discovered it these patients also have uh very high levels of IG the problem is is that the white blood cells can't get to areas of infection they have cold abscesses normally if you have an abscess it's hot because you have the athema you have the basal dilation their abscesses are cold because the fago sites just don't get to where the infection is and they can't release tumin necrosis Factor all the things are required for the vasil dilation and arthemia that gives you the the warmth they get infected with staff strep in Canada and it turns out that we now know that the ideology is a mutation in stat three so and again sta 3 plays a role in the signal transduction required for chemotaxis and the treatment these patients are in chronic antibiotics in contrast to job syndrome another aiic deficiency is chronic granulous disease and these patients also have ineffective phagocytosis but what they're characterized they have hot abscesses and the reason they have hot abscesses is because the fyes can get there they could secrete their factors they just can't kill the bacteria so therefore that's why the infection can't be cleared but the abscesses are hot so that's great because you have a patient that has a lot of abscesses you put your hand on the abscess and if it's if it's cold job syndrome if it's hot chronic round ous disease now obviously you do the lab test to confirm it but it makes you feel good that you can just kind of Lay Your Hands on a patient and quickly make the diagnosis and again the defect is due to the nadph oxidase that's required in order to get this oxidative burst to kill bacteria inside the vacu and the phagolysosome and again treatment antibiotics gamerant seems to increase as you know the fosic capacity of macrophages and therefore you can generate enzymes that pick up the slack and allow it to kill to kill better okay so now to to summarize and put things into context so if you look at the maturational pathway defects at different stages have impact Downstream so if we look at B cell and T Cell maturation if you have a mutation for example that impacts on the capacity of stem cells to become lymphoid progenitors in this case Denine deaminates deficiency gamma chain okay uh uh deficiency then you basically uh lose B cell and t- cell maturation if you have a mutation in in another Gene for example rag which is required for B cell Gene rearrangement and t- cell Gene rearrangement again you lack B cells and t- cells BTK is only required for B cell maturation so if you knock this out you basically have no B cells but you have normal t- cells in contrast if you have lack of thymus which is required for te- cells you lack te- cells but have normal numbers of B cells does that make sense right and now if you have either a defect in class 2 m c then you'll will have defect in maturation of cd4t cells if you have a mutation in tap one or tap two then you'll have a deficiency in in class in cd8 t- cells right and now if we look at fine tuning in terms of uh the differentiation of B cells and T cells clearly there a whole host of defects in in Signal t- cell signal transduction molecule so you'll have normal numbers of t- cells but they won't be able to be activated and proliferate appropriately if you have a mutation in C40 liand you can class switch so you have hyperm syndrome and it's not shown in this slide but also Aid which I'm you know so proud that you remembered uh also would have hyperm syndrome and again there's a whole host of switch Factor mutations in imunoglobulin genes that give you uh uh sub IG subass imuno deficiencies okay so basically uh patients provide many insights into working of the immune system because a lot of the genes that that I've mentioned play a critical role in differentiation actually turn out to be the case because of patients lack those genes they manifest the inability to undergo that maturational stage clinical history helps diagnose ideology in terms of age presentation what pathogens they present with and ignorance of Immunology can kill your patients and an examp example of that for example is patient with hypocalcemia and seizures you want to know that that they have djor syndrome doal chest x-ray because if you would give them a blood transfusion they would develop graph versus host disease okay thank you very much for your attention and now we have one more lecture to go and we're done thanks a lot
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Medical_Lectures
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Immunology_Lecture_MiniCourse_5_of_14_Antigen_Recognition_by_B_cell_Receptors.txt
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okay so today today we're going to start we're now going to be talking about b-cell receptors immunoglobulin molecules and how those are generated so we'll start again with what questions that we want to address in this particular lecture and the first question is how there are so many pathogens out there how could you possibly make antibody to every possible pathogen so again this is a question in terms of how could you have such an incredibly diverse immune response how do you avoid making Auto antibodies so how is there a specificity that only antigens are the ones that that are empathic genes are the ones the antibiotic against and how do you rapidly increase the amount of antibody how do you mobilize an antibody response again it doesn't do you any good to make great antibodies after the pathogen is basically replicated to such a high level that you're pretty much morbid and those antibodies won't be any be effective in terms of protecting you from the infection so you need a rapid mobilization of response and how do you switch from making IgM to IgG and again as I'll discuss when I talk about the different ISO types of mini blonde one IgM is the first antibody that you make but it's not a very effective antibody it's large it has low affinity the gold standard for antibodies it is IgG so how do you make this transition from making IgM to making IgG and how do you increase the affinity of the antibody now I kind of like have to apologize to you because I lied a little bit because when I talked about the fact that once l1 antigen specificity and that's the way it stays it turns out that's true for t-cell receptors t-cell receptors don't change but antibody molecules actually undergo a large level of mutation to become higher and higher affinity and in fact it utilizes mechanisms that are actually similar to those that H utilizes in terms of mutation so the same way that HIV mutates to escape the immune response antibody molecule has the capacity to mutate to become better and higher affinity antibodies to fight infection more effectively so again and you have to think that the genetic rearrangement is completely antigen independent so how could you possibly know what the highest affinity sequence that you can make is before you've seen the antigen and what this affinity maturation allows you to do is make even better antibodies so how does that happen and how do we generate memory clearly B once you've gone to all the work of making a CD antibody that's incredibly high affinity it seems it would be foolish to basically just throw out those blueprints and then wait till you get infected a second time and start from scratch it makes sense that you have a system that enables you to to have residual cells that have these making these great antibodies that you can then call upon to expand the next time you're infected so how does this process occur so just to kind of give an overview again of antibodies the process of antibody production is in this case you have a b-cell it is expressing antibodies discuss in detail how these are generated that recognize epitopes on the surface in this case of a bacteria this bacteria gets internalized will get digested into the constituent peptides the peptides are presented and what MHC molecule class is presenting this class to a very good now because it's an endogenous from outside its presenting to a helper t-cell because it cd4 class to that t-cell receptor recognizes the peptide and this helper t-cell is now going to provide help in order to do this we'll discuss in a little bit detail later on it is also interaction with co-stimulatory molecules so other molecules that interact on the surface and the helper T cell makes cytokines will a little bit more detail and this drives the B cell both to make antibodies as well as to class switch the B cell differentiates into a plasma cell the plasma cell is basically in a middle on one factory makes large amounts of antibody and these antibodies have very very diverse activities so some of these antibodies could neutralize infection so by binding to surface areas on the bacteria it prevents from sticking if the bacteria can't stick it can infect also it functions to option eyes so but binding to the bacteria the antibody now is able to be taken up by receptors on the surface of macrophages called FC receptors I'm also going to discuss in greater detail that allows macrophages to be more effective in phagocytosis bacteria and in addition and this I apologize for you there's no lecture on compliment in this short course but a complement also plays a role in terms of being recruited by antibodies to allow killing of bacteria now if we look at the primary secondary tertiary response again what we see now and it's a little bit more complicated figure than the one I've showed you before we're not only looking at volume of antibody production we're also we're also looking at what isotype is being produced and what you can see is that earlier on the product the and abideth so in purple and green is IgM but as time goes on the antibody that's being produced is the yellow in about an i GG so over time you have switching from the initial antibiotic IgG to IgM that's the top panel but what's also interesting is navona panel the y-axis represents affinity and what we what you can see is that the affinity early on is extremely low because I Jim has low affinity and even when you are making IgG the affinity is also low but again as time goes on the affinity of the IG gee dramatically increases and I also discuss later the reason for that increased affinity is because of the somatic mutation that's occurring in email loud one gene oh I'll use this okay so what does the antibody molecule look like the antibody molecule you know you always see these cartoons and when I first saw them I always found it very difficult to figure out what exactly it means and you kind of like just nod your head and try to look smart but it just looks like a bunch of colors and ribbons to you so so you have the more simple cartoons and then when you look at the more simple cartoons you think that's what it looks like in real life and you always surprise that it doesn't but again this is the simple one that I've shown you before showing the antibody binding region hyper variable region in this other little more realistic view but still a cartoon you can actually see that there's a hinge that allows these two regions in the heavy chain and the light chain to come together to actually combine to where it's binding the antigen and again in this ribbon diagram it also looks like it's apparently the red is the light chain the yellow is the heavy chain and again they come together so think of it coming together like like these two hands so even though there are two separate chains this one is specific this one is specific and now when they come together the combination of the two is that's what's gives you the specificity and again as I mentioned yesterday if you change the heavy chain or the light chain alone you can now by combining them with different heavy chains and light chains dramatically increase your specificity okay so this is a again showing this clonal selection of antigen specific lymphocytes again to reinforce the idea that you have this pluripotent cell that undergoes genetic rearrangements of the immunoglobulin gene I'll discuss that in greater detail but this gives rise to hundreds to millions millions of cells that all recognize but all recognize different antigens because the antibody molecules that they have have different specificities and after you remove self-reactive b-cells we'll discuss in a later lecture how that occurs you have a pool of these mature B cells I mean globular molecule expressing the surface initially it's going to be expressing only IgM on the surface and only when that particular b-cell comes in contact with an antigen will it undergo expansion you get expansion of millions of copies of that particular V cell and and and then you clear the infection and you retain many of those cells as memory cells so if we now look at the genes oh thanks if we now look at the genes okay can you hear me out okay now if we look at the structure so here you have the light chain here you have the heavy chain it's color coded for you so the variable region of the heavy chain is blue the variable region of the light chain is yellow and now let's look at the sequence of large numbers of immunoglobulin molecules and ask the question where is the variability and by doing the sequencing what one sees is you see three hotspots within the variable region so even within the variable region there are areas that we call hyper variable they have even wearability and these three regions basically compromise different sequences hotspots in the genome and the gene of the immunoglobulin molecule I'll show you what what's the molecular basis for these hotspots and this turns out to be where the immunoglobulin binds antigen because what happens here is that even even though these hotspots look like they're at different locations in the gene which they actually are but now when the protein is made the protein now gets folded and the folded protein actually brings the three regions all together so this region is the is the hhb 3 which is going to be over here this region is the 8th is the HB 2 which is here and this is the HB 1 which is here but now these hypervariable regions are actually put adjacent to each other and that's exactly what sees the antigen so in fact this hyper variable region is the most important part of the antibody molecule because this is where all the capacity to recognize multiple different antigens comes from from this hyper variable these three are hotspots of hyper variable region and this is where it is the most sequence variation from antibody to antibody and so now how how does this are generated and what we could basically demonstrate is that we have a multiple gene segment so if you remember the t-cell receptor we had a large number of v genes we had a general region and for the alpha chain for the t-cell receptor we had a V region a D region and a gen region for the heavy chain and or in it for the T cell receptor it was the beta chain but now you have multiple these multiple DS and multiple J's you pick one V one D and one J and then combination of whichever ones you pick like a combination lock different numbers that gives you a high level of variability and what's going to turn out to be the case is that the joining regions of the B D and the J that's where the hot spots are so in addition to having different these but this is where the high level of diversity gets generated and you can imagine again as we've said the combination of the specificity of the Kappa chain plus the heavy chain that's what recognizes antigen because again those two pieces come together to actually bind to the antigen and this is basic with the immunoglobulin genes going to look like you basically have large numbers of V regions so in this case it's showing you 30 30 B regions a large number of J regions and they basically for the light chains you have to have one B and one J come together even though these are the same color the sequence is of this is very different from this and very different of this so don't get fooled by the same color they're very very different sequences and therefore taking one V and one J is going to make one sequence protein taking a different V and the same J is going to give you a different protein and so forth and so on this is what allows you to get multiple high level of variability and again if you want to think about it simplistically it's almost like thinking of a train and that train has has a thousand cars each one of those cars is a different color and by taking different trains and putting them together you can get multiple different combinations well how does this how does this happen how do you get these different V chains to be approximated towards the J chain and this occurs through a process called recombination and if you think in terms of the train analogy you know that trains are constantly being connected and unconnected and then connected again because they have unique coupling sequences at the end of the train and the same thing occurs at a genetic level each one of these variable genes has switched sequences upstream and downstream of it that allows it to readily be coupled and be recombined so in this particular case what you're seeing is that these you have the V reach of the region V region and they're basically being looped out because switch sequences here are getting realign and then this basically gets cut out and now this B region and this J region come together the same way you can take trains uncouple them pop them an out REE couple the trains back together again and when you have a different sequence of train this this is one molecular loop coach is a different looping approach but the idea is exactly the same you loop out the genetic information so it's basically no longer there but you now have the Jane X neck of different Jade next two different bees depending upon how much of the sequence you move out is that clear and if we do if we do the math and ask the question how many different genes that we have again the Capet and and gamma chains have a 40 barrier of 30 variable to cut heavy chain is 40 the D chains 25 for the heavy chain again there's no D chains in the light chain and the J is actually also of six now does that seem like a large number to you not at all so if you look at that number you would basically say there's no way you're going to get anywhere near the high level of diversity that you're going to need for antibodies so if that's telling you is there must be a different mechanism there that's allowing you to get that high level of diversity so let's just we'll go into it to discuss exactly what that mechanism is so again this is just to dumb show you exactly what those switch regions look like again this is like the coupling sequences in a train that there is this recombination signal sequence Colares has a one on one side is 23 bases on one other is 12 these are flanking all the variable regions displaying all the generations and they can recombine with with each other and therefore again this is the mechanism by which the v's and the J's can couple with each other and loop out but again how can you get even more diversity in the process and it turns out that the mechanism by which this occurs is by introduction of random nucleotides in the joints between the gene segments during the middle of and rearrangement so not only are you dealing with what you have pre-existing genes but now you're introducing new nucleotides randomly that can unbelievably amplify the level of diversity and what you this is your your recombination sequences they're put together they make these hairpins at the coding terms they're the coding regions but what ends up happening is a gene that introduces that then while it opens up now the sequence that before you combine the two of them hey you know you need to get a portable mic because that's what you need or a megaphone but what you is an important enzyme called T V Tech T and what TD T does is it just randomly introduces new nucleotides and these new nucleotides can let in this area and now ultimately hopefully you get the red herring region get rid of the gun the unpaired nucleotides and now you seal the DNA filling in and now you have new nucleotides introduced that hadn't been there previously is that clear well two things are very really critical about this process the first process went is is that well now you can introduce new amino acids so if this encodes an arginine and it hadn't been there before now you have a highly charged region that could recognize a different antigen that's really that's not anywhere near the cool aspect of this process because you know about reading frames right so normally you have triplets you go three nucleotides amino acid three nucleotides amino acids three nucleotides amino acid right that's pretty straightforward what happens when you end a single nucleotide into the DNA sequence what happens to you triplets what happened they completely change because one amino acid completely throws the entire triplet squawks and not only have you are you not just changing one amino acid you change all the rest of the amino acids in that local area so you generate tremendous diversity if you put two nucleotides in there you throw up another reading frame so in essence this process dramatically increases the variability of the amino acids that are being expressed and again now dramatically increases the diversity of antigens that can be recognized by those antibodies so this process both of putting new amino acid capacity by putting a triplet in or throwing off the reading frame dramatically increases specificity and guess what this is again one of those hotspot regions that have hyper variable sequences that are very critical of recognizing and recognizing some antigen and now if you start adding up all these different factors and do the math and ask the question how much where the diversity coming from it turns out that the number of the number of B gene pairs with T regions etc gives you this level of diversity in the T cell receptor in your blog one gene by having junctional diversity where the junctions don't exactly match when you add nucleotide you can see that dramatically increases it you multiply these two out and again for immunoglobulin molecules to come up with five times ten to the thirteenth possibilities and for the T cell receptor you come out with ten to the eighteenth possibilities so that's a large amount of variation so well a simple question it may be well why does the T cell receptor have a lot more variability the antibody molecule so if you look at the numbers where you clearly see is there a lot there's more variable regions in a t-cell receptor and particularly a lot more j regions in the t-cell receptor there are 61 j regions in the t-cell receptor why is that why is the T cell receptor have a lot more variability very winning antibody molecule Center that question and we'll come to it when we start discussing somatic mutation any questions okay so now the antibody molecule has to do multiple things one of the important things it does is it neutralizes viral infections well in order to neutralize viral infection it's basically the virus for example if you think about HIV the receptor has a certain level of affinity for whatever the cellular receptor is so for HIV will be gp1 to one a cd4 ccr5 well if the antibody has less affinity for this for the viral molecule then the cellular receptor does the antibody won't be very effective right we can compete if you have less affinity so therefore it's incredibly important for God for antibody to be usable as a neutralizing molecule that it has to have a higher affinity than the virus has for whatever its natural receptor is and in order to get that high level of affinity what the antibody molecule does is undergoing process called somatic mutation and what's notice to adhere is this is the genetic sequence of the heavy chain B region this is the genetic sequence of the light region and what you see is during an immune response days seven of the initial response you really don't see a lot of changes mutations are indicated by these little whatever these are how do you describe this symbol pump put pins okay let me pushpins round pushpins here not a lot now what happens is you look at 14 days you start seeing a lot more then you have a secondary response now by the time era to the tertiary response you can see there's a tremendous increase in the number of mutations every one of those little pushpins and this is from a single D cell has more and more mutations and where are those mutations localized what area of the immunoglobulin region the hyper variable region which is indicated the CDR one two and three by the shaded purplish region so what's happening is is that if you just take a b-cell followed over time sequencing in a broader molecule over time you see it's spontaneously developing these mutations that haven't been there before they're not in the germline they're new and what this is doing is changing the antibody recognition of capacity and presumably it's just going to turn out it's going to making animate as higher and higher affinity because the antibody is actually undergoing almost a Darwinian evolution because multiple mutations are randomly being generated some retentive mutations don't actually reduce infinity if it reduces affinity the antibody molecule can't compete for antigen the VCO doesn't undergo of proliferation and kind of dies off the mutations that result in higher affinity and higher affinity compete better for antigen and those are preferentially expanded so over time now you've selected for mutations that have even higher and higher affinity and that's how you generating these high penalty antibodies that you need per effective neutralization is that clear any questions and this is just to illustrate anatomically how this happened you start out with the pluripotent stem cell the pluripotent stem cell immature B cell a true B cell it sees antigen for the first time gets T cell help and now is triggered to undo first round of proliferation what wins proliferating it migrates into this dark zone and it's called a central blast and this is a very critical time in terms of its differentiation because as we'll discuss in a few minutes and enzyme called AI D plays a critical role in the development of these mutations it undergoes random mutations and now it comes out comes in contact with antigen being presented to it by the follicular dendritic cells and and then there's a selection process so then these cells that have antibody molecules that have high affinity they basically now are selected for some of them become platinum so to make antibody some of them become memory b-cells that will be around for the next infection and these have affinities that are thousand ten thousand one hundred thousand times higher than the initial germline encoded immunoglobulin molecule however you know any story is always a sad part you know you always have winners but there's also losers and you have b-cells that unfortunately they mutated in the wrong direction their antibody molecules had less affinity and those b-cells really are not going to be functionally of any value to you and therefore they undergo a batata or nergic B cell death so kind of like the same way Darwinian evolution can be very cruel so two mutation in the germinal Center will also be cruel but it results in the selection of these very very high affinity producing IgG producing b-cells any any questions yep just these slides earlier when you there was the number of permutations that could occur those negative 5 10 10 that the bottommost number-wise it- no this is actually approximate that's what it's a squiggle oh this year this is a squiggle but is it and thank you for actually making me go back to the slide because now I asked the question why do you have so many more possibilities in the T cell receptor than the immunoglobulin molecule isn't that bad for antibody production and now you appreciate that it's not as critical because the antibodies had the ability to undergo somatic mutation so even though the germline you have 5 times 10 to the 13th because you have the added capacity of undergoing these somatic mutations in the hot spot regions of those hyper variable areas you could now dramatically increase your ability to recognize the antigen many many many many fold to even have MIT I mean again the numbers are ridiculously high much higher than the t-cell receptor in contrast as I mentioned yesterday the t-cell receptor does not undergo somatic mutation whatever rearrangement and under one initially it stays with that forever so that's probably why you need to have a higher starting point in terms of possibility where the V cell because as the capacity for undergoing somatic mutation can dramatically increase many many pull beyond is baseline germline encoded variability okay is that clear Winkler okay how does this happen how is are these mutations introduced and this is through a molecule coal that enzyme called sided needy Amity and if people are familiar with HIV when you were able back 3G which interacts with v and causes mutations that can actually prevent HIV effective 'ti 8id actually utilizes a very similar approach it's a side of being deaminase it takes the adenine off the side of means changes them into a uracil and that your soul ultimately get translated into a thymidine and it basically is associated with different RNA transcription enzymes that allows the introduction of several mutations this is just a picture of what the enzyme looks like and from a mechanistic point of view what happens is that you have side of these that are present in in the in the gene a ID comes deaminated and converts the cytosine to a uracil and there are multiple different mechanisms that play a role for example one enzyme called uracil DNA glycosylase per UNG sees that there's a mismatch because g and you don't match and again there are anal compulsive enzymes they're continuously checking to make sure that all of the nucleotides match it's like you know anyone ever puts on a pair of blocks of one black sock on one blue sock in the morning and dark you don't notice it and then also you know you look around you welcome with a blue sock and a black sock it looks really pretty pretty nasty and and if you're married your spouse points it out to you and then says you've got to go back and change it you know these enzymes do exactly the same thing they're continuously making sure that the nucleotides mat if it sees it so it doesn't match it basically pops it out and forms a nick in it and now you have a nick here that theoretically should be filled in with a with the appropriate citing but sometimes that doesn't happen again this is one of the mechanisms by which the somatic mutation gets generated and these are the different possible things that can happen well one possibility is is that you just don't you don't notice it and if you don't notice it the euro soul is seen by being a little raised as a finding and therefore when it makes the replicate instead of putting the appropriate guanidine over here it says up this is a T what goes here is an A so right now you've now made a mutation you've gone from a C to G to a T to a well why is that important because now that's encoding a completely different amino acid and that now is giving a variability to the immunological molecule in addition is a lot more complicated processes that go on that you excise that and now you introduce randomly the different nucleotide but what gives even higher level of variability is that somehow this Nick which is open gets expanded and now you have a large gap in your DNA sequence well if you'd be using a polymerase that has a high fidelity that wouldn't be a problem you just put back the appropriate nucleotide but let us say if you're like a bad speller you know so you don't put the right things in right there are enzymes that are very error-prone and this particular error-prone little race is specifically recruited by PCA it actually fills in the place but if it builds it over the wrong nucleotides so now it's another generating all these mutations all the starter because the AI dee' made this single mistake then now it's amplified into more and more mistakes now this would happen in a normal gene we would be in big trouble because we'd be making proteins that don't work we'd be making mutations in oncogenes in beginning cancer it seems for some unknown reason that this particular process is only impersonating in globulin gene and abroad Lucien is the old routine where this high level of area is actually encouraged but again an overt occurs in this narrow window of time during d-sub proliferation in when it's in the dark zone and it doesn't occur after the B cells mature and you look at the level on a mismatch repair this is a very low level of mismatch repair 10 to minus 11 that is is how that you're applying because you're having very very high fidelity replication however as you increase the ever rate you can see bad things start happening so when you're at an error rate of 10 to minus 9 that's not compatible with life because your cells aren't replicating appropriately if you have in this region you start getting cancer because you're basically mutating oncogenes but it turns out that when you have this level present in B cells that actually works out well because you actually have this focused targeted mutations that are only occurring in your model molecule but this dramatically increases the capacity to recognize with a higher affinity antibodies now in other T cell dependent activation you undergo mutations and selection and now these cells that you generate well these particular ones are the losers so these B cells they mutated their antibody molecules as I mentioned before but these are lower affinity antibodies on their surface therefore they can't compete effectively intogen the header processes antigen driven if you can't capture antigen to the b cell that visa will stop proliferating and undergo delayed context is the one that a higher affinity antibody those now are successful in competing and may become in the plasma cells that are making high affinity antibody and those also become the memory diesel on that that explains why the second time that you see a an infection you have a rabid respond there are more effective response because you go to this memory b-cells and the memory b-cells are now making this high affinity antibody that you spent all this time selecting for so you can mount a very rapid response with a highly antibody and therefore clearly infection very rapidly so is that so in essence the b-cells is learning from his mistakes utilizing the mistakes that actually were and doing it effectively to allow you to have a more effective immune response Zek clear any questions it's really amazing process that occurred now I mean run like children logical question that you can ask is why dirty cells have the same process of somatic hypermutation that these cells do what are the b-cells need it may be in T cells don't any thoughts it might be that the the intensity of the interaction that you need would be so is far more important that need an intensity of the interaction that you need with key so he says exit each other it is there disappeared okay so let's look at that thought so you're basically starting out by saying that these cells need to have in your glory molecule that's their my affinity t-cell receptors don't have to have as high affinity okay that's point one let's take that the next step why is that what is the effect or molecule of the V cell it's the antibody that antibodies soluble it's floating around well clearly it has to have a very high binding affinity in order to be effective while it's going to blind to whatever binds to so that's major role of the antibody models to buy the things if the major roles to buy the things you want it to bind as tightly as possible right what's the function of the t-cell receptor it's a switch to turn on the t cell it doesn't have to compete for antigen free-floating because the antigen presenting cell brings it to it shows the peptide right in the face of the t-cell receptor and says here this thing is do you recognize it it doesn't have to have a high affinity as long as it recognize it then it transducers the signal so you don't have it the need to bind with high affinity at the soluble antibody dose is that clear okay and a second explanation is actually that T cells are very very dangerous if they recognize self antigen a question that you may want to pause it to me is what doesn't distal Matic mutation lead to the possible generation of autoimmune antibodies because let's say you make an randomly you've let the bone marrow you've left your screening location and now you give you these cells a license to make any antibody wrong with any specificity could you now make Auto antibodies right and the answer is what do you think yeah you can't absolutely and it happens and in fact that's one of the mechanisms by which more no immunity is generated however it turns out that antibodies if just not as deadly as a cytotoxic T cell if you have a cytotoxic T cell as I'll discuss in subsequent lecture and now it's recognizing a self antigen it cannot go on proliferate and make millions of copies of itself and the lod and kill millions and millions of cells have a very very vigorous powerful or immune response so because of the fact that's cytotoxic T cells are so deadly you never would have to be extra-special careful that you don't become hold of you and and never of you dividing you system has made a decision that T cell receptor maturation stops in the thymus only go to okay in the thymus they leave and they never change so those two reasons probably account-wide antibody molecules undergo somatic mutation while t-cell receptor does not undergo somatic mutation okay is that clear is that helpful okay so now this is just to kind of end as a lead-in to the next lecture so this is also just to review so now you I think you now have an understanding of what's going on in this figure because the initial reaction that you have is IgM it has low affinity because the rgm has not undergone any somatic mutation you mean a blob merging it's expressing is the one that was ran generated by vdj recombination that that occurs in the nucleus in the complete absence of antigen so it has no idea what the antigen looks like just randomly generating them so is going to be low affinity however as time goes on through multiple exposure to an antigen you undergo somatic mutation you mutate the vdj region particularly those hot spots and you're undergoing this evolutionary process selecting for higher and higher affinity minimal on one gene sequences and now but you the affinity dramatically increases and again that's also you have higher levels higher affinity much more effectively antibody response which is why by the time you've been infected a second a third time those antibody pathogen really doesn't stand a chance and you're able to eradicate it very very quickly okay is that clear so now how can we make anti to every possible pathogen and the answer is there are always vdj sequences they have to have diversity just by having these different sequences even more diversity by virtue of having the when they rearrange the junctions don't exactly match introduce new nucleotides into those hotspots and then we have the additional benefit of having somatic mutation to further increase diversity as well as benefit city how do we avoid making one antibody we haven't answered that question yet so we'll keep that at home we'll talk about that in a subsequent lecture hungry rapidly increase the amount of antibody begin by having these memory b-cells they're available and because they have high affinity then even a small amount advantage and they rapidly grab on to it and start getting activated very quickly when the infection initiate how do we switch making IgM to IgG that I'm going to be discussing in the next lecture and the way we increase the affinity antibody is through maturation through introductions of mutations AI dee' deaminate side of the mix into your soul and that starts a whole process of repair that's specifically designed to be inefficient and to be full of mistakes in order to allow this diversity to be generated and how to regenerate memory we'll be discussing that in the next lecture again thank you very much for your attention
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Medical_Lectures
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The_Thyroid_Exam_and_Physical_Diagnosis_of_Thyroid_Disease.txt
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[Music] in this video I will be discussing the physical diagnosis of thyroid disease the learning objectives will be first to be able to demonstrate the general approach to the examination of the thyroid second to be able to describe potential maladies of the thyroid on physical exam and last to list other common physical findings in thyroid disease the thyroid exam tends to be one of the more difficult exams for trainees to learn well because the thyroid gland is relatively subtle in most people preventing the immediate confirmation of correct technique the key to a good thyroid exam is knowing the relevant landmarks so I'll start by describing normal neck Anatomy as it relates to to the thyroid the thyroid is a midline structure located in the mid neck it has a right and left lobe joined by a narrow ismus it has a close and relatively consistent relationship to other structures moving from Superior to inferior there is first the hyoid bone which is not usually poppable in the midline next is the thyrohyoid membrane also non-palpable and then the thyroid cartilage which is the most prominent landmark in the neck and what is referred to by lay people as the Adams Apple immediately below the thyroid cartilage is the cryo thyroid ligament or membrane which is the location of cryo thyrotomy a procedure rarely used to obtain an emergent Airway in situations of upper Airway obstruction then moving downward is the CID cartilage which is also palpable but not as prominent as the th thid cartilage and below the CID cartilage just anterior to the second and third tracheal Rings is where the ismos of the thyroid is most commonly located the two loes extending outward laterally the gland wraps itself around the anterior and lateral aspects of the larynx and upper trachea it may have occurred to you already that these structures are poorly named as it's misleading for the thyroid cartilage to not be the one actually adjacent to the thyroid gland but unfortunately these are the names with which we are currently stuck so how do we actually examine the thyroid gland in a patient as with any part of an exam the first thing to do is inspection on occasion a patient may have an obvious enlarged thyroid which is called a goer that is grossly visible however most findings discovered in the thyroid exam are with palpation palpation can be performed from two different approaches the anterior approach in which the examiner stands directly in front of the patient or the posterior approach in which the examiner is behind the patient and literally wraps his or her hands around the neck from the patient's perspective the anterior approach is presumably preferred by most although some clinicians feel that the posterior approach increases sensitivity for finding subtle abnormalities how an individual clinician weighs these two concerns is a matter of personal preference begin palpation with a finger on the patient's chin and slowly slide it down the midline over the non-p palpable hyoid bone and thyro hyoid membrane the first significant palpable structure will be the lenial prominence of the thyroid cartilage keep following this downward until the finger slips into a small horizontal Groove this represents the location of the cryo thyroid ligament the CID cartilage is the firm structure immediately below the groove and the thyroid ismus should normally be just inferior to that although it's not always clearly palpable but once there slide your fingers to either side of the midline and you should be able to feel the two loes with gentle pressure roll your fingers over the loes some clinicians prefer using the thumb on one side and the second and third finger on the other some clinicians instead prefer using the two thumbs and others prefer using the first and second fingers on each hand I don't know if any of these three is necessary necessarily superior to the others what's far more important is correctly identifying the landmarks and thus ensuring that you palpate in the correct location in the event that you think you may have found a nodule or mass but aren't sure if it's within the thyroid or if it's within a more superficial structure you can ask the patient to swallow as the thyroid is Tethered to the swallowing apparatus it rises when the patient swallows and so should the thyroid nodule if it fails to rise it indicates the nodule is in the subcutaneous tissue although they are most commonly located well above the normal location of the thyroid gland thyroglossal cysts can occasionally be mistaken for thyroid nodules these can be distinguished from the ladder by asking the patient to stick out the tongue as far as it will go thyro glossal cysts will rise up with this action whereas thyroid nodules should not move so as you are doing the thyroid exam exactly what types of abnormalities should you be observing for first I already mentioned a goiter also known as thyroid megal which is simply an enlarged thyroid gland some goers will be very obvious just on simple inspection of the neck as seen here with a relatively modest goiter and a substantially more prominent one other goers will require careful palpation rarely a goer will grow downward into the chest rather than upwards and outwards within the neck these substernal goers are more problematic since they are more likely to compress adjacent structures such as the trachea esophagus and great vessels typical symptoms of a substernal goiter include frequent coughing the sensation of food getting stuck in the throat when swallowing and difficulty breathing while supine each of these symptoms are more commonly caused by other conditions making a substernal goer particularly difficult to identify a dramatic anonomous physical finding of a substernal goer is called pimperton sign in this finding when a patient with such a goer raises his or her arms over the head and holds them there for a minute or so the face becomes red or occasionally cyanotic with a sharp demarcation in the neck the patient may feel fullness of the face or have an unusual sensation in the head but they are not typically dnic or dizzy there are competing hypotheses about the precise mechanism of this finding but all involve extrinsic compression of the great vessels against the enlarged thyroid resulting in venous congestion of the head it can also be observed in patients with SVC syndrome another exam finding already mentioned are thyroid nodules one should indicate the number of nodules as well as size and location of each and whether or not it is mobile tenderness of the thyroid gland can occur in the setting of severe inflammation and the last finding which one should check within the thyroid itself is to listen for a thyroid brewey with your stethoscope applied over the thyroid gland thyroid brewes typically indicate hyperthyroidism and are most commonly described in Graves disease though they are not highly specific for that particular condition I personally don't typically bother listening for a thyroid brewey unless I'm specifically concerned about the Poss possbility of hyperthyroidism based on history and other exam findings something that is important to realize about identifying a goer is that its presence tells you very little about actual thyroid function on its own as goers may be present in patients who are hypothyroid hyperthyroid or even on occasion U thyroid consider the following chart of common causes of goers patients with iodine deficiency are typically hypothyroid those with multi-nodular goer can be either U thyroid or hyperthyroid Hashimoto's thyroiditis is usually hypothyroid Graves disease is hyper thyroid patients with Subacute thyroiditis can have any clinical thyroid status depending upon what stage of the illness they are currently in although the hypo and Hyper thyroid stages both tend to be relatively mild and short-lived and lastly goers can be caused by infiltrative diseases such as ameloid dois and Sarcoidosis which can be either euthyroid or hypothyroid so you can see that the fact that someone has a goer tells you actually very little about the specific diagnosis or even what their thyroid status is before we move from this chart I just want to point out that iodine deficiency is the most common ideology of goer worldwide while goers in the US typically caused by either multinodular goer Hashimoto's thyroiditis or Graves disease and although Subacute thyroiditis doesn't necessarily lead to an obvious goiter um in all cases it is the most common ideology of a tender goer when one exists an aspect of the physical diagnosis of thyroid disease that may not be immediately apparent is that the most diagnostically helpful signs are physically removed from the thyroid gland as thyroid hormone has an extremely diverse range of normal actions in the body both hypo and hyper thyroidism cause an equally diverse range of exam abnormalities starting with hypothyroidism in the cardiovascular system Brady cardia is common and in extreme cases patients may be hypotensive regarding the dermatologic exam their skin is often cool and dry with coarse hair brittle nails and rarely a yellowish discoloration to the skin this discoloration is thought to be secondary to decreased conversion of betacarotene to retinol with subsequent hypercarotenemia hypothyroid patients frequently have neurologic findings specifically slow mentation hypothyroid speech which is characterized by slow low pitched and occasionally slurred speech which is partly a consequence of the deposition of mucinous material within the vocal cords and hypo reflexia which is associated with a prolonged relaxation phase of the ankle reflex two miscellaneous findings include hypothermia and non-pitting generalized edema which is thought to be due to the accumulation of mucopolysaccharides in the subcutaneous tissue this form of Edema is occasionally called mix EMA which is similar to but not identical to pre tibial miedema which is actually more classically described in Graves disease in ideology of hyperthyroidism and to add to the confusion the term miedema is occasionally used as an old school synonym for hypothyroidism itself in general individual findings of hypothyroidism are more specific than sensitive thus the presence of exam findings particularly characteristic hypothyroid speech and coarse skin argue for a diagnosis of hypothyroidism however there is no specific finding whose absence argues significantly against the diagnosis it is certainly possible to be symptomatically hypothyroid without any physical exam abnormalities moving on to hyperthyroidism it too has a large number of diverse findings in the cardiovascular system there's almost always Tac cardia which is usually sinus Tac cardia but is occasionally atrial fibrillation patients are frequently hypertensive and they may have a flow murmur on account of increased mardio contractility the skin is frequently warm and moist and hair is fine and abundant the neurologic exam is frequently abnormal patients can exhibit psychomotor hyperactivity which is also known as hyperkinesia they can have pressured speech a fine tremor proximal muscle weakness and generalized hyper reflexia there are also several distinctive eye findings such as lid retraction and lid lag these are variations on the same phenomenon in lid retraction the patient has an unusual staring appearance caused by a widened palpal fissure that is there is more than normal space between the upper and lower eyelids in lid lag the eyes appear normal at rest but as the patient looks downward there's a transient appearance of white Scara between the iris and the upper lid the lowering of the upper lid which normally accompanies downward gaze is delayed by just a moment both lid retraction and lid lag are a consequence of sympathetic hyperactivity affecting the levator papyra superioris and Superior tarso muscles of the upper lid it's the opposite effect of the tosis seen in Horner syndrome where sympathetic activity to the eye is impaired although lid retraction and lid lag are frequently included in descriptions of graves disease unlike other opthalmological manifestations of graves these are not specific to that diagnosis to test for lid lag simply observe the patient with his or her eyes in the neutral position then have them look up and then down watching if a section of scare becomes briefly visible directly above the iris as the patient looks downward I'm going to return to the complete list of physical findings of hyperthyroidism for a moment from a statistical standpoint the physical exam is more helpful in either ruling in or ruling out hyperthyroidism than it is in hypothyroidism first the vast majority of hyperthyroid patients have palpable thyroid glands thus a normal gland size on exam argue strongly against the diagnosis Tremor and tardia are also relatively consistent findings whose absence would argue against the diagnosis the findings which are most specific and which argue most in favor of a diagnosis of hyperthyroidism are lid retraction and lid lag you probably noticed some strong parallels between the findings of hypo and hyper thyroidism I'm going to highlight them because remembering that the two diagnoses have many opposite findings will help to remember each's presentation When comparing hypothyroidism to hyper thyroidism we see there is Broc cardia versus tardia hypotension versus hypertension slow mation versus psychomotor hyperactivity slow low pitched speech versus pressured speech hypo reflexia versus Hyper reflexia cool dry skin versus warm moist skin and last course hair versus fine abundant hair I'm going to conclude the video with a discussion of some notable physical findings that are specific for Graves disease which is the most common cause of hyperthyroidism in the US the first is Graves opthalmopathy which is seen in about 50% of graves patients this is not one specific finding but rather a constellation of related findings the most well-known of these is exop thalos also known as proptosis in exop thalos the eyes are literally pushed forward out of the orbit on account of fluid accumulation in the retroorbital space the distinction between this and lid retraction which is not specific for Graves is much easier done from the side than from the front of the patient other findings of graves opthalmopathy include lid and periorbital edema limited eye movements and something called compressive optic neuropathy symptoms of opthalmopathy include the subjective impression that the eyes look different irritation excessive tearing vetro orbital pain or pressure and in severe cases visual loss the visual loss which is attributable to the optic neuropathy is usually of such Insidious onset and slow progression that patients may not even recognize its happening during its earlier stages examination of the eyes in patients with Graves disease should also include an assessment as to whether the upper and lower Lids can close completely as failure to do so will place the patient at a higher risk of corneal dryness and ulceration the suspected pathogenesis of graves opthalmopathy begins with the TSH receptor antibodies activating tea cells which release certain cyto kindes these cyto trigger fiberblast to creete glycosaminoglycans which accumulates in the extraocular muscles and retroorbital tissues this increases the oncotic pressure in those locations which leads to fluid accumulation and forward displacement of the eye the second physical finding which is more specific for Graves disease is infiltrative dermopathy which is seen in just 5% of graves patients it is also known as thyroid dermopathy and pretibial mix edema this consists of bilateral asymmetric plaques or waxy induration usually on The Shins The pathogenesis appears to be very similar to the opthalmopathy an issu raised earlier in the video is the potential confusion surrounding the use of some of these terms for example it's common to hear clinicians refer to this finding as preal miedema yet the term miedema by itself is a historical synonym for hypothyroidism adding to the confusion is the fact that this entity of pretibial mix edema has been uncommonly described in both hypothyroid and you thyroid patients as well and I've even heard people refer to any lower extremity edema in a patient with hypothyroidism as pretal mix edema irrespective of whether the type of Edema seems remotely consistent with this diagnosis now I won't pretend to know the history behind the confusing terminology but I will recommend avoiding the term prebio miedema in favor of either infiltrative or thyroid dermopathy when associated with hyper thyroidism and just referring to it as plain old edema in any other context with the one qualifier as to whether or not it is [Music] pitting that's the end of this video on the physical diagnosis of thyroid disease if you enjoyed it please don't forget to like it or share it and feel free to leave questions below lastly if you haven't already seen them you may find my other videos on thyroid disease interesting as well [Music]
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Medical_Lectures
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Immunology_Lecture_MiniCourse_11_of_14_Mucosal_Immunity.txt
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this lecture is going to be on the mucosal immune system and I always uh think about as coming from Albert Einstein College of Medicine I always think about Albert Einstein when I think about the mucosal immune system because when you think in terms of quantum mechanics what was so difficult to understand is that quantum mechanics violates all the laws of Newtonian physics and in fact the physicists back then really had this tremendous conflict and cognitive dissonance because they're so well-versed in Newtonian physics we have action reaction that they had couldn't understand how subatomic interactions were occurring in a sense like how could one particle know what another particle was going to do in order to respond to it before it even happened so this kind of uh difference is the same way of the normal immune system or the immune system we don't have to call it normal uh that we've discussed so far what goes on in the mucosal immune system it's almost a completely different kind of behavior and the reason for that is the environment that the mucosal tissue has as opposed to the normal uh think of the lymph node so the questions to consider is how is the mucosal immune system different from the systemic immune system how does the immune system prevent overreaction to antigenic loads well one amazing difference between the mucosal immune system and systemic immune system is think about what the gut looks like what is the gut exposed to day in and day out antigens it's getting and they're foreign antigens I mean you all came back from lunch right you ate whatever vegetables meat whatever you ate I guarantee you you didn't eat human flesh I hope right so therefore everything that you ate was basically foreign antigens and now you dump that into your stomach you dump that into your intestine and you even think what am I doing to my poor immune system I'm overloading it with foreign antigen you know it doesn't even bother you at all now would you inject that kind of stuff into your bloodstream you wouldn't you know you wouldn't make it for for for very long if you did that so clearly there's a very different situation if you would take a lymph node and put it in tissue culture would you grow out any bacteria it's relatively sterile environment almost very you know unless you're infected you don't have anthy if you take the gut and grow it out in culture it's very very very difficult to do because you get tremendous infection because it's loaded with bacteria Etc so it's two very very different environments so therefore you have to do something to your immune system to deal with that so the question is how does the immune system prevent overreaction to antigenic loads how does the mucosal immune system protect itself from infection so you have really these two Divergent th thoughts you have to do you see all this bacteria all this antigen so you say I have to like not be paranoid and I have to not trigger my cytotoxic te- cells antibodies Etc because they're not out to get me they're commensal bacteria they food antigens on the other hand there are pathogenic bacteria that do get into your gut so you have to be able to respond to them quickly and this balance of ignoring Bine antigens or B N bacteria and yet being ready very quickly to respond to infections is really what the M job of the mucosal immune system is and also think for a minute the mucos is actually a relatively thin barrier and if you basically have a pathogenic bacteria coming through the epithelial cells and then you kind of like say oh let me think if that's pathogenic or not let me think think think next thing you know that bacteria has gone through the barrier now it's penetrated it gets into the bloodstream and it's too late so you also have to have this hair trigger ability to rapidly eliminate infection or infected cells before it could get to sem ated so this is a really tough job the mucosa has and as a result it behaves in a very very different way from the peripheral immune system and again this is what I want to cover in this lecture and final and how do pathogens bypass mucosal immunity obviously we all get infected all of us have had gastrointestinal disease salmonella shagel Rota virus so clearly the mucosal immune system is not absolutely impenetrable so how do a pathogens surmount this barrier and what th subtypes are preferentially activated in mucal immune system we spoke a lot about th1 this this morning being important for intracellular bacteria or a lot of bacteria in the gut is that the preferential immune response or perhaps it's the th2 or uh t-reg cells uh or th17 the preferential immune response okay any questions okay so if the want to talk about an overview of the mucosal immune system this is just giving you where it is and this is just showing that you have mucosal tissues lacrimal glands salivary glands mamary glands uh some even in the kidney the GI tract lungs esophagus naso Fingal location so this is all in a nutshell whatever parts of your body are exposed to the outside environment because in essence this is all exposed to the out except for mam gland that that's really where you have to have this unique specialized mucosal immune system so the major components GI tract respiratory tract genital tract now the unique attributes is the first line of defense because this is really where you come en counter with your the outside environment now another tissue what other tissue do we have that's always exposed to the outside environment skin but how is skin SK dealt with that it's become impenetrable for the most part can the gut be impenetrable no CU otherwise we would never be able to have nutrients come through oxygen diffusion etc etc so therefore it has a unique need to basically allow things through but to basically make sure that it doesn't get infected so it it's constantly exposed to antigen that is the Hallmark of the mucosal immune system that makes it different from the systemic immune system and therefore has to come up with completely different regulatory uh mechanism okay uh uh mucosal infections are bad and in fact they cause a tremendous amount of death throughout the world I mean you can see the numbers are incredibly high and uh in terms of the uh fatalities from it so clearly you have to have a well-developed mucosal immune system to protect you from this high level of infection well there are many distinctive features of the mucosal immune system and this kind of gives an overview and as the lecture progresses I'm going to go into more and more detail about how these unique features play a critical role in the behavior of the mucosal immune system well first as I you have a in Intimate interaction between mucosal epithelia and lymphoid tissues and in fact mucosal epithelial cells themselves have certain characteristics of immune cells they make satkin they make chemokines and it's really a very tight partnership between them and in fact they are are lymphocytes CD cells that are interdigitated adjacent to epithelial cells at the front lines similar to lymph nodes there are discrete areas of lymphoid tissue pyus patches uh and lymphoid follicles but they are not encapsulated again they're in very close contact with the environment they but again the immune system has decided that it can't basically shut itself down and not know what's going on in the outside environment it has to have specialized cells called M cells that allow antigen specifically to diffuse through even pathogens but it's basically reinforced that area with B cell t- cell dendritic cells to protect itself from infection the effector mechanisms utilized are a high level of activated and memory te- cells as I said before you don't have a lot of time in the mucosal system to respond to infection and therefore the kind of cells that you want going to want to have there are memory te- cells which as you recall I said very rapidly respond when they come encounter with antigen so therefore you have a large number of memory te- cells there and in addition you have effective regulatory te- cells again that that play a role in terms of modulating giving a rapid immune response and also if an epithelial cell gets infected you want to rapidly be able to eliminate that epithelial cell and another factor of the mucosal tissue is it doesn't mind being a little quick and dirty so if it thinks an epithelial cell is infected it's not going to weigh make sure that that peptide that it has is the appropriate far peptide that the t- cell re recognizes it's basically going to do kind of like Behavior observation so uh you one can think of the mucosal system as a very very busy airport you have millions of people going through day in and day out almost all of them are not any to be D to be concerned about they're normal tourists that are just going from one place to another same way in the gut pretty much all the food that's going through all the antigen is not going to harm you however sometimes you do have uh people who act strangely and you sometimes have to quickly just get those people put them on the side and and and do something so too the immune system has to say this this epithel looks like it's sick looks like it's infected I can't see the forign peptide there but nevertheless I can't take the chance I have to do something about it so we'll talk about that mechanism and it's also critical in terms of imuno regulatory environment because I as I mentioned before you have so much antigen there that you have to tell your immune system don't be paranoid you know it's it's okay there there's germs there there's bacteria there relax it's not going to infect you so you have to downregulate the immune system and therefore you also have to kind of allow this concept of Tolerance namely if you've seen an antigen and the antigen hasn't harmed you in the past then you say I'm going to give it a pass and ignore it and in fact that's an important mechanism by which the mucosal system ignores antigens it just generates this kind of Tolerance okay any questions okay so let's look at the anatomical as we learned morning Anatomy is Destiny let's look at the anatomical structure of the gut and try to use that to understand what the functional activity is so first thing uh that I'll to point out is where are the lymphoid cells and how does their location impact upon their function so the first thing you see in this slide is the fact that all of you are familiar with pyes patches and pyes patches basically is the equivalent of the lymph node in the uh mucos muscle system and particularly in the gut and this basically has the classic t- Cell B cell type interaction that occurs when antigen is seen B cells and t- cells help each other proliferate Etc there's also isolated lymphoid follicles we discuss in a minute these all drain to mesenteric lymph nodes where ultimately they can drain into the thoracic and into the bloodstream but in addition to that if you look at the in this case Villi in the intestine you see actually two populations of te- cells one population of te- cells are present in the lamin propria and they have one function but there's also another population of T cells that are interdigitated into the into the epithelium like studded scattered throughout the epithelium that are closely approximated adjacent and interacting with the epithelium itself so the Air's patches you have B cells and t- cells lymphoid follicles these are smaller mainly B cells also have these in in the the respiratory tract lining of the nose and the appendix is another uh site for mucosal immunity mes and the mesenteric nodes he basically are where white blood cells lymphocytes drain from the ly the the pyus patches and the lymphoid follicles and again these differentiate independently from the immune system during fetal development it's own separate immune uh differentiation that occurs now I mentioned M cells and actually that was the last slide I showed in the last lecture and this is uh both a pathological demonstration of an M cell and the EM is really nice you can see it's this very very discreet looking cell that's present in the intestine they cover PES they they differ from epithelial cells in that they have no microv they have broader folds they don't secrete enzymes mucus they have no surface glyco they're not really involve in the digestive process at all what they do do is they transport organisms from the gut Lumin to immune cells they endocytose or cyose antigen at their anterior surface uh deliver it to dritic cells and in fact as I'll show later some pathogens use the the uh M cells as a way of get inside of the body and people have actually postulated that HIV also uses M cell to penetrate the mucosal system and as I is shown over here and this is just a cartoon showing that you basically have this is the Lumen of the intestine here you have uh antigen perhaps or bacteria here's the M cell if this would come in contact with other areas of the of the uh the intestine it would not be able to penetrate the barrier however the M cell is specialized to allow things through however it's reinforced right behind it by a large number T cells macrofagos B cells it's expecting things to come in and now once it comes in it traps it it digests it it presents it now to t- cells to to amplify responses to the anthen B cells to make antibodies as a way of knowing what's out there in the lumen in order to know what kind of immune response to generate but you can imagine that also pathogens could also use this as a way of sneaking in okay any questions now dritic cells also are present as you know in large amounts in the mucosa what's really uh cool about dendritic cells is as they're very stretchy and if they have these processes and recently it was shown that if you take this is actually I believe this may be taken from a mouse that's transgenic for FP expression using a deck 205 promoter so it's only expressed in dendritic cells and you see this process coming out and this very thin line is actually the epithelial barrier and now a cross-section of that in a cartoon shows you dendritic cell the process is going out between two epithelial cells and you have looks like to me like a submarine you have a periscope you know so if you want if the sub if you're on a submarine you want to know what's going on in the ocean you don't take your whole submarine up and look around you stick this Periscope up look around and then see if everything is okay apparently dritic cells send this process out into the Lumen to ask the question what antigens are out there and then it actually internalizes them and then processes them again to try to educate B cells and t- cells here to know what kind of immune response to generate but you can also Imagine the same way the M cell can be used as a poral so to infectious agents can use this as a way of getting past the epithelial barrier and then down below into the lamin appropriate and in fact HIV has also been postulated to use this mechanism so dendritic cells are also recruited to mucosa in response to chemins they extend these processes capture antigen they also as I'll show you present within lamin propria and several chemokines ccl20 and ccl9 bind receptors and ttic cells and if you actually ironically block ccl20 signaling you block recruitment of dendritic cells you may prevent HIV infection which also is evidence that dendritic cells may play a role actually in dissemination of HIV across the uh epithelial barrier and antigen again loaded dendritic cells May migrate from the Dome region of py patches to t- cell area or draining lymphatics to present antigen again to mount the appropriate immune response now as I said there are fector te- cells that are present all throughout the gut you have resident te- cells that are found in the epithelium and the lamin appropria the but the epithelium itself these epithelial T cells are predominantly cd8 cells whereas in the lamin propria underneath the epithelial layer this has a more diverse population of lucaites cd4t cells cd8 cells plasma cells macf dritic cells and eils and mass cells so this is clearly compartmentalization of what cells you have based on where you are with cd8s predominating in the epithelial cell now just throw this out to the group why do you think cd8 cells are present all throughout the epithelium what are they going to be doing killing because that's what cd8 cells do and in fact as I'll show you in a few minutes that's their job they're almost like the um uh I guess the police or security for the epithelium if they see any of the epithelial cells adjacent to it that look like they're acting funny like they've been infected their job is boom just immediately kill them don't ask questions so that's will turn out to be why this is predominant cd8 whereas in the lamin appropriate this tends to be more of a classic immune response we have the Cooperative interaction helper self inct cells macro prod etc etc and in in terms of the plasma cell what antibody they make as you know predominantly IGA is produced the mucosal tissue as I had well it turns out ironically that in the genital tract and this is actually very relevant to HIV infection you actually make more IGG than IGA and the reason that's important is in terms of vaccine development you probably want to make a vaccine to protect from osal transmission of HIV that actually generates uh IGG neutrophils not only seen in the gut when you have inflammation or infection and how do tea cells get into the lamin appropriate as I'll discuss in a few minutes they express a unique homing molecule on their surface integrant Alpha 4 beta 7 and ccr9 which attracts them into the laminal propri area FL the from the bloodstream whereas epithelial t- cells Express a different homing molecule integrant Alpha eb7 now I have have to apologize these are not exactly the ciest names and it's hard to remember you know which is which so you sometimes maybe they do that on purpose to to to make mucosal immunity seem more complicated but again these are two different molecules that bind to two different lians because integrin Alpha e beta7 binds to e cedrin on epithelial cells and that's what allows these CDA cells to be studded in between the epithelial cells okay any questions so now to summarize mucosal how te- cells get to the mucosal how B cells get to there why they home there and why they come back so as you know naive T cells and B cells leave the thymus and bone marrow and circulate in the bloodstream they're basically looking for a place to land wherein they could see antigen for the first time because they're naive tea cells some of them may enter pyes patches through endothelial venules using eltin which is the same which is and ccr7 as their homing molecules if they don't see any antigen same way in lymph nodes they leave the mucosal tissue through a ferent lymphatics return to the bloodstream perhaps recirculate back to other mucosal tissues or potentially re recirculate back to lymph nodes again they're looking for antigen they're looking for something that they recognize to allow them to determine whether they recognize an antigen that's useful if antigen is encountered however again as seen in lymph nodes the cells become activated they leave the drain into the mesenteric lymph nodes to the thoracic duct and then though they come back to the gut so in essence they become activated but now they localize to a different area they don't localize necessarily to an area where they see antigen now they look such as pyrus patches now they localize to an area where they could be eector cells so either in the lamin appropriate or in inter dispersed among epithelial cells and T cells that first encounter antigen in the mucosal tissue such as in the gut and the gall they now are expressed gut specific homing molecules the alpha 4 B7 and ccr9 that ensures that when they come back in the bloodstream they circulate back to the gut and this is just a pictorial representation of it okay now the expression of these homing receptors is triggered by dendritic cells and here you see just a pictorial representation of it here is the high endothelial ven that the te- cell naive t- cell circulating in it comes out it's expressing eltin and ccr7 it's now waiting to see if an antigen is present antigen now comes through the M cells then dtic cell presented and now the T cell if it recognizes the antigen gets activated it leaves the gut gets into the thoracic duct into the heart then gets pumped into the circulation but now it comes back because of the fact that it's expressing the Homing receptor ccr9 and and the alpha 4 beta 7 that allows it to bind to Mad cam 1 which you recall is the mucosol adhesion molecule and ensuring that now you have te- cells that recognize gut specific anen coming back and again ccr9 binds to ccl25 on the gut epithelium and what's this why is this important for vaccine design because if you want to make a vaccine that's going to be providing you with mucosal protection you're going to want to administer it in a way that you going to ensure that those antigen specific cells go back to the gut it doesn't do you a lot of good to have large numbers of cells that are specific spefic for mucosal an pathogens in a lymph node that's under your arm that's not going to help protect you from an infection that's coming in through your intestine and that's why as you well know there are mucosally delivered vaccines so for uh pulmonary uh pathogens there are inhalation type vaccines and for gut there are oral vaccine so polio we swallow because that's that's where the polio virus lives so again that's important so for HIV for example as you well know in addition to designing vaccines that are given systemically we're also designing mucosally delivered vaccines and you may be familiar with a lot of the Mac studies that that have looked at that in terms of because you want to make sure that your immune response is is present in the gut where the pathogen is going to be coming in okay any questions and the other but the other cool aspect of that is because of the fact that T cells or B cells that have seen antigen in the gut home to the gut they just don't home only to the location where they saw the antigen they home all throughout the mucosal tissue and therefore now they'll populate all the mucosal tissue lung uh GI G GU tract and even going to the breast tissue which is actually why you have IGA specific to enteric pathogens being secreted in the milk because B cells that have seen antigen in the gut now home to the breast tissue they secrete their IGA into the milk now when the baby gets the milk the mayy has IG that specifically recognizes gut pathogens which is what you want the milk in the in the gut to be specific for okay and this is just showing uh higher magnification here you see the lymphocyte uh is expressing eltin and Alpha 4 beta 7 binding to Mad Cam and this is now going to allow it to basically stick to EP to the endothelium of the blood vessel but in addition you also need something to pull the cell out of the bloodstream and the compounds that do that are chemokines and you have expression of chemokines present ccl28 ccl25 that are binding to these receptors ccr10 and ccr9 that now suck the lymy from the circulation into the gut mucosa where and again they could potentially interact with antigen if they don't see antigen they leave they continue to express these adhesion molecules so keep on recirculating trying to see if they recognize antigen okay secretory IGA is a is the critical umal Yeah question how an just keep going around how long can they do that yes the question is how long can they go round and round and you you'll see figures like four to eight weeks tend to be quoted and again I honestly don't know how they did those studies because what really involved doing is taking naive cell tagging it and then kind of like waiting and seeing how long it could last so potentially you could just tag it with the fluorescent D maybe X Vivo inject it back in and see how long it l but basically for for at least one or two months you could keep on doing that okay so secretory IG we discussed this yesterday is it's a dominant class of imunoglobulin in the gut in the respiratory tract in blood there's also IGA but IG in the blood is monomeric it makes sense because what the dimeric structure does it's linked by the J chain it has two act it has and and and IG is generated by class switching from IGM and there are pathogens that can cleave ig1 but the fun of the J chain theoretically is to make it more resistant to proteic digestion and also as I showed you yesterday and I'll show again it's important in the transport of IgA from the B cells making in the lamin propria outside into the gut Lumen across the epithelium and ig2 apparently is more resistant to digestion so now if we look in terms of the mechanism by which the IG gets transported across the gut this is basically the same slide I showed yesterday IGA secreted by the B cell it now has to get across the gut there are tight junctions between the epithelial cells so it can't go that way and therefore it's binds to a poly IG receptor and I'll show in a few minutes this also binds IGM but it binds this it now gets transported inside of the epithelial cell through endocytosis and then ultimately gets spit out into the Lumin where it now could see whatever an pathogen or antigen it's specific for so activated B cells just like t- cells express the same homing Mo molecules the IGA is secreted as a dimer the poly IG receptor recognizes the J chain and again transcytosis and then again IG apparently binds to muin at the epithelial surface to keep it there so it doesn't just get shed out uh in the in the stool what's very cool about this process is it's not just utilized to get the IG out there it's also utilized to get rid of things that are bad so in this situation here we have secreted uh well first of all you one thing it's doing is you're making IGA it gets secreted here let's say there's a toxin that's present in the lum well the same way yesterday we discussed IGG binds tetanus toxin so too if this is a CH toxin or one made by a gut pathogen IG could bind to it and prevent it from causing damage in addition as IG is endocytosing let's say there's something bad inside the cell well the antibody actually combined to antigens that are present in endosome so again this is a a rare situation where an antibi body is actually functional inside of a cell in terms of protecting the cell and finally it's if there are toxins that have somehow penetrated into the gut you need to pump them out what IG can do is it could bind these toxins on the other side of the epithelial barrier find and then using its transportation system go out leave and dump a Meed alumin so I mean it's very elegant again how IGA is really um amortized its ability to transport itself across the epithelial cell layer okay question I well one can make it if you if you've been infected for example with Chala and now you have a lot of Calera toxin that's present potentially that could do it Chala any any any anyone and because you again it's it's a it's a nice system because you basically saying it's gotten across this but we don't give up we'll just try to dump it out and it's like you all those movies where you know they're just taking whatever they have and throwing in the airlock and getting it outside quickly before it could cause damage okay now this is also important to know IG does not activate the compliment pathway why do you think that is why could that be bad you don't need to because the IG is just going to go out anyway okay you don't need to but could it be bad for the gut to activate compliment Dam it could damage the gut again you always have this kind of concern if you have an overz immune response it's bad and why is why are you so concerned about damaging the gut because you have so much stuff growing there ex it might potentially enter the BL exactly you have it's a barrier you want to make sure you don't damage your barrier so you're willing to detune your immune response a little bit because you don't want to damage that barrier so think about it's like having a space suit if you have even a small Hall in it it could potentially be extremely deadly so you know if you if you're if you're having a fight with SPAC suits you don't use knives because they're sharp objects because you're concerned about that little uh small uh wound it also again doesn't trigger inflammatory response you have a lot of IgA for exactly that same reason you don't want to damage the barrier and the other thing that IG apparently does is it restricts commensal FL Flora to the Lumin again the bacteria commensal bacteria we like it doesn't harm us etc etc but other hand we don't want it to start getting into our bloodstream because then it potentially can be toxic okay there are people who have IG deficiencies the most common imal deficiency however most individuals have no clinical problems and this was a real puzzle because on one hand we're telling you that oh IG is so important in mucosal immunity and yet these individuals that lack it usually aren't very sick why should that be it turns out that IGM can replace IG in secretions because migm also has a j chain and it can also be transported by that poly IG transporter so if you lack IGA you may have enough IGN that gets put into the mucosal into the secretions that actually could protect you and um the it's suggested by the fact that if you not have IGA there's really no increased susceptibility to infection but if knock out the poly IG receptor then those M are susceptible to mucosal infection that suggests that IGM may be replacing some of the function of IG a blunting the impact of the deficiency okay mucosal te- cells the overwhelming majority of te- cells are present in the alignment appropria as I had mentioned before the overwhelming majority of them are memory cells cd45 row o is memory as opposed to peripheral blood which mostly have naive te- cells they Express gut homing markers they also Express receptors for inflammatory chemokines and they proliferate poorly in response to antigen amien again the gut has a little detuned immune response you don't want to have too many te- cells to also potentially interfere with uh with absorption of of nutrients they make a large amount of cyto particularly Al 10 al5 and gameron it it's unclear again in the healthy gut it's unclear what their function is but clearly they play critical role in protecting you from infection uh and they may have a regulatory role now as I mentioned before one population of te- cells are intraepithelial lymphocytes they're present studded within this epithelial barrier and there's about 10 to 15 lymphocytes per 100 epithelial cells so what could think about this as a as a front lines 90% of t- cell overwhelming majority of which are CD cells they express these homey markers in order to allow them to bind to e cedrin and therefore home to this specific location they're activated and they have high levels of perforant and Grine so as we said before this means they're arms dangerous and ready to kill and they however do not seem to be this elegant antigen specific cells now why is that because if there this elegant antigen specific ific cell population they're not going to be able to be functional unless they see the pre the antigen to pre-programmed to recognize and the chances of that happening are probably slim so therefore they must be being triggered through a different mechanism to allow them to initiate their killing and the way this happens is in two ways there are two types of intraepithelial lymphocytes one type called type A is a classic cd8 cyot toxic te- cell restricted expresses CDA Alpha and beta chains the type B expresses cda8 alpha alpha dier so it's different from the classic cd8 that we've seen so far in the systemic immune system it expresses a unique molecule called nkg2d which is a type of NK receptor and this binds to two unique molecules MCA and micb these are molecules that are uniquely expressed by epithelial cells when they're damaged and the Act and the activation of these interepithelial cells can be mediated by inal 15 but now let's look at let's see yeah now let's look at under higher magnification the two uh what can happen in the situation so in the case of a classic cytotoxic t- cell Behavior you're the epithelial cell is infected with the virus the virus comes through it now gets loaded a viral peptide in the context of a class one MHC molecule the cytotoxic t- cell sees it in the classic fashion it's already been activated by a dendritic cell because remember it's a Memory te- cell and that's why the overwhelming majority of te- cells in the gut are memory because you want them now to be primed and ready to respond and now it secretes perforant or fast fast liant interaction and kills the infected cell that's the classic way but the type two intraepithelial cells use an alternate me mechanism to make itself aware of the fact that the epithelial cell is infected and basically in this situation there's either a Toxic effect or infection and that stimulates the epithelial cell to express high levels of mic Alpha Beta the mic Alpha Beta binds to the nkg2d and now this T Cell signals perer killing of the intraepithelial lymphoid cell so it's not through an antigen TCR specific interaction it's basically this t- cell sees that this epithelial cell is acting suspicious because it's now expressing M IA AA and B it figures it's acting suspicious because it's been damaged or infected and just kills it okay it's that c it's a completely novel unique mucosal pathway for killing mucosal cells now in response uh to infection well again as as you all know mucosal services are not sterile and the mucosal immune system has to do something very complicated it has to differentiate between a harmless pathogen or actually pathogen by definition isn't harmless harmless Flora from pathogenic microbes now if you see uh a bacteria inside skin you know it's pathogenic because it got there if you see in ly Noe it's bad but this is a new job that only mucosal immune system has to deal with on the other hand the gut is the most fre qu site of infection makes sense because it's seeing the most the the external environment so it has to balance those two issues well in order to do this it has the help of epithelial cells it almost has deputized them to make them assistance in the immune system they have some of the functions we traditionally associate only with immune cells they are are polarized so that the apical area which faces the Lumen has very different activity from the basil area which faces the the uh lamin appropria so apical surface faces Lumin basil faces the uh lamin propria and the polarized expression allows different receptors having different functions to occur so the same molecule expressed on the basil surface may have different functional activity from the same molecule expressed on the apical surface now toll like receptors for example are expressed at both membranes but the effect of the tollite receptor is different commensal bacteria again play a role in maintaining homeostasis and now this is just again to review all the different to like receptors we've done this before so I won't spend much time on it but you know each one of them can specifically recognize a different Motif that's present in different bacteria or pathogens now the way the polarity plays a critical role is in terms of how what it tells the immune system to do if if um if to toac receptor 9 in the basil area so the basil area is is already on the inside if it comes in contact with what it's pre-programmed to recognize then it stimulates activation of nfca B which is going to turn the epithelial cell on and activate it why because if you have something on the other side of the epithelial cell it means it's penetrated the barrier which is bad however if the toll like receptor on the apical surface is activated which is facing the Lumen that's not bad because that's probably a commensal bacteria would you want to activate the epithelial Cell No in fact you want to deactivate it to prevent it from mounting an inflammatory response so it actually upregulates ikb and therefore which sequesters nfca B and therefore basically prevents activation of the cell and therefore that's how apical tlr9 stimulates intracellular tolerance to commensal bacteria in in addition to uh the previous mechanism I showed you interfacing with M interfacing with interepithelial uh uh EP cells with in epithelial te- cells there's another mechanism by which epithelial cells signal te- cells that they've been infected and that's by basically expressing unique molecules that are like hanging out a red flag saying I'm infected you have to basically kill me to prevent spread of that infection and again this whole idea is that you can't depend upon this elegant peptide MHC process when you're on the front lines of potential infection where you have to act very rapidly and in this case you have toll like receptors within intra cellular vesicles you have these mole ules called nod one and Nod 2 which are shown over here they are binding to bacteria so in this case nod one recognizes mural tripeptide on gam negative ronze nod two recognizes the mamal dipeptide on the Pepto pepo glycan of most bacteria and if bacteria now penetrates the epithelial cell it now binds to one of these non molecules this activates the NF cap pathway and now this NF Capa B pathway is activated it stimulates the epithelial cell itself to make a broad range of cyto kindes and chemokines that we normally associate with immune cells and again as I said before this is how epithelial cells get deputized to assist the immune system because now all these factors are going to now recruit immune cells macrophages polyes dritic cells to now help block this infection and basically protect it from being infected now however it's a double-edged sword because you can facilitate further Invasion because a lot of these factors as you know inter cause the intraepithelial Junctions to become looser and weaker and therefore allow more bacteria to come so you again you always have this delicate balance as you pointed out you don't want to ruin your barrier but on the other hand if you ignore the infection then basically uh it's going to be too late so and in addition the inflammation in and of itself can cause symptoms in the gut but again you have to look at the big picture potentially that you need to block the infection and you're basically taking a temporary hit and I mentioned before that some pathogens utilize M cells to actually get across the epithelial barrier this is just an example showing that salmonella can either cross the M cell directly it can get in across epithelial cells or it can could use the process sent out by the dendritic cell to bypass and just get in through the dendritic cell itself and then come out on the other side that's again something for every action that's done in order to monitor the the outside environment there is the potential for pathogens to utilize that to get inside the cell and as I had mentioned previously there is some evidence that HIV also uses this mechanism either dendritic cell or M cell in order to get into the gun okay now tolerance uh the overwhelming majority of antigen like 99.999% that your gut seed day in and day out is basically benign and it's most likely food well you don't want to start making antibodies or immune response against food if you do what do people develop food allergies or or food intolerances where they eat a food and then they get a rip roaring allergic reaction because unfortunately they've mounted immune response against some harm peanut antigen or some harmless egg antigen or milk antigen in order to avoid doing that there's a very powerful way of generating oral tolerance and in fact if you take mice feed them with ovalbumin and then 7 Days Later inject them with oval to generate an immune response what you actually see is that mice that were given control food develop a very nice reaction to the forign O albumin but mice that were fed OV alimin they do not mount an immune response against the OV Alin even though they've been immunized with it and this is basically tolerance because you basically said to the immune system OV alamin is nothing to worry about even if you see it in the context of immunization the immune system will still ignore it so this is good because the overwhelming amount of antigen in the gut in terms of food you don't want to mount an immune response but you have to be a little bit concerned about this potentially for if a pathogen could take over this pathway however this is used clinically as a way of desensitizing people towards antigens that they're specific for generating or oral tolerance by feeding it through the oral root if the MU if the mucosal tolerance breaks down and you start responding to antigens you can develop celiac disease and apparently it's associated with HLA DQ dq2 expression you could see CD4 T cell gamerin production in response to glute to glutens food allergies Cron's disease and again there are some nonto mutations that make those nonto molecules overly active also stimulating inflammation now commensal bacteria are good and in fact you all know that you have large amounts of commensal bacteria however let's say you put on antibiotics the antibiotics kill those commensal bacteria so now you have almost very little commil bacteria now you can become infected with cicil and cicil itself now can produce toxins that cause mucosal injury again you break through the barrier of the gut and now other pathogens can come in and cause infection in addition you see red blood cells can leave and that's why these patients classically can have bloody stools because that's an example of the damage that's caused by this toxin if you hadn't gotten rid of the gut Lumin uh bacteria this wouldn't have happened and that's why C def fail infections are classically seen after someone's been placed on antibiotic therapy so normal gut maintains Health it competes with pathogenic bacteria it also uh inhibits pro-inflammatory signal and the loss of normal gut floor allows other bacterial to grow and then disrupts intestinal barrier and allows uh even allows non-pathogenic commensal cell uh bacteria to invade the bloodstream now this is actually quite relevant to HIV infection because a few years ago um Danny DW at the NIH published a really elegant paper showing that early on in HIV infection you basically Wipe Out the intraepithelial as well as the lamin appropriate te- cells in the gut which makes sense because there C the in lopate the CD4 positive but an an ramification of that is that they lost integrity of the gut barrier and they had high levels of LPS systemically as a result of that and then LPS as you know can activate the immune system and in addition to causing classic toxicity from LPS it also now could cause cells that are HIV infected particularly macres to make even more HIV so that was a really nice example I mean of demonstrating how HIV infection can really have a significant effect systemically in this particular case in breaking through this gut barrier Okay so endogenous floor recognized by adaptive immune system the s s secretory RG and T cells recognize commensal bacteria and make sure you don't uh develop an immune response to it and typically they don't invade if you rais Min in germ free environment they have small lymphoid tissues low IG and reduced immune response and this is basically showing that one of the mechanisms by which dendritic cells by which commensal bacteria you become tolerant to is by virtue of actually affecting the uh dendritic cell uh ability to modulate thtre cell expression versus th1 and th2 expression so in this particular case this dendritic cell is being is being exposed to pge2 and TGF beta that's being made by the epithelial cell itself and this is putting the dendritic cell into this kind of resting stage it also makes TG TGF beta also makes Incan 10 and as I mentioned earlier that will specifically cause production of t-reg cells t-reg cells inhibitory cells and that quiets down the immune response to the commensal uh bacteria however if bacteria invade across the epithelial cell now that dendritic cell becomes activated and now in response to that it could either activate a th1 response or if it makes Incan 12 and th2 response to appropriately respond to the uh infectious agent so again here's where you see how the activation state of the dentritic cell is critical resting dentritic cell TS turn off the immune response activate the dentritic cells to infection turn on the immune response and okay and if the regulatory mechanisms fail then again you start seeing th1 responses now as you know one of the major Lo pathogens seen in the gut are parasitic infections and naive c4t cells are activated uh during infection with parasites T2 responses are protective th1 responses produce inflammatory reaction that damages the mucosa so th1 actually is probably bad for the gut and as you pointed out you damage the mucosa you open the the barrier and you open the floodgates to potential infection Isle 3 and is 9 recruit mucosal mass cells that produce lucrin proteases that again remodel the intestinal mucosa and make a hostile environment for the parasite so as you see over here the uh parasites stick to the epithelial cells that's how what anchors them to the gut if however you secrete factors that SLO off the epithelial cells the parasite the worm could hold on to it as hard as it wants but once that cell is not no longer anchored in the gut it's going down the tubes and going down the toilet literally because that's what holds it in place so that's how the immune system basically uh flushes out parasites the um and so these you turn over these epithelial cells to eliminate them but again as you pointed out it's a double edge sword because that potentially opens up the epithelial barrier on the other hand parasites have evolved next mechanisms that modulate the immune response okay if you have the the if you now have an infection the previously tolerant environment gets changed dritic cells become fully activated present antigen to te- cells and you basically but you can have both of them at the same time now this is a something that someone raised earlier which is called the hygiene hypothesis and this basically is saying that it actually may be good to be infected or early on because if you haven't been infected early on you haven't developed this real balanced approach so many of you know people that have lived very very sheltered lives and whenever you put them in a stressful situation they totally decompensate because they've never been exposed you know something silly happens like they miss a light you know that'll trigger a complete decompensation or they you know something trivial happens they they bump someone else's Fender in a car they basically lose it however people that have a lot of experience and going through a lot of different things they tend to take things in stride because of their large amount of experience so too in the immune system if you've seen a lot of pathogens early on so you if you've had a lot of exposure to parasites you really have a good balance of your immune system you know when to get responses when not and therefore you have a a markedly less incidence of hypers sensitivity responses to harmless environmental antigens if you haven't been exposed you tend to be more allergenic you get more food allergies you'll get more respiratory allergies and that may be the trade-off so in fact it may be good to be infected at least not terminally obviously early on in your life now don't go feeding your kids you know parasites or dirty marshmallows have been sitting in the backseat of a car in order to protect them okay so summarize your Coastal immune system avoids making active responses to the overwhelming majority of anthens because they're benign they're not going to harm you disruption of this balance leads to disease local dendritic cells play a key role either to suppress the immune system or to activate it when appropriate the dendritic cells in pyus patches produce Incan 10 to suppress the immune response rather than proinflammatory aisle 12 local IG produce response to antigen and it's maintained by these factors that are produced by epithelial cells which themselves have some immune function and dritic cells can migrate to the mesenteric node but therefore they they may also contribute to the generation of Tolerance because they don't provide co-stimulatory signals and gut homing molecules are induced on te cells to allow them to relocate back uh to the mucosa so therefore now to come back to the questions I asked how is the mucosal immune system different mean we could spend an hour hour reiterating how it's different but you appreciate it's different in terms of being more tolerant because of the antigenic load but also having Pathways that don't depend solely on TCR MHC peptide Pathways for recognizing that a cell is infected and killing the cell how does immune system prevent overreaction by by basically having cells like T-Rex cells that downregulate it by having apical versus basil polarization of epithelial cells to give you this either inhibitory or or uh ex activation uh response depending upon which side of the epithel y'll sees the the bacteria and how does the mucosal immune system protect itself from infection it does have clearly te- cells macrophages and the interepithelial lymphocytes to basically the first sign of infection the first sign of tissue damage to basically make sure that it's it's it's minimized and how do pathogens bypass mucosa immunity by actually using the M cell to get inside of the cell or potentially to use the dendritic cells to get inside and what th subtypes are preferentially activated in the mcal system predominantly going to be th2 because again IG is going to play a critical role in terms of in terms of uh parasite parasite uh protection and again th1 responses may be so ere exuberant that they may actually be toxic to the epithelial barrier and ultimately be bad for you okay thank you very much for your attention
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Medical_Lectures
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Basic_Immunology_Nuts_and_Bolts_of_the_Immune_System.txt
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This program is a presentation of UCTV for educational and non-commercial use only. I'm very excited to introduce to this week Tony DeFranco. After spending significant time going back and forth between East and West coast I've learned tonight, he very wisely settled on the West Coast in 1983, coming to UCSF. But he spent undergraduate time at Harvard, and then he was at Berkeley for graduate school, and then was at NIH, and then came back out to UCSF. When I asked him how he wanted to be introduced, he mostly very modestly said, please introduce me as a teacher of immunology. Well, that is certainly an understatement because he has been absolutely core to the teaching programs not only of medical students, but of everyone here learning immunology. Compared to my basic overview that you got last week of clinical immunology with a little bit of basic thrown in with just mostly introduction and terminology. Tonight, you're set for the nuts and bolts of immunology so that you can learn a little bit more in depth and more precisely about the immune system. I feel very privileged that he's here tonight, and I'm going to turn you over to Dr. DeFranco. Thanks, Dr. Gundling, and thanks for that fine introduction. It's great to see everybody here tonight. I gave a very similar talk about two weeks ago to another group of medical students. That group is really busily studying for a really big exam. That's our second year medical students and you all don't look nearly as nervous as that group. Nuts and bolts are in the title and I hope you don't get too afraid by that. I'm going to give you a few nuts and a few bolts. But what I'm going to try to mostly do is give you the principles and use those nuts and bolts to illustrate the key principles for how our immune system helps protect us. Of course, I think most people, even before last week's lecture, knew that the immune system is to protect us from scary things that want to eat us. No, not these scary things. These scary things. On the left, we have an electron micrograph of a virus particles. This is the virus that causes AIDS, Acquired Immunodeficiency Syndrome. This is a bacterium here. That's the bacterium that causes tuberculosis. Over here, we have a red blood cell that's got some parasites inside. Those are the parasites that cause malaria. Those, of course, are the three big killers worldwide. Infectious disease killers of people worldwide along with influenza, we probably should include that to be the big four. We're going to hear something today about how the immune system protects us about these kinds of threats. Now, these are numbers from the US Centers for Disease Control, the CDC, who keep track of this thing. This is the number of types of viruses and bacteria and fungi and worms and parasitic protozoa such as the one that causes malaria. These are the number that cause serious disease in people. You can see, it's not just a few things, but the immune system has got to protect us from a continual barrage of things that want to eat us. They don't eat us the way a lion or an alligator would, but they want to make use of our goodies and grow in us. That really should be in the very big type, and then some other functions of the immune system I've listed here should be a much smaller type. The immune system also promotes the normal functioning of the body by helping clean up our tissues and repair wounds. It helps remove abnormal cells, including the beginnings of some cancers. This is now well established that the immune system does cure cancer. To some extent, it prevents cancers from coming up, and it's also in some clinical settings does cure cancer. I'll talk about that toward the end. Then this is the good. This is why we need our immune system, and this is why individuals who are born with a big malfunction in their immune system, really are going to die in a very short time unless we do some really drastic treatment. We have a lecture coming up on bone marrow transplantation, which is one way that some of those diseases can be completely cured. Which I think is very exciting and some of that really was pioneered here at UCSF. Some of those therapies. I don't know if anybody saw the Chronicle, I think it was on Saturday. There was an article about the pediatrics department here is working on getting faster diagnosis of those kinds of individuals because the therapies, the bone marrow transplantation works better if you catch the people before they have serious infections. This is an ongoing area that UCSF is a leader in. But then there's also the bad part of the immune system and a couple of the lectures are going to touch on that. Our immune system sometimes confuses all of these bad guys for things that are not so bad, like hay fever pollen and cat dander and things like that, and can cause allergies, which is Dr. Gundling's specialty. Even worse, it can mistake some of our own components for foreign invaders and cause auto-immune diseases. We're going to have a lecture from Dr. Andrew Gross in our rheumatology division about some of the exciting new therapies that are coming online that make use of this information, the understanding of the immune system that I'm going to talk about today, to develop new therapies for some of these very nasty diseases where we really need a really strong intervention. Then of course, another example would be transplantation. If somebody needs a new kidney, or a new lung, or a new heart, or a new liver, we have immunological rejection as one of the problems. Those procedures only became feasible when we learned how to suppress immune responses enough to allow those organs to remain and not get destroyed by the immune system. A slider to remind you of some of the things that Dr. Gundling covered last week. First off, I just want to remind you of the players. The Giants have pitchers and in fielders and outfielders and a great catcher, what does the immune system have? The immune system has sentinel cells that are sitting in the tissues waiting for an infection to come along, and there's really three types of immune cells sitting out there. A cell called the dendritic cells, and I'll talk a little bit more about them. This is a type of cell that immunologists are really excited about right now. We're learning about these dendritic cells as being very important players. There's the macrophage, which is also an important player out in the tissues, and there's a cell called a mast cell, which is important for allergies and asthma, and also for protecting us from worms and blood-sucking insects like mosquitoes and ticks and things like that. Then these cells that are in the tissues waiting for an infection to come along, when they find one, they call out for help from their friends which are circulating around in the bloodstream, and they come in two flavors. The cells that are good at killing things, the phagocytes are good at eating things and the granules are good for killing things. You saw some movies of neutrophils last week. Neutrophils, monocytes, and eosinophils would be the three types of cells in this category. They're circulating around in the blood waiting for an infection, and then they're going to go into the infected tissue and help get rid of that infection. Then we have our lymphocytes. These are what we call our adaptive immune system. They are the B cells that make antibodies, T cells that do cell-mediated immunity. You heard a little about these last time, and I'm going to go into these in a little bit more detail today. Some of these immunodeficiency patients that really need a bone marrow transplantation are because they're missing these components here. Although that therapy is also being considered for defects in these kinds of cells as well, genetic defects. Then finally, I heard that you had some questions last week about a type of cell called the natural killer cell, and it's really good at killing virus infected cells. It is more in this category than in this category. But morphologically, it's a lymphocyte, which is what does it look like in a light microscope type of definition. Anyway, you'll hear a little bit about this. One of the world's experts on natural killer cells is Lewis Lanier, and he's going to be part of that bone marrow transplantation lecture coming up in a couple of weeks. Those are the players. Let's then look at what happens when we get an infection. Again, this is something that we heard a little bit about, we're going to go into some more detail about it tonight. The sentinel cells in the tissues, and that would be the macrophages and dendritic cells in this case, if we're talking about a new infection. They would recognize the microorganism that's coming in. Let's say you get a cut in your finger and you get some bacteria in there, or you breathe in something and it gets past the barriers in the lung and gets into the tissue, then we would be having an infection, and then our cells in the tissue would be recognizing that. This process of recognition by these types of cells as opposed to the lymphocytes, we call that innate immunity, and they utilize evolved receptors that recognize broad classes of microbes. They would recognize that this is a virus, or this is a bacteria, or this is a worm. They recognize broad classes. They recognize molecules that are characteristic of broad classes of organisms. Not necessarily pathogens, but any bacteria, or any yeast, or any worm. Some small fraction of the bacteria in the world are pathogens of people. Most of them are benign and live in the soil, and help decompose dead organisms, and so forth, they provide a very beneficial role. Most of the bacteria are good, but then occasionally they are really ones that come after us and are bad. These sentinel cells in the tissue, the dendritic cells and the macrophage, what happens when they see these agents? Well, they realize that they are in limited number and they won't be able to really take care of the problem all by themselves, they got to call out to their friends in the bloodstream. The way they do that is by secreting proteins. They secrete proteins that then go and diffuse over to the neighboring cells and to the neighboring blood vessels and cause those cells to now bring in the cells from the blood. These proteins that are made by immune cells and act on other cells, we call cytokines. Cyto for cell, and kine because it induces an action on the part of the cell that it's acting on, so cytokines. The cytokines are really important in immunology. I'm going to mention them again and again in this lecture. I'm not going to give you the names of too many of them, the actual nuts and bolts. We know of about 50 cytokines. We don't even make our second-year medical students, and the people who write the board exam questions, don't make them learn the names of all 50 of those cytokines. They have to learn about a dozen of them. I'm going to mention about three or four or five tonight, so you're not going to get quite as many as our second year medical students. But you're going to hear about some more from Andrew Gross because some of them are targets of new therapeutics. We'll talk about cytokines and what they do. These cytokines go over to the neighboring blood vessel, the blood vessel cells then respond to that. What do they do? They put adhesion molecules, sticky molecules, on their surface that's facing the bloodstream for this white blood cells in the bloodstream to attach and then come into the tissue. I'll explain how that works in just a minute. They're going to attract circulating immune cells to the site of the tissue. This is a very efficient system. The cells recognize the infection, they say to the neighboring blood cells, "Bring in some help." The blood vessels bring in some help, and here it comes. The blood vessels also allow fluid from the blood. This is the cellular help. They also allow the fluid to come in with proteins that are put directly protective as well, such as antibody molecules that you heard about last time. I'll talk in a lot more detail tonight about antibody molecules. In general, this response of bringing in immune cells and bringing in fluid from the blood, we call this inflammation. Everybody is familiar with inflammation. You'd get a cut in your skin, you get bacteria in there, it swells up, it's red, it hurts. The definition of an inflammation was made by the ancient Romans. We've known for many years. You've all experienced inflammation. What I'm telling you is inflammation is good. We think of inflammation as bad, in this context, inflammation is good. It's doing what it's supposed to. It's getting the immune system there to fight the infection. When inflammation is bad, however, is when it's prolonged and chronic then you get tissue damage. Those immune cells that are coming in, their goal is to kill the bacteria, kill the fungi, kill these other things that are trying to eat us, and they're going to cause some damage to the underlying tissue if we give them the chance. That's where inflammation is bad, is when it's prolonged and chronic. This is just an illustration of the process I just went through. This is the tissue side. This is the layer of cells that line the blood vessel called the endothelium. This is the bloodstream here. We've got our white blood cells flowing through here. On the interstate, they're going through it 70 miles an hour. Out here in the tissue we got these bacteria. Our sentinel cells are recognizing those bacteria, they're secreting cytokines, and I've listed two cytokines on this diagram, interleukin-1, IL-1 and TNF. Among about the 50 cytokines, about 35 of them or so have a systematic name called the interleukin-1, interleukin-2 all the way up to interleukin-35. That's good for being systematic, but it's hard to remember them. What does interleukin-22 do again? We immunologists struggle with that. They originally named interleukin because it was thought that they're being secreted by one immune cell and acting on another immune cell. We now know it's more complicated that they can act on non-immune cells, like blood vessel cells as shown here. That's when the name was changed from interleukin to cytokine. But instead of calling them cytokine-1, cytokine-2, cytokine-3, we've stayed with the interleukin 1, 2, 3, etc. About the other 15 all have names that were too popular to give up. This one, for example, the TNF, that stands for tumor necrosis factor. Now, put yourself in the place of the scientist who discovered tumor necrosis factor. He had this thing that he found that he injected into a mouse that had a tumor, and the tumor cells all died rapidly by a process of cell death called necrosis, which is what happens if you starve the blood supply. The cells don't get any blood then they die by necrosis. It's a violent death. It's also what would happen if you take blood cells and stick them in a glass of water and they would blow up because there's no salt around. He had dollar signs and visions of fame flash before his eyes when he gave this molecule this name. Turns out, you can't use it to treat cancer. You cannot give people TNF. The reason you can't give them TNF is because this happens too much and you get inflammation too much, you go into shock, very bad news. However, somebody is making money from TNF, but by doing the opposite, by blocking it. This is really good for some nasty inflammatory diseases such as rheumatoid arthritis, then blocking TNF can be a very good therapeutic. You're going to hear more about that from Dr. Gross in a couple of weeks. IL-1 and TNF. It turns out we have therapeutics that block TNF, they are good for some diseases. Therapeutics that block IL-1, they're good for some other diseases, not quite as many diseases, so they're not making as much money from that. Somebody got the bright idea. What if we block both of them at once? That clinical trial stopped really fast because the people got severe infections. These two cytokines have overlapping functions and if you block them both, you have bad news, but you can block one of them and there's a therapeutic value that you can get from that. Our host sentinel cell sees the bacteria, secretes TNF and IL-1 that acts on these blood vessels, they put up several adhesion molecules. This one here grabs the leukocytes as they're coming by and they start rolling on the endothelium instead of zipping by 70 miles an hour, they get on the off-ramp and say, "Do I want to exit or not?" Whether they want to exit or not is determined by this little green ball here, which we call a chemokine, because it's a kine means that the cells respond to it. The kine refers to the cells responding, and the chemo is referring to the fact that there's a chemical that it's attracting cells. Basically, what the chemokine does is it controls which immune cells come into that site of inflammation. The reaction started here is determining which chemokine, which is determining which blood cell is coming in. The immune system has the ability to customize the response, and we'll talk a little bit more about how that happens a little bit later in the talk. You can customize the response to fight the type of infection that is being sensed here. The type of chemokine determines what blood cell comes in. If this cell sees the right chemokine that it can respond to, then its other sticky molecules, adhesion molecules become more sticky. It then stops, instead of rolling along, it stops and then it squeezes through between the endothelial cells and comes in here and then helps fight the infection. That's that sequence of events to get inflammation going. As I indicated there, the neutrophil is the immune system's first responder. If we have an infection with bacteria or fungus, yeast type a cell, it's the neutrophil that's going to come in and do the heavy lifting. Neutrophils are typically the first cells that come in, they are really good at killing things. They make a lot of nasty chemicals, including bleach molecule. Basically, what we use as bleach is one thing that the neutrophil makes. They're very good at killing microbes. They can also damage tissue in the process. If you get an infection, you'll probably notice you get some pus, and a lot of that is due to the dead neutrophils that are coming there. The neutrophils make nasty chemicals. They're very short-lived cells, because they make these nasty chemicals it's not good for their own health, but they come in and they fight the infection. Because they don't live very long, they need to be continually replenished from our bone marrow. This is, again, another medical correlation here, is that, if we have a situation where the bone marrow is not putting out blood cells adequately, the first place we feel that would be two things, the red blood cells and anemia, and the other thing would be the neutrophils and getting more infections or more severe infections, if we don't have enough neutrophils, because the neutrophils have got do this all the time because we're always getting some bacteria or fungus getting in there whether we got to kill off. Some chemotherapies for cancer treatment have this problem, and luckily, we have a couple of cytokines that make the bone marrow produce more neutrophils, and so since they've come online as therapeutics, that's helped cancer patients getting chemotherapy to do better that we can boost their neutrophils back up to where they should be. The neutrophils, now, because they're pretty nasty, this basic inflammation generally doesn't bring in neutrophils for very long, maybe about a day, and then it switches, and the next cell up is the monocyte. The neutrophil comes in first and then we start making different chemokines on the blood vessel that attract now the monocytes instead of the neutrophils. The beauty of the monocyte is it's bipotential. It can either be a good killer like a neutrophil, not quite as good as a neutrophil, but it can still be pretty good, or it can clean up the mess and help us repair the damage and get back to normal healthy tissue and not being all red and swollen anymore. What controls whether the monocyte is going to fight or going to clean up? Cytokines. Whatever cytokines are getting made when the monocyte gets in there, if we've already gotten rid of the infection, the cytokines shift over and we get the ones that promote wound healing. If the bacteria are still multiplying, we haven't gotten them all yet, then the cytokines are getting produced by the other cells in the tissue are going to be the ones to tell the monocyte to keep fighting. The monocytes comes in, it's sort of, "What's going on guys? Tell me what to do." It can respond to cytokines to do what we want it to do. Now, as I said, the neutrophils make a lot of nasty things. Another important principle about how we kill microbes is that we eat them. Our neutrophils, and our monocytes, and our macrophages, and our dendritic cells are really good at eating things. That's a process that we call phagocytosis. Phagocytosis means one cell eating something, in this case, eating something big. The definition of phagocytosis is something big, at least a bacteria size. That would be eating a microbe. The neutrophils and the monocytes can do this to some extent on their own but it's really greatly helped along if we make antibodies against that bacteria or that virus or that fungus particle. If we coat that microbe with some soluble immune components, can be some innate components work this way, but for sure, antibodies are really, really good at this. Now, our phagocytes really will eat them much faster and much better and kill them much better. That's the cooperation of the circulating phagocytes that act early with antibodies which are going to get produced a little bit later against things that we don't kill off right away. Now, I think this is a pretty important point here. Those things I gave you from the list, I just gave you the numbers, I didn't give you the names. Those things that cause human disease, mostly, and I think there's probably very few exceptions to this. Almost all of them cause disease because they're good at evading the immune system killing them. They have figured out ways to avoid getting killed by the immune system. Microbes that cause illness in healthy people either resist phagocytosis or resist killing inside the phagocytes or have some other related strategies to get by. Some good examples of the latter, some of them get taken up by the phagocytes, but then they keep the phagocyte from killing them somehow. A couple of good examples of that are Salmonella, which is a common cause of food poisoning, and also causes Typhoid fever if you have a breakdown in the water supply. Another great example would be Mycobacterium tuberculosis, the organism that causes TB. Tuberculosis, as I said, it's one of the big four. It's a huge threat to human health in many parts of the world. Although it's not a terrible problem in this country, there are people who get infected in this country as well. It's just much less, but in many parts of the world, it's a big threat. An example of an organism that evades this principle here of the antibodies coating it and making it really easy for our phagocytes to eat them up and kill them, an example of one that evades that is Streptococcus pneumoniae, which is a bacteria that is important cause of severe bacterial pneumonia. This one, it changes its outer surface so that if you make antibodies against Streptococcus pneumoniae, it's got a lot of very close cousins that your antibodies won't recognize that slight variant. There's lots and lots of really close cousins of Streptococcus pneumoniae. We've tried to help this along because we have a vaccine. One of the main vaccines for this takes 23 of these different variants and puts them all together and immunizes people. That takes care of about 90 percent of the cases, there's still about 10 percent that are relatives that are enough different that these antibodies don't help. But it does help about 90 percent of the time. Now, other examples of common organisms that change their coat so that the antibodies don't work again and protect us from a second infection, influenza virus is another flu, is another example of that. Another common example would be the common cold, rhinoviruses. Again, come in about 100 different varieties. Eventually, you will have had almost all of them, but it takes a while before you've had that many colds. Unfortunately. Yeah. Our adaptive immunity is good, but the bugs can figure out ways to evade it. That's a good transition for me now to talk about the B cells and antibodies and the T cells and cell-mediated immunity. They're part of what we call adaptive immunity. We've been talking so far mostly about innate immunity, the ability of our neutrophils and monocytes and dendritic cells and macrophages to directly recognize infectious agents. Adaptive immunity uses a completely different strategy. Innate immunity, those innate cells, they just have a bunch of different receptors that recognize very conserved elements of whole classes of organisms and are hard for those organisms to change because it's part of their lifestyle, certain aspects of their cell wall, if it's a virus, most viruses have RNA as their genetic material and when they replicate their RNA, they make a double-stranded RNA that the two strands paired against each other, we can detect that double-stranded RNA. It's hard for the virus to change how it replicates its genetic material. There aren't very many viruses that can directly avoid doing that. There are a few viruses that have switched to using DNA as their genetic material but most of them use RNA. Anyway, that's how the innate immunity does. Adaptive immunity uses a completely different strategy. Here now instead of all of the cells having a dozen different receptors, each of which it recognizes broad classes of organisms, our T cells and B cells, each one uses a similar molecule but slightly different. So it can recognize a different pathogen. We'll have one cell in 10,000 that will recognize influenza virus, one cell in 10,000 that will recognize rhinovirus that causes colds, one in 10,000 that will recognize the organism that causes tuberculosis, etc. That's all before we get infected. When we get infected, those ones that can respond, they expand up and we get more of them so that there's now enough to help fight the infection. Adaptive immunity uses a very different strategy. It starts out with each cell having one receptor and being able to recognize one type of thing, one individual rather than a broad class but then you have to expand those up to make them useful. Now, there are some other points here which I'm going to come to in a minute. The way in which this works for the B cells and the T cells, is that they do it at the DNA level. They start out with their ability to recognize these pathogen molecules. The gene encoding, those recognition elements exist in pieces spread out across the DNA of one chromosome. Then as part of the development of each individual B-cell, it picks one of these several 100 elements, and sticks them next to one of these, in this example, four elements. Some of the genes have three different types that come together, some of them have two that come together. I'm just trying to be very schematic here, but any given individual B-cell as it develops, it picks at random one of these 1, 2, 3 up to several 100. It puts it next to one of these four, you're getting a multiplication to give you the number of possibilities. This example I have given you only get up to a few 100 different possibilities, but it turns out, there are various mechanisms that are a little more complicated than I want to get into, but we can get up to millions and billions of possibilities for our B-cells and T-cells. Really, our B-cells and T-cells can recognize almost anything. There were studies done by an American by the name of Landsteiner in the 1930s. He made lots of organic chemicals and tested the immune system's ability to make antibodies against them. Really, whatever he made, he was able to get antibodies against pretty much. The B-cells and the T-cells, they can recognize whatever we throw at them. As I mentioned, these cells start out pretty rare because we've got each of which recognize different things. The principle here is that once we get an infection, those B-cells and those T-cells, which can recognize with their receptors, can recognize molecules of the infectious agent, they multiply many times to expand their number, and then some fight the infection, others are left in reserve as memory, and that's one of the main reasons that we don't get nearly as sick the second time around is that we have more of those lymphocytes. Instead of one in 10,000, we have one in a 100 or several in a 1,000, we have a lot more. That's what vaccines do, vaccines give us a fake infection that gives us more of that immunological memory and antibodies so that if we get the real infection, we're more prepared, our adaptive immunity doesn't take as long to get going, it gets going faster, it beats back the infection faster before we get as sick. Adaptive immunity is very important in a first infection and it's even better in a second, it learns from the experience of the first. It changes how many cells there are waiting around for the infection to give us more of those to be able to respond faster. I've illustrated that a little bit in this slide here. This is now lymphocytes, these happen to be T-cells, based on the shape of this molecule, but they're each a different color here, and that is supposed to represent the fact that they each have a little bit different shape and they can recognize a different pathogen. We start out with many different possibilities, now, because of the way this is generated, the B-cells and the T-cells, they are randomly bringing these gene segments together. It turns out, not only can they recognize any molecule we can synthesize, or any molecule that a pathogen can throw at us, but they also recognize our own molecules as well. That is the downside that comes with the good side, the bad with the good. That's the reason when we have an autoimmune disease or an allergy against food or against cat dander or against pollen, our immune system, our B-cells and T-cells have made a mistake, they've made an immune response when they shouldn't have. In most of us, we make that decision correct so often that we don't notice that we ever make any mistakes, we do make mistakes at small levels, all of us do, certainly, not rare, but only in a few percent of people do those mistakes get made badly enough that it becomes a disease or a really bad nuisance. Now, I guess probably, for allergies, it would be higher than the few percent, that would be probably, many of us experience allergies in springtime and things like that, but the point is that we randomly generate these lymphocytes. We then have to have a way of trying to get rid of the ones that are bad. Part of that is done during their development, they go through a phase in their development where if they contact their antigen, they're inactivated. That's illustrated here, some of the colors have dropped out, the point of that is that our self molecules are always there, whereas infections are not there and then appear. We have a timing device. The cells are developing all the time, the ones that develop before the infection, well, if we're going to see that infection in the future, they could just develop, they wouldn't see our self molecules, they would make it to the mature stage, and now when we get the infection, they can respond. The ones that saw ourselves, our self is there all the time, it would be there during the immature phase, and a lot of those cells would die off. That is one of the mechanisms which we get rid of the self reactive cells, it's not the complete mechanism, and I'll come back later in the talk to other mechanisms that the immune system uses. This is one of the important mechanisms, but there's also a subsequent mechanism that occurs. Some cells sneak past that checkpoint and then we have to silence them later to avoid auto-immune disease. If we get an infection, in this example, we're getting an infection with a virus that is seen by this cell here with the orange receptor, and so those cells expand. The other colors don't expand, they stay the same in number, but we give more of these. Now, at this state, if we get infected again with this orange pathogen, well, we've got a lot more cells to deal with it. We call that clonal selection. This would be referred to as a clone of cells that came from the original mother cell divided many times and gave rise to a clone of cells. They're all identical or nearly so. Again, vaccination works by generating memory T-cells and memory B-cells. Also, the cells that make antibodies, the B-cells during an immune response, some of them become antibody secreting factories and make large amounts of antibody, during an immune response, some of those can become very long lived cells, and continue to make antibody for many years, certainly, up to a decade, if not longer. We don't know completely how long they live, but certainly, for a decade, they slowly go down over time. We have antibody right there, right at the beginning when the pathogen comes back to deal with it right away. I've been talking about antibodies. What I want to do for the next five or so minutes is, probably, more like ten minutes, I want to tell you a little bit more in detail about antibodies. You all have heard about antibodies, but let's talk about some details so you understand a little bit better how they work and how we can use them in therapeutics. As a person, I can make millions or billions of different antibodies, but any one B-cell is only going to make one antibody. That's because it rearranged its DNA, each B-cell did that individually. Now, the molecule, I want to define another term here that immunologists always use, antigen. Antigen is what an antibody recognizes. That's an easy way, we have to have a word for what does an antibody binds to, it binds to the antigen. The name antigen comes from the fact that the antigen generates the production of the antibody. We inject an antigen into ourselves or into an animal, our immune systems produce the antibody against that molecule, they generate an antibody response. It's a circular definition. An antigen is something that causes you to make an antibody that recognizes it, that's where the name comes from. Then by analogy, we call it when a T cell recognizes also an antigen, although T cells don't make antibodies, but by analogy, we call that antigen as well. Now, as I mentioned, we have many B cells in our body and each B cell is making a little bit different antibody. But when we have, let say, a flu virus come in, we have different B cells that will make an antibody recognize different parts of that flu virus. It's not just one B cell that's getting activated, it's a handful or 100 or maybe even 1,000 B cells are getting activated and each expanding in number and then each making antibody. Each of those is a clone, but it's many clones because there were many different B cells to start with, each of which made a different antibody. We call that antibody polyclonal, and I'm going to distinguish that in a minute from a monoclonal antibody which we use in therapy, so this is a clone. Now, another important principle here is that B cells are often helped by T cells. Although T cells do cell-mediated immunity and B cells make antibody, they also work together and I believe you've got a movie of that last week from Dr. Gundling. That's the T cells helping the B cells to make antibodies. When that happens, the B cells make a higher quality antibody. What will happen in an immune response is actually some B cells will go off and make some antibody quickly on their own and that's an antibody that sticks a little bit to the pathogen, but not really well, sticks a little bit, so we want to get some of that out there fast, just to help right away. But some of those B cells, they're going to take the more long-term investing strategy, they're going to collaborate with the T cells, and they're going to make better antibody. They're going to go through a process in which they introduce mutations and then pick out the one, the mutants that have the highest affinity for the antigen. There's a slow, but ultimately useful process in which we make higher-quality antibodies that bind more strongly to the pathogen and therefore are more protective. We make some lower-quality antibody quickly, they help out right away and then we take some of our B cells and we invest them in a long-term bond and at the end of the day, we have higher-quality product. The B cells and the T cells see antigen in a fundamentally different way, as I'm going to explain. But yes, the way this works is that T cell has to see the same virus or bacteria or whatever that the B cells sees and only if they both are against the same molecular antigen, perfect, the same particle or, it doesn't have to be exact, immunologists would usually use antigen to refer to a single molecule. But in a virus particle, there are several types of molecules in there, and this will work for two different molecules in the same particle as well. Now, we call this the germinal center response and you don't need to remember that. But the key point is that this gives you these long-lasting antibodies secreting cells that go to the bone marrow and they secrete antibody for years and years and years. This makes sense if you think about it, that B cells that make the quick response, that's not as sticky, but it's going to help a little bit, we don't worry about, we just make that those cells only make antibody for a short time, but the ones that get the T cell help and make the really high-quality antibody, those are the ones we want, those antibody secreting cells to last a long time and to really help us over time so that when that pathogen comes back, we've got the best quality antibody to fight it. Almost all the vaccines work by this principle of producing this high-quality antibody that you're cranking out all the time. At least 25 of the 27 licensed vaccines work by that mechanisms, there is a little bit of argument about the last two, but almost all the vaccines work by this mechanism here, this is what we know how to do. Now, I want to say that actually this understanding of immunology was put to use in the 1990s to develop improved vaccines. It turned out some of the vaccines against some bacteria were using a type of antigen which could not engage the helper T cells, and so you only got the quick low affinity response that was only protective for a short period of time because you didn't keep making those antibodies six months later. It was found that those vaccines in very young children were not protective, it was only protecting against half, when the trials were done, the number of cases of that disease only went down by about half, not more as you would see with other good vaccines. They said, "Well, wait a minute, this vaccine doesn't have any way to activate the helper T cell, let's add that in there." That's a type of vaccine called the conjugate vaccine. Those came into being in the 1990s. Those are a case where this basic understanding of immunology has been applied to making new vaccines. Just to step back a little bit, I would say that where we stand with vaccines right now is we have some really good ones, some ones that are sore cells, and some ones that just don't work. The three big ones that I mentioned, AIDS, tuberculosis, and malaria, we don't have their vaccines there obviously. There's a lot of thinking that we need to really apply, we need to learn more about the immune system and apply it to our vaccines because what we've learned to do is this kind of immunology and what we need to do is to learn, probably to boost the cell-mediated component of immunology better in order to get vaccines against the things where we don't have them now. I'm an apple guy, but I have to give credit to Bill and Melinda Gates Foundation is really spearheading the effort to develop these new vaccines, particularly for things like malaria, which are problems in third-world countries. Now, I want to show you some molecules here. This is what an antibody looks like if we look at the actual structure. I've got one version on the left, and one version on the right. The version on the left shows all of the atoms in the antibody, the antibody is in purple and the antigen, which is a viral protein is here in red. In this diagram here, every atom is shown by a little ball, and the size of the ball depends on which atom it is, whether it's a carbon or a nitrogen or a hydrogen. Anyway, what you can see here very clearly, I think, is how close the fit is between the antibody and the antigen. The antibody mimics the antigen in its shape and that allows it to stick to it tightly, like a hand in a glove or a key in a lock, those are some analogies people like to use. Now, that's what we will learn from looking at this picture. What we learn from looking at this picture, now, what this shows is not all of the atoms, but just the traces, the path of the protein molecule. Protein molecule is a long thin molecule with amino acids stuck together, many, many, many. Two protein parts stuck together here, there are about 200 amino acids each. This is tracing out from the beginning to the end. It's a linear molecule in terms of the sequence, but it folds up into a compact shape. Now, what I want to illustrate here is that this part of the antibody and this part of the antibody, they are very similar from one antibody to the next, so the shape of antibodies is very similar from one antibody to the next. What is different is these little loops up here at the end that make the part that actually contacts the antigen. We get all of them, not all, but almost all of the variation occurs out in these loops. The structure is very similar, but the actual part that touches the antigen varies a lot from one antibody to the next because the variation goes into those loops. This is a case where the detailed structure lets us see the principle of how the whole system works. Yes, each B cell makes a different antibody, but they're not that different from each other, they're very similar to each other, except in those loops. Only the loops are different and it's the loops that give a very exquisite hand-in-glove type of fit. Now, actually, this is just a piece of the antibody molecule, it's about a third. It corresponds to this part here in this diagram. An antibody has two equal parts that each bind the same antigen because one antibody molecules made by one cell and one cell just makes one antibody, so the two halves are identical and can each bind to the antigen. Then this back-end is going to help the antibody be useful to the immune system, it's a tag for the other cells in the immune system. Then in the middle, we have a flexible part, this allows the antibody, if you can imagine my two hands of the part that grab onto the antigen, and my shoulders are the flexible part we call the hinge, the value of that is that my two hands can grab on to that virus particle regardless of its geometry because if I have to, I can do like this or I can do like this or I can do like this. I have a lot of flexibility in my ability to latch on with both hands. Again, that makes for a tighter binding, I can grab on more tightly if I use both hands than if I just use one. Now, the other point I want to make from this slide is that we make actually five types of antibodies that differ primarily down here, which is the part that the immune system uses to latch on to help do other business such as phagocytosis, the phagocyte taking up the antibody coated bacteria. That's illustrated here. Antibodies can help in several ways. They can help by just gumming up the works. This is called neutralization, where we have a virus particle, the antibody binds to the virus particle and now that virus can't get into the cell anymore because we've grabbed onto the part that it needs to get into the cell. Now, that virus particle we can destroy it, our phagocytes can eat it up and chop to pieces, but it can't even infect our cells in the meantime. This is the holy grail if you're trying to make a vaccine against the virus, you'd love to make a vaccine that makes an antibody that neutralizes, that can really just directly prevent the virus from entering. However, an important part of antibodies, as I said, is connect to the phagocytes. If the antibodies in this case they're coating of bacterial particle, that lets the phagocyte eat it up much more easily. Here, we've got an eosinophil which is good against defense against parasitic worms. Here's a worm, it should be much bigger than that, this is obviously a worm that's got to say, a few thousand cells compared to one cell here, so it should be 1,000 times bigger but if you'll ignore the artistic license there, if it's got antibodies on it, then the eosinophil is going to secrete its granule contents, which make nasty things that are going to kill that worm if we do that up and down its length. Then there's some other mechanisms as well. But you can see that the antibodies can work directly or they can work together with our immune cells to help protect us. Few other quick comments about antibodies. Vaccines cause us to make our own antibodies, we call that active immunity. That is good because it's going to last a very long time. The tetanus vaccine, we're supposed to get boosted every 10 years, so slowly your amount of antibodies go down over a 10-year period, you want to boost them up every 10 years. Now, the disadvantages it takes a week or two weeks to make some good antibodies. If I may needs some antibodies right away, they just got bit by a snake. What are we going to do? We can give them antibody, we can inject the antibodies into them so that the antibodies can just float around and grab the snake venom and inactivate it. That's the alternative then, we call that passive immunity. The advantages is fast. The disadvantage is not going to last very long. Antibodies in the blood have a half-life of about three weeks so within a couple months they're pretty much gone. Examples where we would use that would be if somebody gets exposed to tetanus and has not been immunized, hepatitis A, there's now use of vaccine. But about 10 years ago I got a hepatitis A shot when I was visiting India, for example, that was antibody against hepatitis A, that was passive immunity. Protection against snake venom, and also the mother imparts this type of immunity to newborn. When in utero, there is transfer of antibody across the placenta. That's our most important passive immunity is what we get from our mother and help us through those first few months of life. Finally, I want to mention that as we said, that the antibodies taken from a person or an animal, such as these examples, those would be polyclonal antibodies. They would be made by the progeny of a number of different B cells. But we can also make monoclonal antibodies. We can take a single B cell, we can immortalize it and then have it turn into antibody secreting factories in the laboratory and make lots of antibody. Then the advantage of that, we can use that for as a therapeutic or diagnostic. The advantage is that they're all identical, enhance very standardized, and we can, know what they're going to do time after time. Whereas these products really have to be carefully tested and make sure it's good in that kind of thing. One, I think really big change is that in the last 10 years, we're getting new therapeutics based on monoclonal antibodies that are coming and getting approved by the FDA and it's I would say on the order of 3-5 new ones every year and I don't mean me too types of therapies. I mean novel therapies where it's a new target is a new indication and so I think this is a really exciting time for the field of immunology. We're seeing it translated into new therapeutics. The key to this was the monoclonal antibody technology got discovered in around 1970. But this really is taken hold in the last 10 or 15 years and that's because we've learned in the meantime to make those antibodies as similar to human antibodies as possible, they were originally made in mice. The problem was if you injected a mouse monoclonal antibody into a person, our immune system would see that as foreign and make an antibody and then we'd get rid of that as a therapeutic. That still happens with these new generation, but in a much smaller percentage now, less than one percent typically. That was the key breakthrough and that's why we now have a lot of these coming online. This is just a very short representative list. These are two molecules here that are used in cancer immunotherapy for breast cancer and for B cell lymphomas. These are made by Genentech, which is in South San Francisco, and I don't own any Genentech stock. I should say happy to say, but probably would have made a lot of money on it. But these are two that are made by a local company that was started by a UCSF faculty member, Herbert Boyer. Then some of these are used for inflammatory diseases blocking immune responses and here's one that's used for coronary disease just as some examples. I wanted to say a few words about cell-mediated immunity and then I want to finish off by the news you can use, some thoughts about what do we know about the immune system that we can apply to our everyday lives? I've got a few slides on that at the end. I want to switch over to cell-mediated immunity. The T cells recognize, instead of recognizing an antigen in its normal form, the way it would be sitting in a virus particle, the way it would be sitting on a bacterial cell cell surface. Instead of that, the T cell recognizes a piece of a pathogen the immune system is extracted out of that pathogen and displays. A piece, and we call it a peptide, which means it's a short piece of a protein. The advantage of that is that the T cell can do what an antibody can't do. An antibody can only see a pathogen when it's outside of cells. Antibodies are secreted from cells, they float around on the outside of our cells. However, T cells can see those pathogens when they're hiding inside our cells. Those bacteria that get into the phagocytes and keep the phagocytes from killing them, the T cell can see that. The virus, when it infects a cell and the cell is now producing virus particles, well our antibody can act against the virus particles that get released. But the T cells can see this infected cell and they can kill the infected cell. The T cells are very complimentary to the B cells. They see the antigen or pathogen when it's inside our own cells and they can root it out and get rid of it. The way they do that is they see little pieces of the antigen that our immune cells extract from antigen and put it onto our own proteins that display it. Now our T cells come in two types. The T cells to recognize antigen, which is up at this peptide, bound to one of our own proteins, they use a molecule very similar, to the antibody molecule. We call it the T cell antigen receptor, or TCR for short. It's very similar type of molecule that's got the loops and it's very parallel system, it looks very similar. However, it's never secreted, it's always on the surface of the T cell, the T cell never secretes it. It doesn't float around. It's always stuck to the T cell and that's because the T cell is always acting locally. The antibody can act all over the body. The T cells only acting locally because it's going after the infected cell so it doesn't want to get away from the fact that I'm here and you're here and I'm going to attack you. That's the idea of the T cell. There are two types of T cells. There are the killer T cells, their purpose is to kill virus infected cells, and there are the helper T cells, and their purpose is to secrete cytokines to direct the action of other immune cells. The helper T cells are the cells that are infected by HIV and get depleted, and that's when we get the immunodeficiency for the acquired immunodeficiency syndrome, is when our CD4 T cells get too few to protect us anymore. If you know anybody with AIDS who is on antiretroviral therapy, their doctor will take some of their blood, send it to the lab, and measure the number of helper T cells in the blood. If that's falling instead of rising, then that means the virus is getting resistant to that drug and I better switch them to another drug because once that number falls below a certain level, they're going to start getting all sorts of infections, and that's when life becomes threatened in those individuals. The helper T cells are needed to protect us by secreting cytokines and boosting our immune responses from other cells. The cytotoxic or killer T cells instead of CD4, they have a molecule called CD8. The CD4 and CD8 are molecules on the surface. They play important roles in the function and that allows us to distinguish cytotoxic T cells from helper T cells. This is just a picture of what I just told you. The killer T cell kills the virus infected cell. It sees a peptide from the virus that is displayed by a protein of the infected cell. In other words, the infected cell take some of these proteins of the virus that it's making inside, chops them into pieces, sticks them onto this protein, sticks the protein on the surface. Even though the virus is not putting its own proteins on the surface, this cell is putting that piece of the virus on the surface, the killer cells sees that and it kills. It secretes toxic molecules that make holes and induces cell to die. This won't necessarily kill the viruses that have already assembled and haven't yet been released. But the point is that you want to cut off the factory that is producing new virus, they block the virus that's already been released with antibodies. The killer cells and the antibody work in a complementary fashion. Helper cell is a little more complicated. Here what happens is that instead of the pathogen molecule being made inside the cell, now the immune cell is taking it up from the outside by phagocytosis or endocytosis, so related process with smaller molecules, and then that cell chops it into bits, takes those peptides, sticks them on a very similar molecule but a slightly different molecule that binds those peptides. This type of molecule takes peptides that the cell brought in from outside. This type of peptide binding molecule takes peptides that were made inside the cell. Some elaborate cell biology that matches the peptide to the type of binding molecule that's going to bind it. That's illustrated here that we have two types of peptide binding molecules, and they look like hot dog bun. It's got this part, this part, and it's hard to see the three-dimensionality. This is sticking out and this is in the back. It's a table with two sausages on the surface here, and then the peptide goes in the middle. We have two of these types. The red type, what we call class 1, is going to pick up the peptide that's made by the virus inside the cell and stick that on the surface, and the blue one we call class 2, it's going to take a peptide that was brought into the cell from outside by phagocytosis, then we chop that pathogen into pieces, load the peptide on, and then stick it back on the surface. The T cell is interacting with cells that are displaying peptides for it to see. There's a cooperation between the cell, it's got the peptide and the T cell that can then visualize that because it's displayed outside for it. That's very different than antibody. The antibody just binds directly to the pathogen molecule. Here we have elaborate process for finding these, not letting the virus hide it inside. We're not going to let it hide it. We're going to show it out there so that it can be seen. Now, one more point before I go to the immunology you can use is that immune responses are tailored to the type of infection. I already mentioned this a little bit. We can have a microbe infection or we can have a virus infection. We can have an extracellular microbe or an extracellular microbe like the TB microbe or the Salmonella food poisoning microbe. We can have the location. The immune system can specialize to immune response in the lung versus in the gut. There's some different properties to the immune responses. Worms and biting insects versus microbes would make a different type of immune response against the worms and the biting insects, then against the microbes. Defense against most microbes is antibodies and the phagocytes. We've already gone over that in a bit of detail. We get a neutrophil rich inflammation, and that can be prolonged by the helper T cells. Normally, I said that you get neutrophils first and then monocytes. Some circumstances, the helper T cells will prolong the neutrophils and keep bringing them in. That's of course a more destructive type of inflammation. But sometimes, that's what happens. Normally, it would switch over to monocytes within a day. Defense against the microbes that can survive and replicate inside phagosomes. Now, here it's mostly the monocytes and macrophages they're surviving in. Because remember, the neutrophils came in for the first day, after that, the monocytes came in. A lot of times what happens is these things gain a foothold in the monocyte or the macrophage. It doesn't really do them any good to get a foothold in the neutrophil, because a neutrophil dies in a day or two anyway. They're not going to stay there very long. But the macrophages are long lived cell. If you can establish an infection in a macrophage, it can stay there for quite a while. That's what happens with tuberculosis and salmonella. The defense against this type of infection is what we call type 1 immunity, and the helper T cells then specialize to detect these infected cells, they release cytokines that promote killing by the macrophages. Defense against viruses, I think we've already covered this, the early defenses, the innate mechanisms that restrict the replication of the virus that would be primarily a cytokine called interferon, one of the first cytokines that was discovered, interferon, and it's called interferon because it interferes with virus growth. It makes cells able to resist the replication of viruses to some extent. That's our early defense and that's really critical, and then adaptive immune defense comes in and we make neutralizing antibodies, and we have the killer T cells. I think we've covered that. Then finally what we haven't really covered very much is defense against worms and biting insects. This is a different type of immunity called type 2 immunity. Here the helper T cells, do two types of things, they promote a different type of antibody called IgE, which works together with a mast cells, and another type of cell called the basophil to strengthen our barriers at the scan and in the gut and in the lungs. At the barrier between the outside and the inside, the epithelial layers, will strengthen that barrier. Also these type 2 helper T cells, they attract eosinophils to the site of infection which are very good at attacking worms. Some manifestations of type 2 immunity, which you probably don't like, would be things like sneezing, coughing, itching, diarrhea, tears. What these things all have in common is we're trying to get rid of stuff. We're trying to blow it out. We're trying to wash it out. Diarrhea, were trying to wash it out that end. Coughing, we're trying to blow it out by coughing. We just had somebody coughs. That's your immune system trying to get rid of something. Now, the reason we don't like that is because we live in a world that doesn't have very many parasitic worms, and so our immune system, the one theory that you're going to hear about from Dr. Boucher in one of the other lectures in this mini Med school is that one of the theories that has a lot of support is that our immune system gets bored with not having enough worms to deal with. It goes after pollen and cat dander and things like that, which if we had worms and other nasty stuff, we would just ignore that stuff, we would concentrate on the important stuff. That's called the hygiene hypothesis. You're going to hear about that from Dr. Boucher. I've left you with a big problem, actually is a big problem that immunologist don't fully understand. This is one of the really active areas in immunology research right now, including at UCSF, and that is how did the B cells and T cells know what's an infection and what's pollen and cat dander and the food you ate that you had never eaten before? The answer is that, that's a very complicated thing, but I can give you at least some of what we understand. One thing is that endorse innate recognition mechanisms, the dendritic cells and macrophages, the sentinel cells recognizing bacterial cell walls and things like that. They then put molecules on their surface that promote T cell activation, so that innate recognition of microbe is translated into promoting the T cell activation by putting molecules on the surface of those innate cells, particularly the dendritic cells. We call this process something called co-stimulation. The co being, the stimulation would be the antigen, and this molecule that the dendritic cell is making in response to it recognizing an infection. It's saying I see an infection, I'm going to tell a T cell I've seen an infection. I know this is an infection. That's one of the most important principles. We now have a therapeutic that directly attacks that co-stimulation, directly blocks that co-stimulation to try to suppress T cell in some nasty immune diseases. Another mechanism we have is something called the regulatory T cells. Some of those CD4 T cells make cytokines that promote immune responses. Some of them are now going to be the stop T cells instead of the go T cells. They're going to make cytokines that slowdown immune responses, that inhibit immune responses. We call that type of cell a regulatory T cell. The rare individuals have defects in the ability can't make any regulatory T cells. They get horrible autoimmune disease of all their organs at a very young age, is a very horrible disease. Luckily, only a very few people ever experienced that. But we really need these regulatory T cells. Again, they arise during development. When they see self-antigen during development, instead of being gotten rid of, they actually stay around and suppress the immune responses later. Then finally, that's what I just said, the defects in making regulatory T cells is very bad. This is another area that's a research area that we have some people at UCSF doing, can we harness regulatory T cells for therapies? Can we take a person's regulatory T cells, expand them in the laboratory, and then put back where there's now more of them, and now maybe they can block an immune response in a very specific way? Instead of just blocking all immune responses, maybe we can block just the rejection of that kidney transplant. That's a very exciting area for the future. Quick couple of comments about chronic inflammation. Sterile inflammation, we mostly talked about the innate immune system responding to actual microbes, but it also responds to just tissue damage. That really, I think, is a strategy that we use because some pathogens have figured out how to not be seen by the recognition of bacterial cell wall components, etc. Just if we have tissue damage, that will induce some inflammation on its own. Many of you probably experienced this with sore knees and sore elbows after doing too much basketball or something like that. Chronic inflammation, if an antigen persists, then you can get chronic inflammation because it's driven by the T cells. This is true in autoimmune diseases and some allergic diseases as well. There is speculation and I think there's some evidence is becoming now very well accepted in atherosclerosis field that inflammatory processes participate in the progression of atherosclerosis, and I can talk about that more if people are interested afterwards, and then this is now more speculative. There's something in the Alzheimer's disease may also probably not be initiated by inflammation, but propagated and made worse by inflammation. There's a lot of hope that maybe we could learn to ameliorate these things a little bit and hold off some of these diseases and not let them progress as much. That's an exciting thing for the future. The immune system in cancer, as I mentioned before, there's good evidence that the immune system removes early cancerous cells in many cases and reduces cancer incidence. The immune system is actually used to cure some leukemias together with chemotherapy, and this is a process called the graft versus leukemia effect. When bone marrow transplantation is done, we use bone marrow from a different individual, and then they leave the T cells in there, and then they can attack the residual cancer cells. There are some side effects to that, but this is very commonly used, including at UCSF now because it really does work and that is the immune system killing off the residual cancer cells. We have some man-made killing cancer cells with monoclonal antibodies. I mentioned that before. Then there are ongoing efforts to boost the patient's own T cells to react against their cancer, and there's been some real progress in recent clinical trials on that front, but there's still a long way to go for that, I think. Anti-inflammatory therapies, some of you may have, well, probably all of you are involved with this one way or another, or maybe many of you anyway. Many of you probably have taken aspirin and ibuprofen. They're good painkillers, but they're also anti-inflammatories. They're used to inhibit inflammation as well as pain. Now, they'll work when the inflammation is not too strong. If it's too strong, then you got to go with the bigger guns, the glucocorticoids steroids like cortisone, and the more potent versions of that. They were developed originally in the 1950s, and they were really life-changing for people who had severe inflammatory diseases like rheumatoid arthritis. They're very effective immunosuppressive drugs and anti-inflammatory drugs, but the problem is they have very significant side effects of long-term use. We've been trying to find better agents, and then you're going to get a whole lecture on the newer biologics, monoclonal antibodies, and related molecules to block the inflammatory cytokines. These are really very effective, not every patient, but in the patients that they work in. They worked really quite well. You'll hear about that in another lecture. Last three slides. Keeping your immune system in good working order. Now, we're getting into the areas where the immunologists have, it's more speculation and less hard fact, but let me give you a few things. Nutrition and the immune system, you'll hear a lot about nutrition and the immune system. I'll tell you my opinion. It's very clear that real malnutrition, if you're really so poor that you're not getting enough calories or not getting enough protein, your immune system will suffer from that. That's what we call macronutrients. If you don't getting enough calories or enough protein, your immune system won't function as well as it should. There are some micronutrients that are important, some of the vitamins are important. A normal, diverse diet would do that fine. You don't need to take vitamin supplements unless there's something about your diet. You're not getting your vegetables or what have you. Then finally, you'll sometimes see a lot in the news about fish oils. Fish oil is anti-inflammatory, that is pretty well established. You have to take a fairly high amount of it to achieve that effect. But fish oil is one of those things that will be anti-inflammatory. That has to do with the fact that inflammation is regulated by a variety of lipids, and we're now changing the lipids that are present. So we're changing a little bit the balance between pro-inflammatory lipids, which would be blocked also by ibuprofen and aspirin to some extent. Again, if you take enough of them and which again, speak to your doctor about that. I'm not telling you how much to take. But there's also anti-inflammatory lipids, so the thinking is the fish oil helps boost the anti-inflammatory lipids and protects us in that way. It reduces the chronic inflammation in that way. Again, if it's fair, this would be a case where it's relatively mild, I think not a really strong inflammation. Stress and the immune system is another popular topic. We actually have a lecture for this, for our medical students. What I can tell you is that chronic stress and I really mean, over several year period being stressed, a caregiver for a spouse who has Alzheimer's disease or for a child who has leukemia, something like that where you're really stressed a lot for a substantial period of time, that has detrimental effects on immune function, but it's complex. It really depends, and again, these things will correct. There are some circumstances where chronic stress will affect your ability to fight infections. Mind over matter is a very popular topic. What I can tell you is there is some science that relates to this. There is a nerve that controls the heart beat, the vagus nerve. It also enervates many of the immune organs, and it does modulate immune responses up and down to some extent, it won't totally shut off an immune response, but it can influence it up or down. When people do clinical trials, you really want to do something to account for something called the placebo effect, and that is if somebody is getting a pill that they think is going to help them, they actually will get better with some diseases. That's called a placebo effect. If you're doing a clinical trial for a new therapy, you have to give half the people something that's the same pill, but one of them has got the real pill and one of them is not the real pill and neither the patient nor the doctor really knows. That's called a double-blind placebo-controlled, and you really need to do that with immune system diseases because they have a substantial placebo effect in some cases. I think there is some mind over matter when it comes to the mind and the immune system. It's not something you can necessarily use except that maybe try to be less stressed. Aging and the immune system, I think this is something where a good diet and trying to keep yourself healthy is going to help your immune system. It is well established that as people get old, they don't respond as well to vaccines, but it's quite variable from person to person. Some people show that immune aging, other people not so much. We don't really know all the details, but it would be a good guess that a healthy lifestyle is good there and then vaccination. I want to say a few words about vaccination. I really appreciate your willingness to stand around. But I should have put this in earlier because this is maybe the most important thing I want to say today and that is in terms of keeping your immune system in good working order, there's one thing that we for sure no works and that is vaccination. Not every vaccine works, but a lot of the vaccines we have work really, really quite well. What we don't have, we need new vaccines for some things, but the ones that we have that work really do work. Childhood vaccination, again, get the advice of your doctor, but I really think vaccinating children is the right answer, and I think there's a lot of fears being spread on the Internet, which really just are not true. There are British physician claimed, this as more than a decade ago that mercury preservatives in vaccines caused autism. This has been completely disproven. There's even people who think that this was fraud to start with, and it wasn't just bad science, but he may have had an extra grind. But in any case, it's clearly not true. The one that had a little bit of truth was there was a small incidence of side effects from the old DPT, which is the diphtheria, tetanus, and pertussis. Pertussis is whooping cough. That vaccine, the formula used prior to 2002, did have high fever. It's uncommon, but not really all that rare side effect. It really had to be monitored. The new formulation has much less problem there. It's a more purified version of the pertussis that does not cause nearly as much fever. This is something that's gotten quite a bit better, but nonetheless, there's a lot of fear surrounding this, and whooping cough has been making a comeback among unvaccinated children. It's a very nasty disease. If you have any little children at home or grandchildren, this is my advice. Don't believe the fears on the Internet about these vaccines. Finally, vaccines aren't just for children, they're for us as well. Few things, tetanus booster, you should get a tetanus booster every 10 years. The best advice is that, do it when your birthday has a zero. So 40, 50, 60, then it's easy to remember when the 10 years comes up. Because who can remember when you had that tetanus vaccine last? Every 10 years really, that's great advice. You don't want to get tetanus, I guarantee you. Flu vaccine. Yes, for most people, influenza can be life threatening, particularly as people get older in life. Most of the guidelines say 50 and older really should get vaccinated every year. Then it changes its coat so that the antibodies that we make against this year's influenza won't help us very much against some of the strains that are circulating the next year. Each year, they try to figure out what the most important flu that's circulating is, make the antibody against that. They're all very close cousins of each other. Those of you traveling to foreign countries, vaccination may be helpful. Plan ahead and I can recommend from personal experience, the city of San Francisco Health Department has an adult immunization and travel clinic, which is really excellent. Also, you can check out the Center for Disease Control website that will tell you what part of the world you're going to, what vaccines you should be thinking about maybe to help you there. Then as we get older, even if we don't travel, there's two vaccines that are specifically aimed at people getting near retirement age or after retirement age. That would be the shingles vaccine, which is basically, if you had chicken pox as a young child, that can come back as you get older and your immune system wanes a little bit. It never went away completely and you can get a very painful condition called shingles. This vaccine's been shown to ameliorate that greatly. There's another one called Pneumovax. That's against that Streptococcus pneumoniae that you had. It's got 23 different cousins stuck in that vaccine and it protects really quite well. Again, this is something. Ask your doctor, but if you're 60 and older, you might want to consider these things. I really thank you for your patience, and I will stay, and take your questions. The question is, how do we immortalize an antibody producing cell to make monoclonal antibodies? There's two ways. The way that was discovered in early 1970s and for which the Nobel Prize was awarded to a man in England, Cesar Milstein and George Kohler, two men in England. That was they would take a cell that came originally from a cancer of an antibody producing cell in which they had manipulated in the laboratory to lose its own antibody producing genes, and then they would fuse it with a B cell so that the B cell in the fusion part though, the one cell gave you immortal growth, the other cell gave you the antibody genes to make the antibody you wanted. That was the original method and then more recently, there's direct DNA cloning method of pulling out antibodies and screening through to find the ones that you want. There's two methods now. There's the fusing two cells or just going straight in and pulling out the DNA and expressing it in a new cell. The question has to do with autoimmune disease called lupus. The long form is systemic lupus erythematosus. That's an autoimmune disease that preferentially hits women usually in their 20s, and 30s, and 40s when it starts and then it's never really goes away. You just try to keep it at bay as best you can. I understand there's a new therapeutic that was just approved for this last week and we're going to get Dr. Andrew Gross in one of the future lectures, and that is he sees lupus patients. He'll give you all the up-to-date thing. What I will tell you is that for most of the autoimmune disease, we don't know what triggers it. There's a generally thought, and you're going to hear this for allergies as well, is that there's a genetic susceptibility which is complex and we all have lots of genes that influence our immune response, it's a little bit this way and a little bit that way. If we get the wrong combination, then we would be susceptible to a particular autoimmune disease, and then there's some trigger. The trigger is thought to be probably some infection or another. It's a combination of genetics and some trigger. That would apply to all the autoimmune diseases. In the case of lupus, the disease manifestations are primarily due to antibodies. Bacterial spores. Does immune system recognize bacterial spores? Yes, it does. Now, so the outside of that is a really thick carbohydrate and so the antibodies can recognize it, but the T cells really can't recognize it very well. Again, that what I was mentioning, the conjugate vaccines, what was done was to take those, that coat of a bacteria and the original vaccine was just to use the carbohydrate part. Because it was known that the antibody to the carbohydrate was protective, but by coupling it to something that T cell could see some protein component, you can then get this higher quality. Question's why would the virus in a virus infected cell allow the immune system to take a peptide and stick it onto that MHC molecule, the HLA molecule? That's because that's what we're trying to do, it's not what the virus is trying to do. We have a machinery that's doing this all the time with our own peptides. But then that we're already tolerant to those because we've developed in the presence of those peptides all along. It's when the new peptides show up that the killer T cells can get activated and can come and kill. Now, there are some viruses that do block that process. I don't know if people have heard of cytomegalovirus. Cytomegalovirus is a virus that specifically blocks that process. CMV, exactly. CMV is a virus that blocks that process. Most viruses can't block it, but that's one that does.
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Medical_Lectures
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31_Cancer_3.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Last time I told you about oncogenes. Oncogenes-- we discussed the fact that these were gain of function mutations that occur in cancer cells-- that occur in normal cells as they develop into cancer cells, and then occur in cancer cells. And we discussed the fact that these are dominant mutations. And this was exemplified by the Weinberg experiment in which he transferred a dominant oncogene into a "normal cell' and caused that cell to become transformed despite the fact that it had normal copies of the same gene in its genome-- definition of a dominant mutation. We also talked about tumor suppressor genes. And tumor suppressor genes importantly, carry within them loss of function mutations in the context of cancer. Loss of function mutations, as such, these mutations are recessive mutations at the cellular level. It's necessary to inactivate both copies of a tumor suppressor gene in order to give rise to a cell that's lacking that function altogether. And that's a cell that is on its way to becoming a cancer cell. And these loss of function mutations can be various. You might find nonsense mutations in a tumor suppressor gene-- blocks the production of the protein. You might find a deletion. It takes out the entire gene, or a big portion of the gene. You might find a frame shift mutation-- again, that blocks the ability to make any protein, or much protein. And I also told you about chromosome loss events. The loss of the chromosome that carries the normal copy of the tumor suppressor gene as a frequent second event to lose the remaining wild type copy of the gene. OK, so this is just a little bit of review. Now, I framed the discussion about oncogenes and tumor suppressor genes from the point of view of cell division control and the production of more cells through the action of these mutations. And indeed, cancer is a product of inappropriate cell division. And the two genes that we discussed in some detail regulate this process. The RAS gene and the product oncogene stimulate cell division. The RB tumor suppressor gene inhibits cell division. When we consider the kinds of mutations that we have in these genes that regulate the cell division process-- in the case of the RAS gene, we find activating mutations. And in the case of the tumor suppressor gene RB, we find inactivating mutations. RAS is in oncogene. RB is a tumor suppressor gene. Activating mutations in the oncogene. Inactivating mutations in the tumor suppressor gene. But cancer isn't only about cell division. It's really about cell number-- inappropriate cell number. And there's another important process to remember in this context. Apoptosis-- program cell death, which I've referred to many times in many different contexts. Apoptosis, which results in dead cells. And failure to undergo apoptosis properly will likewise result in too many cells, which again can be cancer causing. We have genes that regulate this process. For example, the P53 gene, which we'll discuss in some more detail, positively regulates apoptosis. And another gene called BCL-2, which negatively regulates apoptosis. Is P53 an oncogene or a tumor suppressor gene? Oncogene? Tumor suppressor gene? Good. P53 is a very important tumor suppressor gene. It is inactivated in the context of cancer because you want to get rid of apoptosis as you are developing cancer cell. Is BCL-2 an oncogene or a tumor suppressor gene? It's an oncogene. We find gain of function mutations in BCL-2, producing more of this inhibitor of apoptosis in the context of cancer to block apoptosis leading to an increased number of developing cancer cells. OK, so oncogenes and tumor suppressor genes can regulate these two processes differently. As you can see, inhibitors in this case, stimulator's in this case, in the context of tumor suppressor genes. All right, a little bit more about P53, which is probably the most important cancer gene of all. It's mutated in at least 50% of all human tumors. And P53 is known to function as a molecular policeman of sorts. It's sensitive to various perturbations within the cell. For example, DNA damage. DNA damage will feed into the P53 regulated pathway, as will a fascinating process, which is still incompletely understood, where the cell can recognize that it's dividing inappropriately. Abnormal proliferation-- cells dividing when it shouldn't. This gets detected, and this, too, can feed into the P53 pathway. And this leads to an increase in the levels of the P53 protein. P53 is a transcription factor. It regulates the expression of other genes. And the genes that it regulates fall into two broad categories, some of which cause the cell to undergo cell cycle arrest. When the P53 pathway is activated under certain circumstances, the cells are instructed to arrest. During which time, whatever the damage is can be fixed before the cell continues to cycle. If there's DNA damage, the cell might arrest, fix the damage, and then continue on. In addition, P53 regulates apoptosis, as I mentioned a few moments ago, causing the cells to die. Causing the damaged cells to die-- and that's good because dead cells make no tumors. This is a sacrifice by the individual cell who has been damaged, basically saying, I'm in trouble, killing itself, and allowing the organism to survive. OK, and this is a very important process in cancer prevention we now know. And I'll tell you some examples of how we know that momentarily. OK, so I've introduced you to a couple of different oncogenes and a couple of different tumor suppressor genes. I want to focus a little bit more on the tumor suppressor genes in one respect. We touched on this last time, but very briefly. Tumor suppressor genes is sporadically occurring cancers. And sporadic means that the individual has no family history of that particular cancer type-- sort of a chance event. And tumor suppressor genes in sporadic cancers require two mutational events. And I described these as hits. And we typically do describe them as hits in the context of tumor suppressor gene inactivation. Sometimes it's mutations. Sometimes it's mutations coupled with chromosome loss as the second hit. But regardless, two mutations are necessary to get rid of both copies of the tumor suppressor gene. And I also briefly introduced you to the fact that tumor suppressor genes are often-- not always-- but for the most part, familial cancer syndromes, where the individuals have a predisposition to developing a particular type of cancer are caused by inherited mutations in tumor suppressor genes. So one of the hits, one of the mutations, is inherited from one parent. Meaning that every cell in that person's body carries one of the mutations already. Meaning that only one hit, one mutation, is required somatically. That is in the individual's own cells in their body. And this is why these individuals are cancer prone, because they're one step away from lacking the tumor suppressor gene altogether, whereas in most people two mutational events are necessary, and it's rare-- not never, but rare-- to get those two mutations. And I showed you briefly this pedigree, which is a pedigree of familial retinoblastoma, where the individuals inherit a defective copy of the RB gene, and as such are predisposed to the development of this tumor of the eye. And in fact, they will develop the tumor typically in both the eyes, and typically multiple foci of tumor. So they inherit a defective copy of the gene from one parent and they go on to develop the disease at high frequency. When you look at this pedigree, remembering back to your lessons in disease genetics, this looks like an autosomal dominant disease. If you inherit the disease allele, you have a very high likelihood of developing the disease regardless of who your other parent was. It looks like an autosomal dominant disease with actually incomplete penetrance, as we can see here. And we'll talk more about this in a second. An autosomal dominate disease, but interestingly, we've been talking about the fact that the mutations are actually recessive. So this seems confusing. So who can explain it? How do we have a recessive mutation at the cellular level causing what appears to be an autosomal dominant disease? Who can answer that question? Yeah? AUDIENCE: Predisposition is the dominant. It only needs one mutation. PROFESSOR: Exactly. Exactly. You are predisposed. And there's a very high likelihood that in some or in fact, many of your cells, this second event will occur. It's almost guaranteed. And since it's almost guaranteed, if you inherit just one mutation, you will develop the disease, and therefore it appears to be at the organismal level-- autosomal dominant-- because your predisposition nearly always guarantees that you will develop the disease even though the mutations at the cellular level are recessive. And so with that in mind, as we consider a pedigree similar to the one I just showed you, where one parent is heterozygous for the mutation-- the loss of function mutation-- passes that onto the next generation and beyond, in this individual, as well as the other ones shown with the dark symbols where a tumor did develop, what is a necessary second event? The loss of the wild type of allele, either by a second mutation, or a chromosome loss event. And that happens at very high frequency. In the case of retinoblastoma, it typically happens in a dozen cells in the developing retina-- leading to an average of a dozen independent tumors in those kids. OK, but now let's think about this individual here who did inherit the defective allele because he actually passed it on to his three sons, but he himself did not develop retinoblastoma. How can we explain that? Why was he spared? Well, there are two general explanations. First, he was incredibly lucky. Although it's highly likely that some cell or cells in the developing retina will mutate the normal copy of the gene. In him, it just didn't happen. He got lucky. None of his cells mutated the normal copy of the gene. His eyes developed normally. And after that point, actually, the cells are much less sensitive to mutation, and therefore after about three or five years of age, you typically wouldn't develop the tumor. So he might be incredibly lucky. Or it might be that in him, because of some other inherited allele of some other gene, even if he were to lose the wild type alleled of RB gene it wouldn't matter, because he's got some other gene that's functioning maybe in the place of the RB gene, leading to him to be protected. And these two possibilities exist, and we have examples of how both can be important. OK, final question on this slide-- what would happen if an individual inherited a mutation from both parents and was therefore homozygous for a mutation in a tumor suppressor gene. What do you think would happen? The answer was they would be stillborn or they wouldn't make it out of embryogenesis. And that's usually the case. And we know that not so much from the study of people who are homozygous for these mutations, because it's actually quite rare to find people who are heterozygous who have children-- so the number of such examples is few-- but we know it from knock-out mice. All of these tumor suppressor genes exist within the mouse genome as well. And my group and others have made mice with mutations in these genes. And actually we know that, in fact, for many of the tumor suppressor genes that we care about, like the RB tumor suppressor gene, if one creates a homozygous mutant mouse, or the APC tumor suppressor gene, which is important in colon cancer prevention, or the BRCA1 tumor suppressor gene, which is important in breast and ovarian cancer prevention-- in all of these cases, the embryos don't survive. And they die at different points along the way of embryogenesis. And it's actually not because they develop lots of cancer as embryos. Although, you might have thought that was true. They die because these genes are actually important in normal development. They're not there just to protect against cancer. They're there because they play a role in regulating normal cell division processes, normal cell death processes, normal physiology, such that when they're missing, the embryo can't survive. There's actually one exception to this. Well, I shouldn't say it that way. There are a few exceptions to this-- but one notable exception to this. And that happens to be the P53 tumor suppressor gene that I've introduced you to. My group and others have made animals that are mutant for P53 either heterozygous for the mutation. What would you expect the phenotype of these mice to be? Are they going to be totally normal mice, do you think? They are, in fact, cancer prone. They look like those people with inherited predisposition to cancer. They carry one mutant copy of a tumor suppressor gene. And they're one mutational event away from lacking the tumor suppressor gene altogether. And that happens at some frequency in their cells. And the mice will go on to develop cancer and die early for that reason. We cross these mice together with an expectation that again, they would not survive embryogenesis. But in fact to our surprise, you can make a fully P53 null mouse. And what do you think the phenotype of that mouse is? It's very cancer prone. Because this is a mouse, that in all of its cells, this very important tumor suppressor gene is lacking. Normal mice will live about 2 and 1/2 to three years. These mice will live about 1 and 1/2 years and die from cancer as a consequence. These mice will live about four to six months and die from cancer. So P53 is actually not important in normal development. It is a true tumor suppressor gene. It probably evolved to protect cells against the kind of damage that is inflicted on cells in an individual's lifetime. And if that damage is sufficiently great, the cells are eliminated or arrest permanently so they will not develop into cancer cells. It's a true tumor suppressor gene in that case. OK, so I've told you about oncogene and tumor suppressor genes. We are now entering an era over the last couple of years really, where many, many more cancer genes are being discovered through the application of genomic sequencing in the context of cancer. Cancer genomics is all the rage, including here at MIT, for example in the Broad Institute. And there are many papers appearing in the literature like these looking at the complexity of the genome of individual cancers, like this study out of the Broad on prostate cancer, and this study out of the Sanger Institute in England on small cell lung cancer. All the genes or the entire genomes of lots of different cancers are being sequenced and compared to the normal DNA of the same individual to catalog all of the mutations that are present within an individual tumor. And although there are differences depending on the cancer type, the average cancer genome actually has 100's and sometimes thousands of mutations. Small cell lung cancer, for example, has tens of thousands of mutations compared to normal cells of the lung. Why? Because they arise following years of exposure to cigarette smoke, which carries mutagens that bathe the DNA in mutation causing chemicals leading to mutations. Now, not all of the mutations that you find in a cancer cell are relevant to the cancer phenotype. In fact, we think that there's only amongst all those mutations that are found that there's only about 5 to 20 or so mutations in oncogenes or tumor suppressor genes. And we call these mutations driver mutations. Driver mutations, meaning, they actually are participating in the development of the tumor, causing some aspect of the cancer phenotype. And the remainder, we call passenger mutations. Silent mutations-- mutations that actually don't do anything to the cancer cell, but they just happened to occur at the same time, or in the lifetime of the cancer cell when another important mutation took place. So that clone of cells that develops carries those mutations too, but they're not actually contributing to the cancer phenotype. So among the hundreds and thousands of mutations, some of them really matter, and some of them are just going along for the ride. It makes the analysis of the cancer genome much more complicated, actually, because it's hard to tell what's a passenger, and what's a driver. But increasingly recognizing what are the important ones and focusing our attention on them. OK, that's all I'm going to tell you about cancer genetics. And I want to now turn my attention for the last 25 minutes-- and I may run over a little bit, so I ask for your patience if I run over a little bit, because I do want to finish this-- to talk about cancer therapy. Before I make the switch to cancer therapy, I want to first introduce the concept of cancer prevention. A lot of us work on cancer genes and cancer genetics to understand how to treat cancer better. But in the future, hopefully we'll have many fewer cancers to treat, because we'll be able to prevent them. If people would stop smoking, we'd have 80% fewer lung cancers to treat. If you use sunblock and stay out of the sun, we'll have fewer skin cancers to treat and so forth. Better diet and excess exercise will prevent a lot of other types of cancers. So there are lifestyle things that can lower the number of cancers in the context of cancer prevention. There's also ways to prevent agents from producing cancer in your body. And the best example of that is Gardasil. Gardasil is an example of cancer prevention involving a particular virus-- human papillomavirus-- and specifically, human papillomaviruses, which are described as the high risk type. Human papillomaviruses or HPV of the high risk type can cause cervical cancer. They are the main reason that women develop cervical cancer. And they're responsible for other types of cancers as well, including in men. And what we know is that in a normal cell of the cervix, when infected by the human papilloma virus, will at some frequency, and after a period of time, develop into cervical cancer. This suggests that the virus-- some genes of the virus-- are changing the cells behavior in such a way that it develops into cancer. Yes? AUDIENCE: I don't know if you're talking about our class or something else, but it's very rude to talk. It makes so much noise during the lecture. PROFESSOR: Thank you. So something in the virus is causing the cells to divide abnormally into a tumor. And this has been studied at length. And we now know that there are two genes, which are responsible for causing cervical cells to become cancer cells-- two genes of the virus called E6 and E7. And it's been learned that these genes in fact encode proteins that inhibit cellular proteins that we're actually quite familiar with. And we now believe we understand why the virus causes cancer. E6 inhibits P53. And E7 inhibits RB. So the virus for its own reasons of viral replication, takes out these two tumor suppressor genes. And as such, the cells are lacking these two critical tumor suppressor genes, and they're well on their way to becoming uncontrolled cancer cells. OK, so that's how HPV high risk types cause cancer. And what was developed in the context of Gardasil is an HPV vaccine so that individuals cannot be infected productively with this class of viruses, specifically it's a component vaccine made of recombinant proteins that are present in the high risk types of HPV. As a component vaccine, this vaccine does not produce a replicating virus. It's just pieces of the virus. So there's no risk of a viral infection here. But an potent antibody response can be elicited, including neutralizing antibodies that will prevent an individual from being infected by the real thing at a future time. OK, so that's an example of cancer prevention, eliminating an ideological agent that is responsible. There aren't many examples of virus caused cancers in humans. So this is kind of a special case. But HBV-- hepatitis B virus induced liver cancer will be another one before too long. OK, so now let's talk about cancer therapy. I'm going to talk to you in a few minutes about some new cancer therapies that are based on our improved knowledge of the genes in cancer cells. I'm going to tell you more about anti HER-2 antibodies. I'm going to talk to you more about a small molecule inhibitor of this kinase. There are actually many theorems that are based on mutations that we know occur in cancer cells. There are processes that I've mentioned to you, like angiogenesis, the recruitment of new blood vessels. These two have led to new therapies for cancer to block that process and inhibit cancer development. In the case of tumor suppressor genes, individuals are trying to develop gene therapy to put the genes back. If the gene is lost in a cancer cell, perhaps you can restore its function by gene therapy, and normalize the growth of the cancer cells. And although I won't talk about it, there's a lot of promise for immunitherapy for cancer. Cancer cells acquire a lot of mutations. As such, they produce a lot of antigens. In theory, your immune system should recognize those as foreign and eliminate the cancer. But in general, the cancer wins, the immune system fails. And we think that there's ways the cancer actually inhibits the immune system from functioning properly. But that's being figured out now, so it's possible that we'll be able to develop new cancer therapeutics based on the immune system. All right, but before I get into the cool new stuff, let me tell you just a little bit about cancer therapy more generally-- what we would consider to be conventional cancer therapy. The most effective form of cancer therapy is surgery. If you can get to the tumor early before it has spread, you cut it out, the person is generally cured. Another very effective form of cancer therapy is radiation therapy. And this is good because you can focus the radiation beam directly on the cancer cells and eliminate them by causing a lot of damage to those cells. And the third is chemotherapy. And chemotherapy is used when you think that the cancer has spread, so radiation can't work because the cancer cells are somewhere else-- and you need a drug that can diffuse throughout the body and hopefully kill the cancer cells. Radiation and many chemotherapies act in the same general way. Adriamycin, which you will have heard of, cisplatin, which you will have heard of-- these are well used cancer therapeutics-- and many more function by inducing in the cancer cell DNA damage. And the damage can be sufficiently severe that the cell will die. And these therapies can be effective. There's another class of cancer drugs for which Taxol is the best known, which are described as mitosis inhibitors. These drugs actually bind to microtubules, block the formation of the microtubular spindle, and that way prevent cells from dividing. And since cancer cells divide a lot and you want to inhibit their division, these drugs are used and can be effective. In fact, they're used because they were tested and shown to be effective first in cell-based studies, where one looks at the growth or survival of cells in a Petri dish, scoring the number of cells or the percentage of cells that are alive at any given time, or in any given dose of drug when the concentration of drug is increasing in this experiment. What we find is that for certain normal cells, they will survive to a certain concentration of drug and then start to die off. And for certain cancer cells, the concentration required to kill them is lower. And this difference is called the therapeutic window. In theory, this looks good, because it suggests that the cancer cells are more sensitive to the drug than are normal cells. And that's why some cancer therapeutics work for some cancers. But there are problems. Some normal cells in your body, unlike these normal cells, are very sensitive to the drug at low concentrations-- the DNA damaging agents, for example. This results in side effects. And I suspect you are all familiar with the side effects of cancer chemotherapy. Your hair falls out. You get nauseous. You get anemic. This is because cells in your hair follicles, or your intestines, or your blood system are dying at low concentrations of the drug. They're actually dying by apoptosis. They're actually dying by P53 dependent apoptosis. So that's why cancer drugs cause many bad side effects, because some cells in your body are very sensitive and will kill themselves in response to low concentrations of the drug. The second problem is that some, in fact not a small percentage, are very resistant to the concentrations of the drug-- even high concentrations of the drug. And one reason for that is that many cancer cells are lacking P53. I told you that P53 is mutated in about 50% or more of human cancers. I also told you that P53 was required for cells to respond to DNA damage and die. And these cells lack that protein. And therefore, are very resistant to dying. So many therapeutics don't work, because this important machinery is lacking. So we have problems with standard therapies based on both kinds of issues. So the goal then, is to find drugs that don't cause these kinds of side effects and can work even in a P53 deficient cell, which leads us to developing new types of therapy. OK, so I want to introduce you specifically to two. And they are probably the best known and highly effective actually, great examples of molecularly targeted anti-cancer agents. The first is in the context of breast cancer. And the gene in question here is a gene called HER-2. In a normal breast cell, there are two copies of this HER-2 gene, as there are in virtually all of your cells. And those produce amounts of RNA that produce amounts of protein that lead to the decoration on the surface of these cells-- the certain number of receptor molecules called HER-2, which are a growth factor receptors. They bind to specific growth factors. And when they are engaged with their growth factors, they send a signal into the cell. And the product of that signal is for the cell to proliferate. And this is necessary in normal development and in other times. So this is normal regulation, normal signaling in response to a growth factor in the normal levels of a growth factor receptor. 30% of breast cancers carry a mutation that results in amplification of the HER-2 gene. So we don't have two copies anymore, we've got 10, or 20, or 50, or 100 amplification. This is a mechanism by which all good genes get activated. Too many genes, too many proteins. This cell now has way too much of that growth factor receptor on its surface. And in the presence of the same concentration of the growth factor itself, we get a much stronger signal-- much higher levels of proliferation. This also affects the ability of the cells to survive. It keeps them alive at times when they shouldn't be. Too much proliferation, too much survival. Given this situation, a logical therapeutic would be something that blocks the function of this growth factor receptor. And what a company called Genentech discovered was that they could make an antibody. An anti-HER-2 antibody. And that led to a drug called Herceptin, which binds to the growth factor receptor and prevents it from functioning. And in women who have this alteration, and only in them, the drug is actually highly effective. In the metastatic setting, it will lead to multiple additional years of life. But it is actually not curative in that setting. Recently, individuals are being typed for this mutation at a much earlier stage in their disease course. And when women are given the drug then, it's leading to some cures. So this is a targeted agent which is highly effective in the context of a specific mutation. And actually only then-- other breast cancer patients given the same drug have no benefit whatsoever. So this is what it looks like. This is actually the drug package. And this is what I just described to you-- normal cells, cancer cells, Herceptin antibody binding to and blocking the function of this abnormal number of growth factor receptors. OK, let me now turn to the second classic example. And this comes in the context of chronic myelogenous leukemia, which is a leukemia-- a blood cell tumor. It's a particular type of blood cell-- the myeloid lineage type of white blood cell. You can diagnose this disease by looking at blood smears and you can see that this is the normal blood smear with a single of these myeloid cells. And here is a cancer situation where we've got too many of these white blood cells circulating. This is a disease that's been studied for a very long time. And it's been discovered that in the vast majority of this type of cancer, there is a specific chromosomal event-- a specific mutation caused by a particular translocation. And that translocation was actually identified a long time ago in the city of Philadelphia. And as such, it's called the Philadelphia chromosome. It's the product of a specific translocation-- chromosome 9, which has a gene on it called ABL, which encodes a protein that is a kinase involved in phosphoralating other proteins. And chromosome 22, which has another gene called BCR-- chromosome break events occur here. Chromosome break events occur here. And a translocation results, which produces a new chromosome-- the Philadelphia chromosome, which has the BCR gene and ABL gene inappropriately fused to each other. This produces a fusion gene, which encodes a fusion protein. And that fusion protein is referred to as BCR ABL. And it was discovered that the BCR ABL form had increased kinase activity. And this resulted in increased proliferation within the cells that carried that translocation. And so the question was, could one develop an inhibitor? An inhibitor that blocked the kinase? And this resulted in the development of a drug called Gleevec. Gleevec, which is highly, highly effective. This is the idea, here's the BCR ABL fusion protein. It binds to ATP, which it needs to transfer the phosphate group onto a substrate protein in this signaling cascade. The idea is that one could make a small molecule drug that could fit into the ATP binding site very specifically, and block access of ATP, therefore shutting off the kinase. And if that were possible, then the cancer cells would be deprived of this signal, and may stop proliferating, or even die. That was the idea. In fact, they were successful in making a small molecule drug. And that may surprise you, because you might think there are a lot of kinases in this cell. They all bind to ATP. How could you ever find one that was specific to this kinase? But in fact, it was possible. You can make kinase inhibitors, because not all the ATP binding pockets look the same. And you can therefore get some specificity. And when this drug was used in patients, it showed remarkable activity. If we looked at white blood cell number, normal individuals would have a certain low level. And in case of CML, the level would be high. And actually, it goes higher still as the disease progresses through a phase called blast crisis, where additional alterations take place and the cells begin to divide even more abnormally. In this context, however, if you give the drug Gleevec, in the vast majority of patients, the numbers drop precipitously. And the drug is extremely well tolerated-- has almost no side effects. The patients take the pill with their orange juice in the morning every day, keeping their cancer cells at bay, leading to what is called clinical remission. Clinical remission-- the disease has gone into remission. And it can stay in remission for a very long time. And it's sometimes curative. But sometimes the disease cells come back. And this is a phase we call relapse. And even though the drug is present throughout this disease course, the tumor cells are dividing again. Can anybody tell me why? Mutation. The cancer cells have acquired additional mutations, specifically within DCR ABL. If we imagine DCR ABL, it can bind to Gleevec, and be shut off. At some frequency, however, mutations can occur, which change the active site in such a way that Gleevec can no longer bind. And this is still an active kinase. So now the cells begin to divide again. So the question now is what can you do about it? And the answer is, you can make a new drug. And this has actually been done successfully. A new drug that can bind specifically to the mutant form of ABL kinase. And there's a drug called SPRYCEL, which is now also FDA approved for the treatment of drug resistant forms of CML. So before you run away, this is what I've just told you, here's ABL kinase. This is where the drug binds. This is the structure of the drug. But at some frequency, mutations occur within that ATP binding site. And different mutations will do this, as shown down here. And those mutations will block the access of the drug. And the good news is that one could make new drugs that will overcome that form of resistance. So this is a good news, bad news, good news story. We'll stop there.
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Medical_Lectures
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My_Aching_Joints.txt
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Stanford University well good evening everyone glad to see you again so first of all I want to thank you for being here tonight not only because it's a great session but also because of course it's the national championship and so here's the deal I have I've TiVoed this show and I don't want anyone to say a word right I don't want to see hands go up I don't want to see anything now here's what happened to me this is true last week I had my one of my early lessons in tivoing this was the very close game I think it was with Xavier and I T voted as well and I got down to just about the end and of course I didn't extend it being illiterate in terms of TiVo technology so now I've added an hour to mind so I think I should be okay so so that's the first deal and of course we all hope Stanford wins we know that the women are superior to the men as is well evidenced by year after year of repetitive performance and at least that gives us some comfort but tonight we'll think about their joints as well because they're going to be jumping all over the place right but there's a big difference between you know the compression on your joints from exercise I do it as you likely know every morning versus what happens when your immune system begins to react against your joints so last week we had a very detailed tour of the immune system and as I reminded you it was pretty complicated right I mean even a little bit more than the nervous system all these different elements you know of innate and adaptive cellular and humoral all the different intricate components of cytokines and interleukins but they play an important role not only in defense as you heard last week infectious diseases but also when the other side of that system becomes operative and it becomes overactive it could lead to serious disease and much of that comes in the realm of what we call autoimmune disease so what you learnt about last week is actually quite important to this week's session and it will be two next weeks as well on vaccines so you'll see how these things relate one to the other so tonight I'm very pleased to have dr. mark Genovese who's a professor of medicine here at Stanford talk about he says my aching joints and I'm glad they're his and not mine because mine are doing okay for the moment and it's about autoimmunity and degeneration now mark comes from a big football and somewhat basketball school called Notre Dame and then he went to the absolute nerd school that has no athletics called Johns Hopkins for medical school and then came back to Stanford which is kind of a mixed story where he's been ever since doing his residency training as well as his fellowship training in rheumatology and joined the faculty where he has really been a pioneer in leading the area that we call clinical and translational research focusing on inflammatory disorders in particularly rheumatological disorders in addition to his many original contributions he's also one of the editors of the major textbook in rheumatology so clearly a world authority in this important area and I look forward to hearing his presentation tonight so mark please join us no applause if Stanford wins or loses remember well thank you for the kind introduction and thank you for coming out tonight I know that there are other things that compete for your time so I'm gonna do my best to try and provide some knowledge about the way we think about arthritis our joints and how we tie that together with immunity as well I don't know that I'm gonna be entertaining but I promise you will not fall asleep so I like to start my lectures here I think this provides a fairly straightforward viewpoint about the way most people think about arthritis they think about somebody who's older they think about deformity and they think about disability and they hope that this won't be them and I assure you while you may look around and see some individuals in our society that still have this as the end outcome of their disease this absolutely does not need to be the end outcome of someone who develops arthritis in 2010 in fact this is a rare outcome nowadays from patients who develop significant and debilitating arthritis in fact we rarely send these types of patients to surgeons anymore you look at our orthopedic surgical colleagues they rarely see patients that require any type of intervention for this type of arthritis so tonight I'm going to tie together the immune system and what we see as far as the physical manifestations like you see on the right-hand side of this line and tie that together with some of the manifestations that take place in the bone and you see that on the left-hand side of the slide so let's first talk about how we define arthritis because it's a term that's fairly thrown around fairly loosely but we've tried to find it maybe a little bit more specifically as inflammation of the joint and this inflammation of the joint is specifically related to a set of symptoms often stiffness pain and tenderness and then as well related to another set of signs those include warmth redness swelling a decreased range of motion and then for many patients a deformity and for those of you who are writing now or taking notes I promise you I'll put these slides on and eventually the lecture will show up as well importantly though you need to think about the development of arthritis really as a result of an imbalance between a process of joint destruction and a process of reparation or restoration of normality within the joint so remodeling has to fare fairly well against a breakdown that's taking place within the joints it's the reason why when we injure ourselves at 12 or 15 we get right back up and keep going and when whingers ourselves maybe when we're 40 we're a little less aggressive about getting back up again our body's reparative processes don't necessarily work as well as they did when we were younger but that's going to be part of the lecture later on when we start talking about aging later this quarter so let's talk for a second about the epidemiology of arthritis what do we know about the diseases and how common are they well the rheumatic diseases including arthritis and soft tissue joint symptoms really are the most prevalent chronic condition in the United States in fact they're the leading cause of disability we often think about cancer cardiovascular disease or infection but the reality is this is the most common disease that we will encounter it's characterized by progressive pain and progressive physical impairment that involves the soft tissue in the joint itself and we're going to talk a little bit about how that takes place there are 43 million Americans affected by it in the late 1990s and estimated to be at least 60 million Americans suffer from disability and other deformity related to arthritis by 200 2020 more importantly though there are other components to it there are social there psychologic and their economic impacts to this and being able to address it early on and prevent these problems from occurring can make a major difference as far as both the economic but as well as the social and psychological impacts so for a second let's try and at least for the sake of argument break arthritis down into two big categories I listed them here is autoimmune based or traditional inflammatory based arthritis and at the bottom you'll see not immune based or non inflammatory arthritis and we're going to spend the first as much you're talking about the immune based arthropathy we're going to build on what we learned about last week from David Lewis and talk about how immunity and autoimmunity come together to result in a disease we're also going to speak for about 30 minutes about the non autoimmune based or the non inflammatory arthritis it's the kind of arthritis that everyone in this room has or will get before you die even the impedes oh you're welcome so we're gonna start here there are a family of inflammatory diseases that take root in the joints that have clinical manifestations that end Oregon is destruction of the joint and it can be in any age group it can be in the chronic juvenile patient that develops a Northrup ethey that results in deformity and destruction changes that we can see quite evident in the foot the toes of a child it can be in patients that suffer from other features of autoimmunity such as psoriasis and the development of psoriatic arthritis it can occur in patients that have other deviations of the immune system that develop a disorder called systemic lupus erythematosus another disease of an aberrant or an abnormal immune system that can also result in end organ damage in the joints can take place in other disorders such as systemic sclerosis also known as scleroderma and the last picture here is that of gout and for many of you men in this room this will be a problem that you encounter before the end of your lifespan as well well not a traditional autoimmune disorder it's a metabolic disorder that results in inflammation in the joint all of these have common underlying principles that the immune system has become activated in an aberrant way in an abnormal way resulted in inflammation and the destruction within the joint so let's talk for a second about inflammatory arthritis and when we talk about inflammatory arthritis the prototypic one we discuss is often rheumatoid we do that because it has a high prevalence rate it is fairly common it happens between one and two and every hundred individuals and a room this size we have somewhere between two and four of you that either have it or will have it before the end of your lifespan so let's start here with a Stanford patient this is a patient who presented to Stanford in 1974 who was her initial onset of her presentations and she was in her 30s this was an x-ray that was done by one of the rheumatologists here at the time this x-ray looks fairly normal I think to most individuals except there's some subtle features here that a Rheumatologist or an orthopedic surgeon or radiologist might notice and that the bones they look kind of washed out they shouldn't be so washed out in here we call that minor peri articular osteopenia it's a big word that just implies that the patient has lost a little bit of bone density and that's not normal for a young woman this is the same patient one year later patients being aggressively treated with standard care standard of care for the day this individual has some notable changes that have taken place in just in one year if you look carefully where the arrows are there's evidence of swelling in fact if you look here where I'm putting the laser pointer there's a big bulge around the wrist and there's a similar one on the other side obviously the physicians that we're treating the patient didn't need the x-ray to see the swelling but the x-ray provides a great deal more context to what's going on underneath the skin in fact if you hone down carefully on just a select area of joints you get a real sense for what's taking place in one year's time while you may not be familiar with what normal joints lose it usually look like but I assure you they don't have pieces missing this patient has large erosions from each of the joints and that shouldn't be there in one-year time frame this patient has lost a lot of bone and then general this bone does not come back so as you might imagine I'm now going to show you what the patient looked like six years after presentation this is a Stanford patient and in six years she's gone from a normal healthy woman in the middle of her life to a patient who's got severe destruction and deformity and unfortunately disability well how did this happen and how did the immune system orchestrate a response that resulted in destruction and deformity as devastating as this so now let's bring this back to a discussion of the immune system so I'm at immunology one Oh and we'll tie this together with last week's discussion so in general the immune system is broken down into two compartments on the left side you see that of the innate immune system cells that are responsible for the body's first line of defense relatively dumb cells the right hand side of the slide is what we think of as the adaptive immune system this is the intelligence of the immune system this allows the immune system to understand what it's interacting with it allows it to react not just at the first time it encounters a danger but also allows for immunological so here are some of the reviews from last week the adeg innate immune system is nonspecific for an antigen it doesn't really care what it's encountering it's very good at attacking and it's very good at removing pathogens like infections it initiates an inflammatory reaction like you'd see in a joint in response to a variety of problems and it releases a number of cytotoxic mediators very inflammatory mediators that cause inflammation the right-hand side of the slide or the adaptive immune system is specific for antigens it recognizes very unique things that it will encounter it possesses immunologic memory and allows us to react again when we see something later in our life it's the reason that we only get certain infections once it's the reason that we don't keep getting chickenpox it's the reason when we get vaccinated that our immune system is able to respond to that vaccine and keep immunologic memory hopefully for the rest of our life but also importantly it's responsible for self tolerance how is it that my body learns not to react against my endogenous my intent intrinsic or internal immune system or my internal antigens how is it my body doesn't react when I eat lunch I ingest the food but part of that will be the cells that line my mouth I'll swallow them when i swallow hair i swallow something else during the day how is it my immune system in my gut decides whether or not it's self or foreign well it's a process of self caller ins its ability to recognize self from foreign and not to react against self well in someone who's got an autoimmune problem the autoreactive that ability to ex to recognize self from foreign disappears and self tolerance can be broken so let me think about the adaptive the immune system probably the most important part of autoimmune diseases and why we develop arthritis and someone who's got an inflammatory condition it's because the adaptive system has gone wrong ordinarily we would like T cells and B cells to help quarterback orchestrate an immune response tell these cells what to do it's a lot like if you were running a university you'd want the university president to be able to tell all of the rest of the faculty what to do or if you're running a medical school you'd like your Dean to be able to provide good guidance to how the rest of the minions will orchestrate what takes place well the immune system is done in the same way you need to have a good set of T cells and B cells to help decide what self and what's far so here's how the system comes together we experience our life through antigens whatever you wait for dinner or for lunch gets ingested it gets broken down into its individual components the antigens and the epitopes and it gets expressed on the surface of what we would call a dendritic cell an antigen presenting cell it's a cell that engulfs this particle breaks it down into the most minut components and then spits it out on its surface for the immune system to react to we express this antigen in the context of our own genetic structure so her genetic structure is different than his genetic structure and the antigen is presented in our unique genetic groove that's why some patients will have a genetic predisposition to disease while others will not because they express these antigens in their own genetic context well this antigen whatever you ingest it whatever you encountered in the environment this morning or this evening is expressed on the surface of an antigen presenting cell and it needs to be recognized by a t-cell the t-cell that smart component of our immune system will then look at it decide is it self is it foreign should I react to it or not if the immune system decides to react to it it'll set in motion a cascade of events while interact other parts though b-cells the other smart cells of our immune system ll then interact with a lot of other cells that rapid recognize they're either not so smart said of our immune system the innate part the part that's really good at destroying things really good at reacting these like polymorphonuclear sites and a whole host of cells which I showed you on the previous slide in this case what's taking place in the synovium with a joint is an orchestrated astre sponsz such that we now result in excretion or production of a number of inflammatory particles a number of things that will result in pain swelling and inflammation but the process all starts back here and what we see is the inflammation is the end of the line it's the end organ result of an orchestrated process that started much earlier and in a much higher article procced much a higher on the hierarchy hierarchy so how does this take place in a joint someone that's got an autoimmune or inflammatory based arthropathy well we start off with a normal joint on the left side we start off with a normal lining two bones that oppose each other a lining that's constructed of synovium a thin layer of tissue around it and a layer of cartilage which lines the bone itself and provides a nice cushioning interface this is what a normal joint should look like but over time with patients that have an inflammatory arthritis the snow view becomes a lot thicker it begins to accumulate we call it hyperplastic but there are a lot of cells that begin to emigrate to this area they get recruited from all over the body we recruit a variety of cell types T cells B cells but a lot of those other cell types that are part of the innate part of the immune system and over time they continue to grow and they release a number of toxic mediators that cause the cartilage to break down and the bone to break down and over time we've got a destructive arthritis this is what a normal snow beam would look like so if I were to take any one of you bring you up to the table take a needle stick it in pull a piece of the snow via mouth I wouldn't do that that would hurt but this is what it should look like this is what dr. piezo synovium should look like today right now it's only a few late thick there's no evidence of any inflammation this is nice and healthy now how does that differ from someone who develops an inflammatory arthritis this is what it looks like so just as easy as it was for you to recognize this on the x-ray you can recognize an apology you're all pathologists you pass there are thousands of cells in this case all the little black ones represent inflammation and these large holes that you see they all represent a process we call angiogenesis the development of blood vessels the proliferation of blood vessels that will bring blood and nutrients to this area to allow this growth to continue to perpetuate most of you I suspect would prefer not to have this as the lining of your joint but instead to have if I can get there this so the question is how did this turn into this process and how do we potentially get that back well I'm going to start off by reviewing rheumatoid arthritis for you we'll talk a little bit about how common it is and then we'll talk about the process that mediates the destruction so rheumatoid arthritis is a prototypic autoimmune or inflammatory arthritis that we think about it is a systemic disorder that affects approximately 2 million Americans as I alluded to before about for people in this room it's of an autoimmune ideology and unfortunately we don't know what that inciting trigger is if we knew what incited it we knew what started it we could develop a vaccine against it and we could prevent it from happening but vaccines are next week's lecture for now since we don't know what it is there is no vaccine available or it's just random well I'd like to think it's not random we do know that there are patients that have certain genetic underpinnings genetic types that make them more likely to develop it but it just having that genetic type is not a fate of complete not a guarantee that you're going to get their arthritis we're not aware of any behavioral patterns or diet or any other environmental activities that would necessarily result in this or increase your risk the peak age of onset is really between ages 20 and 45 and that's very unfortunate that's fairly early and really in the peak productivity of our lifespans unfortunately as well this illness tends to occur more frequently in women than it doesn't men and that's true of most autoimmune diseases and most inflammatory arthritis they occur more commonly in women and unfortunately to date we still do not understand why and while there are a lot of easy theories about maybe it's just related to estrogen and progesterone and hormonal differences but that has not panned out today certainly one of the most important things to keep in mind when you have this much inflammation going on it results in inflammation elsewhere in the body and this is in fact a systemic inflammatory disease you only say the end organ results in the joints but the reality is there's inflammation elsewhere in the body and it results in a decrease in lifespan between five and ten years mostly because of cardiovascular risk high blood pressure there was a stroke heart attack these patients also and interestingly at an increased risk for the development of infections there's immune system is overactive but yet they're at increased risk for infection it's because there are components of the immune system that are overactive and inappropriately responsive or inappropriately non-responsive to a variety of components they may encounter so how does this disease develop well for most individuals it's a slow and insidious process it develops over a period of weeks or months most often in women in early to mid life they complain to pain in feet stop wearing heels they start wearing flats they complain of discomfort predominantly stiffness same is true of the men though they don't change their shoes as readily they then continued to have discomfort that results in them getting nonsteroidals or other types of analgesics over-the-counter they often will see their primary doctor who will look at them and say you know it just doesn't look bad it's probably just the fact you're getting older over a period of several more weeks or months it becomes more readily apparent that there's overt swelling and there's more than just an age-related change taking place it's at that point where they're often referred for additional care rarely someone will present with an abrupt ER and acute polyarticular arthritis meaning multiple joints it's rare that someone will come to me or another physician and say on February the 13th I woke up my joints are swollen and I got this that's very uncommon what we see is this gradual change over a period of time we see joint swelling pain and stiffness hallmark signs and symptoms of this disorder and other types of arthritis it then results in a limitation of motion and then ultimately a loss of function and muscle grip muscle weakness and disuse atrophy meaning loss of some of the muscle size and strength are common features so how does this come about and if we bring it back to the immune system how does this occur well this slide is one of my favorites it may not be one of yours but it's one of mine it's one of mine because I think it very well adequately suggests just how complicated the process is but if you look at it carefully it actually explains the way the immune system communicates when we grew up we learned what hormones were we learned how the Anderssen endocrine system communicates we either learned it in school or we learned it because we all went through puberty and we all figured out what hormones were very quickly but the immune system is never really talked about in that context there are in fact hormones of the immune system and those are the cytokines the cytokines are markers they're proteins that are excreted in various areas of our body and they send signals throughout the body either pro-inflammatory or anti-inflammatory well this slide adequately describes that with all the black arrows you see and all the different interleukins and interferons communicating between cell types that's the first way the immune system communicates the second way that the immune system communicates is through a process of what we call cognate intelligent cell-cell interaction this cognate cell signal interaction can be broken down simply into a receptor binding to a ligand receptor binding to whatever's going to stimulate that receptor and this is epitomized at the bottom of the slide where you see two cells connecting with one another this is being the receptor and the ligand binding to it so effectively the immune system has to communicate either because cells touch each other and send a signal between them or because they release cytokines throughout the body well how does this wind up resulting in an abnormal immune response and destruction in the joint well normally we're all excreting a small amount of cytokines that are released throughout the body bind to a receptor and cause a signal but we're also recognizing nowadays that we can block that there are a lot of intelligent ways that we can down regulate an immune response we can do that by trying to block the ligand getting to its receptor or in this case a cytokine arriving at the signal we can do that by pharmacologic manipulation by providing a monoclonal antibody a antibody that's produced outside of our body it's given as an injection or infusion and we'll target specifically what we want in this case the cytokine of interest or we can pharmacologically manipulate the bottled body to neutralize cytokines by creating a soluble receptor and infusing the body with extra soluble receptors so what's the most prominent cytokine in the body and the one that results in the most deviation of our immune system well it's predominantly this one here it's called tumor necrosis factor it's produced by a variety of cell types in an inducible way so what does that mean inducible so we're excreting a small amount of tea and half all the time in our bodies cells do that in a very constitutive way a small amount periodically but under the appropriate situation the right stimulation a variety of cell types are induced to excrete high concentrations of this and when the immune system becomes activated it'll stimulate cells to release high concentrations of tumor necrosis factor and that has a variety of effects elsewhere in the body that further stimulates a positive feedback loop that results in additional cytokines being excreted it results in increased inflammation it changes the endothelium the lining of this of the blood vessels that's why we see change and increased cardiovascular risk factors it changes what our liver produces it lowers our cholesterol level and it raises a lot of other inflammatory products from our liver that results in a variety of other changes within our bloodstream markers of inflammation go up importantly high concentrations of this and other cytokines stimulate the lining of the joint and when that happens a number of very unpleasant enzymes are released enzymes or proteins that are good at breaking down things detergent is nothing but enzymes and soap that combine together to break down the dirt and other things that on your clothing enzymes are very effective at stimulating and breaking down things well in this situation high concentrations of cytokines stimulate the lining of the joint to release a number of enzymes that result in break down a bone and break down a cartilage and further it stimulates certain unique types of cells within our bone to start breaking things down so how does this play out we've got a patient that presented in 1974 that looked healthy but by 1980 had very little left as far as her joints in the small joints and the bones here of her hands well let's think for a second about the way the bones work so this is a process we call hasta clap osteoclast oogenesis it's occurring in each and one of each and every one of us all day long it's a process of bone remodeling we start with normal bone over time our bones will recruit what we call osteoclast cells that break down bone the bone is then resorbed by these osteoclasts and other cell types called osteoblasts are induced the osteoblasts in many ways are the good cells they come back they take these holes that it tunnels that have been dug by the osteo class and the osteoblasts come back and they rebuild bone that deposits mineralized PHA features over time this mineral eise's we've got normal healthy bone but in patients that have too many cytokines floating around the process gets skewed it gets skewed so that we start producing too many of these cells the osteo class and we down regulate and don't have enough osteoblasts to build it back and the areas that have the highest concentrations of the cytokines get the highest concentrations of the osteo class so our bones become deviated in unusual ways that's how we wound up with such destruction in the previous slide so I'm going to appeal to a variety of audiences here so for those of you who want to understand well how did the cytokines really do that well let's think about the process inside the cell so high concentration of a cytokines such as tumor necrosis factor or interleukin 1 will stimulate its receptor this is the ligand this is the receptor and it'll stimulate a process to occur with inside the cell it'll drive another cell type in this case bone and macrophage precursor cells will be stimulated to differentiate these are the cells in our bone marrow it'll drive these cells to repopulate in an abnormal way it'll turn these quiet increasingly populating cells into very activating destructing cells so just the mere higher concentrations will trigger a process inside cells stimulate and drive a process in our bone marrow so that quiet cells become differentiated and then ultimately activated and that results in destruction it's a very orchestrated response where the cells come together they stimulate each other through cytokines the cytokines go on to drive other new cells to be brought about they're recruited to the area that you don't want them the joints in this case and they caused the bone to break down how about the cartilage how does the cartilage wind up breaking down because those cells were just in the bone well through a similar fashion a variety of cell types these being the B cells this being a family of T cells all releasing cytokines all affecting the lining of the joint cause a release of enzymes that then trigger these chondrocytes the cells inside the cartilage again to break down a similar process orchestrated response using the adaptive immune system the T cells and the B cells the innate immune system macro shades macrophages and monocytes and then orchestrated by cytokines and these cells touching one another all resulting in the joint to become deformed in a sometimes long process sometimes in the devastatingly short process we also recognize now that we've become more sophisticated at potentially altering the way the immune system works this is true whether we're talking about inflammatory bowel disease whether we're talking about psoriasis whether we're talking about rheumatoid arthritis or a host of other immune diseases that all represent a similar aberrant or abnormal immune process we can potentially block cytokines like tumor necrosis factor by delivering a number of different drugs these are all commercially available drugs their names aren't really that important but the fact is that we can through technology specifically by - and block some of these in this case cytokines that are existing at too high a concentration in our body this is a patient who presented at Stanford in 1997 this is a young woman she's in her 30s and she's had three months of symptoms of rheumatoid arthritis again I'm gonna ask you to take a look at this x-ray this time this is a foot this is a right foot this is the first toe second third fourth you get the gist is there anything wrong with this x-ray right again you don't have to be an orthopedic surgeon or a radiologist to see it but there's one awfully big hole here right this is a very bad sign someone with early disease who presents with an x-ray like this is going to have a bad outcome certainly you learned that from the first set of slides I showed in 1974 well the question is what happens to the osteoblasts isn't anybody stimulating Jim to come back in the equation well the reality is there isn't a good anabolic way of stimulating the osteoblasts yet there is an osteoporosis you can give drugs like parathyroid hormone and treat some patients with osteoporosis with that but in patients that have arthritis we haven't be able to come back in a good way to bring the osteoblasts up to a higher level instead we have to down regulate the cytokines which stimulated the osteo class to begin with so starting more hierarchically we can prevent this process from occurring and I'm going to show you just that so one would have predicted that this young woman was going to have a very bad outcome but she started on a new therapy in 1997 as part of a clinical trial here at Stanford and this is her four years later so not only did she not worsen but if we block the right cytokine in fact you can prevent further destruction and she even a gruesome bone so it suggests to you that you don't need to upregulate the osteoblasts to get things to stop you have to remove the perturbations the abnormal driving factors that cause the destruction to begin with now I have to be honest here I've cherry picked some slides I showed you the worst case from 1974 and I showed you my best case from 1997 this is much better so this is what she looked like before treatment and this is what she looked like four years later that there has been no worsening and that makes me very happy she didn't get worse in a patient that should have and not only that she got better well I'm gonna leave the treatment out because there are nine different drugs that are on the market now and I we as a profession can get good results with any of the nine so I don't want to appear for lots of reasons to be endorsing any given therapy fair enough all right so let's come back to the cascade of events we're gonna take dr. David Lewis's a date and adaptive immune system put it together and talk about the various points that we can interfere in the immune system but potentially result in improvement ideally you'd like to know what this little orange dot is because that's the antigen that's the offending culprit that's being presented if we knew that we'd have a vaccine in lieu of that we need to stop this presentation right here stop that interchange between the antigen presenting cell and the t-cell and we can do that now we'd like to down regulate the B cells the other folks that are inappropriately quarterbacking us we can do that now or we'd like to interfere with a host of other inflammatory products at the end of the line and I already showed you that there were three commercial drugs that block TNF these three in fact there are five now but the slide gets really complicated if you put five on there so let's talk about the t-cells themselves I alluded to the fact that we can interrupt this process right up here now and potentially get benefit for doing that well it's not quite that simple because in fact the t-cells as David Lewis may have alluded to in fact have multiple different lineages they can be developed and go down different paths if you have too much of type of t-cell you get cell-mediated immunity you get autoimmune diseases if you've got too much of this type of t-cell with the right stimulation you get too much humoral immunity and you get too many allergies and you get asthma and you get HIV if you're a child if you got just the right number of regulatory or suppressor cells you're free and clear you don't get anything and nowadays we're recognizing there's a fourth family another set of T cells called th17 cells that can really result in autoimmunity predominantly inflammatory bowel and psoriasis so there in fact there are a variety of these different T cells that can become abnormal under the right perturbations and driving in our abnormal immune response in fact I simplified this equation a little bit too much I suggested that all the things that have to do is the T cell just has to bind to an antigen presenting cell that's true but how does the T cell decide what to do well a T cell can choose to ignore an antigen a T cell can activate and recognize managing or T cell can undergo what we call apoptosis or programmed cell death and just die well if it's reacting against my own protein I'd like it to just die and go away because I don't ever want that to happen again if it's encountering a foreign antigen and infection I want it to react and in certain situations I'd like to choose it to ignore so how does the body decide well there's a whole bunch of others other what we call co-stimulatory interactions that take place all these little ligands and receptors these are all the things that are taking place on the surface right here I only have two lines there but the reality is it should be about 50 and all of these different interactions can either have a positive effect on the T cell or a negative effect on the T cell and if it decides to act or not so the more we learn the more we realize more we don't know and the more opportunities we have in a very sophisticated way to begin manipulating the immune system to our own advantage here's one way that we're doing it now this is the first signal an antigen-presenting signal this is often where the equation goes wrong and antigens presented in the context of our genetic structure and it provides a signal to the T cell the T cell needs to decide what to do and it meets up with another interaction a secondary signal well this secondary signal then causes the T cell to become activated I've suggested we can interrupt this now and we can we can do that by giving the body a new protein and blocking this interaction between some of the coasting military receptors and ligands so we can interrupt this process and deactivate T cells this faire therapy is effective in autoimmune disease it's effective in psoriasis it's effective in transplantation you can down regulate too many active T cells we have another way of doing things now we can block these b-cells these b-cells that I alluded to earlier on were very important we can remove them for the equation in very selective ways through immune therapy so again we're becoming more sophisticated removing the cells that are quarterbacking these events but I also want to leave you with the idea that nothing comes without a cost not just a monetary cost but significant costs because as we begin to perturb eight excuse me try to intervene in the way the immune system is functioning we recognize that there are unintended consequences so even as selectively as we believe we are when we interfere with the immune system in an attempt to down regulate an overactive system we recognize that sometimes we put patients at risk for infections we also recognize that we sometimes put them at risk for opportunistic infections unusual things like TB but probably more important to the group is the fact that sometimes we'll put them at risk for malignancies because our immune system is in fact at some level responsible for our own host defense responsible for tumor regulation and will begin to play with the immune system in an attempt to down regulate one compartment sometimes the unintended consequence is that we lose termer surveillance or at least a part of tumor surveillance so sometimes we can see increased malignancies and sometimes when we skew with the immune system we skew it one direction or than or another we can switch from one autoimmune disease to another so while I've alluded to the fact that we are really quite sophisticated in what we're doing I'm going to take a step back and say we're really not as sophisticated as we'd like to be we have a good conception as to what the bad actors are but sometimes modulating that just right in an individual patient can prove difficult and I would suggest to you that I've really only touched the tip of the iceberg as far all of the potential ways and therapeutic strategies that can be used to try and make this complicated immune response improve and this is just a short list of the different types of therapies that are under investigation now from the bone from the cytokine and from the cell level it's a good question what about exercise and its role in either exacerbating or improving arthritis well there are two fundamental things to keep in mind there in general we encourage exercise because it helps keep the bones the cartilage and the surrounding soft tissue structure strong the stronger the muscles are the less pain you'll have and the better function you have but you have to weigh that versus the derangement or the abnormality within the joint if your joints is already badly deformed you've got degenerative arthritis in there too much activity across that joint can cause further damage so in part we like to encourage exercise but we encourage exercise in a type of activity that would be best for the joint involved if it's your knee running is probably not a good strategy but swimming and cycling might be an excellent option if it's your hand bowling is probably not a good thing to be doing with it but there all kinds of other exercises you can with your hands so it has to be tailored to the type of arthritis you have and the joints that are involved but exercise in general is a good thing in moderation yes yeah I wish there was so the question is is there any type of score that we can use to assess how normal or how aberrant the immune system is well in fact we're trying to work on that now we're collecting specimens from normal individuals and specimens for patients that have autoimmune diseases and we're trying to give some very unique assays into what's called the human immune monitoring core something that dr. piso has been funding to help us understand what is a normal immune response and what is a abnormal or deviated immune response and can we predict how bad it is in general we haven't become that sophisticated yet so I'm gonna shift gears now and talk a little bit more about non inflammatory arthritis why my joints hurt degenerative change and what happens as we get older this is the type of arthritis that everyone encounters it's type of osteo origin generative arthritis it is more common than are the immune based arthropathy is that I just discussed but before I get too far into it I recognize that anytime I treat medical students or teach medical students in the course I have to provide them some guidance about self diagnosis so what I'm most worried about is that when you leave here at 8:20 that there will be an untold number of you that are going to say oh my god I have one of those conditions so let's take a few minutes and just talk about how we ascertain what type of arthritis we have and if you're uncertain take these down and present them to your doctor and ask them to go through them with you in general what we try and teach our medical students here is that it's fairly straightforward to differentiate what type of arthritis individual has if you can use these basic five tenets and the first one is are you stiff so raise your hand if you're stiff in the morning oh you're lying come on put your hands up come on we're all stiff in the morning well maybe not your daughter Mike but everybody else is so I put stiffness up there but stiffness is a unique thing if you're stiff for more than thirty minutes I start to worry prolong stiffness and stiffness that comes back every time you're sedentary after a car ride after you've been sitting in a lecture hall for prolonged periods of time if that stiffness comes back and it comes back and it doesn't go away right away stiffness is important component stiffness for more than 30 minutes suggests you may have an inflammatory arthritis stiffness less than 30 minutes don't worry about it what about joints that are involved most of the time individual types of arthritis will take on a characteristic distribution for patients that have rheumatoid arthritis they'll often develop a certain type set of joints that are involved that big set of knuckles on your hands but never the little set of knuckles on your hands while osteoarthritis it always gets you in the little knuckles and in the base your thumb but rheumatoid arthritis never does so they're characteristic joints that are involved if you've got an immune or inflammatory based on throb a--they you'll often have involvement that's symmetrical you'll have it on both sides of your body and often above and below the midline because it's a systemic disease it doesn't stop just based on ill-defined boundaries while someone who has osteoarthritis will often have it in a single joint here or there in an asymmetric fashion could be lower extremity could be upper extremity so based on the pattern may make a difference how about other issues though do your symptoms worsen or do they improve with activity this is a hallmark question so when you injure your ankle your knee or you throw out your back does more activity make it better or worse worse right is anybody that feels better when they throw out their back and they do exercise then I had nobody to worry about here if you feel better when you do exercise I want to see you if you don't feel better if exercise makes your arthritis feel worse you're normal there are other components of the immune system that helped us identify whether you've got an inflammatory arthritis or not we can look at markers in the blood so there's certain things that we can look out in the blood that helped us assess the presence or absence of information and when in doubt put a needle in it and take some fluid out of the joint I know that sounds arcane it sounds really devastating but it's remarkably easy it hurts less than having your blood drawn really it does it's relatively simple and the number of cells that are present in the joint will tell you whether it's inflammatory or not so the last issue though to try and distinguish what types of an arthritis somebody has are there radiographs or the x-rays x-rays don't lie if you've had it or thright us for a while they are hallmark features they're very different in rheumatoid they're very different in lupus they're very different Channel and osteoarthritis they're very different in gout they're very different in psoriasis you throw up an x-ray if somebody's had the disease for a while you can figure out what type of arthritis they have often x-rays are not necessary and I'm not encouraging anyone to run out and get x-rays often the distinguishing factors can be made really on the first three bullet points right in the office so I doubt that any of you have any significant inflammatory arthritis but if you had more than three things that were positive here see me after the lecture so now let's start talking about osteoarthritis this is what we started thinking about as far as bone change in the hands osteo arthritis can be common in a lot of joints but hands are typical particularly the women particularly women in this room if you look down at your own hands and the men are looking to the reality is it is a very common it's often an age associated issue and women are more likely to inherit this from their mom than are the men it's not entirely clear why but let's talk about arthritis in itself osteoarthritis was previously thought to just be a normal consequence of Aging and in part it is but more than that though it really is a complex interplay of multiple factors why is it that some people get it and some people don't well it has to do with our joint integrity how lacks our joints are it has to do with our genetic predispositions what type of anabolic repair Ettore processes do we have and what type of catabolic breakdown processes did we inherit from our parents what about local inflammation that plays a role and mechanical forces maybe I didn't inherit a great set of genes from mom and dad but I decided not to be a manual laborer that may have preserved my joints for longer if I inherited a bad set of genes and I've been a manual laborer and I'm obese I may have a far worse outcome than I did if I had a different set of skills ultimately though there are cellular and biochemical processes that interplay here and can be difficult to predict that all interplay again into the reason why some of us will develop it earlier and some of us later it should also suggest to you that some of these factors may be modifiable through our own lifestyle but also in the future through chemical or pharmaceutical interventions so keep in mind that unlike inflammatory arthritis osteoarthritis are non inflammatory arthritis is not a systemic illness it is not primarily an inflammatory disorder and in fact it's more of a process than it is a disease it's a process of interplaying events that result in degradation within the joint and first and foremost it is a disparity between our ability of our joint to withstand stress and the body's balance between destruction and repair yes so our bunions part of osteoarthritis they most definitely are its osteoarthritis of what we call the first metatarsal phalangeal joint of the foot it causes the bone will break down that joint breaks down and it will turn to the side you get a nice big bulge out the side so why is it that women have it more than men well men are more likely assuming women are more likely to develop osteoarthritis to begin with unfortunately but also men tend to be not all of us but men tend to be less fashion conscious necessarily than women and men tend to stick their feet into shoes that may not necessarily conform as well as women we tend not to wear heels it we tend to apply unusual mechanical stresses across our feet in the female population that most men don't encounter and that causes deviation in those first metatarsal phalangeal joints so that's an interplay between genetic predispositions mechanical forces and the local cellular biochemical processes that cause you to get that bunion had you never worn the heels then the local mechanical forces played less of a role indoors that's right when in doubt so let's talk for a second about how we divide up osteoarthritis primary and secondary in primary there's no obvious or prior injury like our colleague here in the second row there's no prior injury she doesn't remember wearing shoes this is wearing unusual shoes there's no obvious prior injury but yet there's secondary types of osteoarthritis it's those individuals that have a previous injury or abnormality or due to a joint the individual that twisted the knee blew out their ACL twisted their ankle threw out their back but yet 5 10 20 years later they now have arthritis in that area that's a secondary phenomenon related to the trauma and the unusual mechanical stress that joint bore so let's think of it a little bit as we divide osteoarthritis into some additional categories this would be primary osteoarthritis and it can be localized and the most common areas it's localized to are the hands the feet the hip the spine and less commonly some of the other joints that I won't read to you and in some situations in unfortunate individuals you can develop a generalized osteoarthritis where you get three or more sites involved for the very active individual for those individuals that had lots of trauma when they were earlier or younger and for those individuals that may didn't get the right set of genes for mom and dad generalized osteoarthritis can occur secondary osteoarthritis again almost always related to trauma sometimes related to developmental or congenital disorders can also occur secondary to an unique phenomenon going on in the body so if you have a crystalline disorder like gal or pseudogout here it causes an injury to the joint and once the joints injured it's much like getting trauma it start to unravel just the way rheumatoid arthritis starts off as an inflammatory condition can injure the cartilage in the bone and even if you turn it off sometimes the joint will continue to erode because it's much like a traumatic event it causes things to do right so you ever gone to the store maybe you saw a ball of yarn and you buy the ball of yarn at the store it's nice and tight the end is tied up not a problem what happens when you snip the end of the ball of yarn well you can never really stop it from unraveling it's like a ball of twine or anything else it starts to unravel and it continues to unravel well osteoarthritis can be that same way once the integrity of the joint starts to erode once it starts to unwind it becomes become difficult to stop it so let's talk a little bit about osteoarthritis it is a universal disorder it is in fact the most common of all joint diseases and the most frequent cause of physical impairment age is the number one factor associated with its development in fact when you take a look at previous studies that have been done recent studies have suggested about less than 1% of individuals between 25 and 34 have active evidence of osteoarthritis while it's over 80% people over the age of 55 previous studies in 1958 and again in 1973 would suggest that just under 10% of individuals between ages 15 and 24 had osteoarthritis already and look at the rate between those 55 and 64 83 to 87 percent in 1973 97% of individuals over the age of 65 had osteoarthritis it is ubiquitous it is a universal disorder that we all will have to come to terms with so what are the risk factors that predispose to this well age is the most common but it's certainly not the only one what is a congressman Lana how does it relate so the question was what is chondromalacia and how does it relate to osteoarthritis condor malaysian is softening a break down of the cartilage how does it relate to osteoarthritis well sometimes chondromalacia can just be a break down of the cartilage it doesn't result in true osteoarthritis but often it will over time I'm going to show you a few slides in a few minutes that'll show you what takes place within the cartilage how it will soften in crack and then how osteoarthritis develops that be okay in the back hypothyroidism and how does it relate well as we change starrwood conditions we change what's taking place in the way the immune system develops we also affect what takes place in bone and you can also affect what takes place in cartilage there are some patients who become hypothyroid who will develop an active arthritis there are some patients that are hyper thyroid hyperthyroid to develop some features of the active arthritis as well the body is a complex organism and as we perturb one set in this case the thyroid we can also result in inflammatory conditions in the joint the diabetes and also a charcot-marie-tooth the question is how does that cause the osteoarthritis or is it the well let me specifically and try and address the question I didn't cover really the bottom part of the slide but I will now how does how does a disorder like diabetes or a neuropathic such as charcot-marie-tooth arthropathy develop is it an autoimmune condition well the reality is that diabetes over time causes problems with nerve endings we lose a sense of sensation that can occur in the toes and in the hands and sometimes the larger joints similarly if you have a Charcot joint you have lost nerve and Inter intervention innovation in that area you can't feel what's going on same thing happens in frostbite so if I can't feel what's going on the joint loses its proprioception it doesn't feel pain the same way I'm more apt to do things that will cause perturbation and destruction in the joint so the phenomenon is very similar between diabetes charcot-marie-tooth tooth or at neuropathy and frostbite where the nerves become damaged and the activities I do become unusual I continue to damage the joint because I can't feel the problem that I'm causing so why is it that some patients will have a localized osteoarthritis and others will have more of a generalized well we understand it in a general sense but we can't break it down to an individual so we have a generalized sense but not an individualized sense the problem is that every one of us is so unique that we do activities 40 years ago that may not have been the ideal one for us you jumped out of a tree that you shouldn't have you did activities that don't come home to roost for 20 years and our memories are imprecise as well other activities that go on throughout our lifespan can certainly affect what would be the normal equilibrium within those joints so why certain people go one direction others and other is difficult to know okay so what happens with cracking your knuckles or cracking your neck or your back I need a show of hands how many think that that's bad how many think that it's okay how many think that it's good it turns out not to make a difference at all there's no correlation between cracking your neck cracking your knuckles cracking your back and developing any osteoarthritis so the take-home lesson tonight no problem not a not a risk factor at all yes okay so the question was about chilblains and Ray nodes so let's start first with ray nodes ray nodes is a spasm of some of the capillaries and small blood vessels will often result in a change in the color and the temperature of our hands you might see your hands go from normal color to purple to red to white and then back again that's characteristic of many of us it's a vasospasm that occurs often accompanied by cold and many of us will encounter that in our lifetime but it has no direct correlation or prognosis for arthritis at all and chilblains is a variant of that chillblains is is often a thought of is a is a vasoactive process that it can occur in a variety of areas often around the nail beds and our fingers our toes sometimes our legs sometimes also related to unusual neuro sympathetic changes that go on in our body the nervous system and our vascular system interacting the result in unusual color changes temperature and even sometimes skin lesions yes so if you have a vascular DIC neuropathy an inflammatory condition that affects the blood vessels that then results in some injury to the nerve endings does that put you at a greater risk of developing another autoimmune process in general no it shouldn't put you at a significantly greater risk developing any other type of arthritis or autoimmune disease but the question would always come back to why did you develop the vascular DIC process to begin with that can happen because of infection it can happen from medications it can happen from our environment and sometimes it can happen from an autoimmune cause so risks idiopathic should not put you or your children necessarily at any greater risk for problem yes yeah I'm gonna cover that in a little bit so the question was is there any evidence that supplementing our diet with nutriceuticals other types of supplements will result in an improvement or a change in the in our arthritis the short answer that is no are there any good medications to treat osteoarthritis yes I'm gonna and no I'm gonna I'm gonna cover treatment in a few slides I'll be at to brief but we'll cover that in a little bit so if you treat a thyroid abnormality does it still a risk factor for developing osteoarthritis and the answer there is it would not be if the thyroid condition is appropriately treated it puts the body back in equilibrium okay so risk factors again women greater risk factor than being male obesity every pound we gain above our ideal body weight puts us at a greater risk for osteoarthritis predominantly in the weight-bearing joints the hips the knees the ankles the toes if we starve our so even better yes the less the impact across your joints the less mechanical stress you provide but there's a breakpoint there right if you don't provide the normal metabolic building blocks your joints as well as the rest of your body are not going to perform correctly so we do not recommend starvation diets or unusual fad diets no cure of grapefruit only diets I mean we like folks to eat a well-rounded diet with a reasonable calorie intake sports activities many are familiar with that occupations manual labourers are more likely develop osteoarthritis than non manual labourers and this means doesn't mean just the person that may be agriculture related but the person who's in construction the person that does a variety of lifting activities during the day may find themselves in an increased risk for development of osteoarthritis one of the biggest correlations are between contractors and the development of osteoarthritis of the hip yes that is not impact swimming okay so let's talk about exercise for a second because this may be sort of take-home lesson number two tonight exercise is a good thing if you are a runner and you're not having pain in your joints and you have a good biomechanical alignment running is a good thing there have been study here it was done at Stanford it's followed patients for 18 years looking at marathon runners high-intensity runners versus non runners they're not at any greater risk for developing a joint injury or arthritis over an 18 year time frame in fact they have less morbidity and less mortality that's if your joints are normal they're not deranged they're not hurting when you run on them and you've got a nice normal alignment if your joint hurts and you run anyway you've got an abnormality in your joint and you run on it anyway do weight bearing activity across it you are potentially risking worsening that joint and hastening the arthritis overtime so the most important lesson is if it hurts don't do it so when it comes to arthritis no pain no gain is absolutely wrong okay if your joint hurts don't do activities that make it worse if you're in the gym and you're weightlifting and your muscles start to hurt that's okay I don't worry so much about muscle pain in that case no pain no gain if you want a big bill bulk and grow strong and increase your resistance that's fine for muscles but you have pain in a joint find another activity so other activities we've already mentioned proprioceptive defects diabetes and a few other conditions acromegaly that's an unusual condition I'm not going to touch and crystal disorders like gout or pseudogout as these crystals deposit in your joints they cause inflammation and that causes destruction your joint yes okay so there's an inverse correlation there more the harder and stiffer your bones are the better bone density you have the more likely it is you are to get it osteoarthritis I will I'm going to say the inverse is the lower the bone mineral density you have the thinner your bones are and the more osteoporosis you have the less likely you are to develop arthritis yes you're right we still have osteoporosis so you're sort of damned if you do you're damned if you don't right a little bit of exercise goes a long way it helps restore more normal bone density normal bone density is a good thing in general but keep in mind for those folks that have really good bone density you're probably more likely to develop osteoarthritis sure why I don't sure that we really know other than that we recognize that a harder and stronger the bone is around the joint sometimes those changes may result in a greater likelihood injure the cartilage at least it's a potential mechanism is that right I don't know there was another hand up there's several hands up well more women get osteoporosis yes more women get osteoarthritis yes the question you asked is well that doesn't seem to add up correctly it does because women get more of both men don't get as much of either so yes there's a larger proportion of women on the auste product side but not everyone that's a woman in this room gets osteoporosis but everyone in the room as a woman gets a hard to deal gets osteoarthritis right even though not everyone gets osteoporosis the reality is that on women our women are unfairly targeted here yes does the avoid joint pain rule apply what you're doing rehab and the answer there in general is yes if your joints are hurting while you're doing rehab your therapist or your physiatrist your physical medicine doctor and whoever is treating you needs to help redesign the exercise regimen so that it's a more pain-free range of motion it may not be possible to completely avoid or abrogate any pain but in general therapy should be designed to avoid pain when possible yes why are women more subject to arthritis than males you don't know yes and viral infection caused arthritis there are a number of viruses that can in fact trigger and cause arthritis we know that exposure to hepatitis B and C can trigger an immune reaction that results in an inflammatory arthritis we also recognize that there are unusual viral protons such as v s-- disease or parvovirus b19 that can cause a fire alarm itis v disease is named because it is v viral exam from of childhood it causes a slapped cheek appearance and if you're unfortunate enough not to get as a child and you may get it as an adult it may present with arthritis which is often self-limited over a few months there are unusual illnesses virus's chikungunya can cause it I can go through a list of about 20 other viruses that can cause arthritis usually self-limited but we recognize that there's probably far more out there than we understand as the immune system interacts with a foreign pathogen can reacts against it and then stimulates the immune system to act in unusual ways yes so what about osteopenia osteopenia is mild loss of bone density over time it's differentiated somewhat arbitrarily from osteoporosis which is a diagnosis when your bone density drops below a certain threshold and we go from being osteopenic or add bone loss to being in a very low level does osteopenia in of itself change our risk factors for development of osteoarthritis it's possible but I can't we don't often look at it that way we we look at it at women that across the third certain threshold and the likelihood of them developing osteoarthritis so I don't know but that's been well-defined okay so they come together as systemic factors genetic factors bone density nutritional factors they all relate to our underlying susceptibility to osteoarthritis and then playing into that or the local biomechanical factors of joint injury obesity are joint deformity muscle weakness and that real it results in specific site and severity of our osteoarthritis so all of these come together to create our individual and unique risk factors for developing this this does that just mean that the joins get stressed because the muscles are so the question was did he miss muscle weakness or did I just avoid discussing it I didn't really cover it well individuals that are stronger will tend to have less joint laxity will tend to incur less injury or damage to a joint and more importantly will function better and have less pain over time those individuals that have more muscle weakness are more likely to develop laxity and injury to a joint so maintaining good strength is important at least as they local or biomechanical factor towards the prevention and the progression of osteoarthritis so this is an x-ray of a normal hand this is what it should look like with nice normal spaces between the small joints the distal what we call the proximal and the metacarpal joints this is a nice normal hand but in someone who develops osteoarthritis the changes can be quite visible I alluded to the fact that our thright ax steaks on characteristic patterns well in this situation you see all of these bone spurs we call these osteophytes osteoarthritis is a prolific bone response the bone responds by trying to grow it's our body's attempt to immobilize the joint the more the bone grows the less well we can move it and it tries to prevent us from incurring anymore injury to the joint it's our body's normal reparative attempt to stop injury you see that here and all the way across many of these small joints but yet this may not be readily apparent to you but these joints are still nice and normal because they rarely have ever involved in osteoarthritis why not clear but nonetheless osteoarthritis is a proliferative bone response and that's different than rheumatoid arthritis which is an erosive process where the Bona roads and disappears here it grows and it grows in spades here's another example of osteoarthritis this time be the knees this is the middle compartment or the medial compartment of the knee and you can see it's completely worn out its bone-on-bone and you see this increased white area around it you see how it's so white here and here but it's nice and medium textured there this is increased bone hardening or subchondral sclerosis that's occurring around the joint this is the body's attempt to harden the bone an attempt to prevent more destruction from occurring so invariably the process of osteoarthritis is not just the break down of the cartilage which is now completely missing but is a change in the underlying bone as well so let's talk about how this happens so forgive me these are somewhat older cartoons but the graphics aren't quite as good as what we have in some of the other diseases we see the first stage of osteoarthritis is the death of some of the cartilage cells we call them chondrocytes this is the layer of cartilage it overlies the bone as some of these cells die the cartilage doesn't continue to replace itself it doesn't continue to grow to maintain a normal homeostasis as it should underneath that we see normal bone levels and believe that is the what we call the subchondral bone plate and the bone marrow underneath that this is normal bone and this is the inside of the bone so what happens in the second stage of osteoarthritis we've already lost a few chondrocytes the cartilage starts to crack you get chondromalacia the bone gets softer and it starts to crack with those cracks you develop an inlet for fluid cells other inflammatory meteor's blood to seek in and creeks through this it's a lot like if you had a basement in your house I grew up with a basement many of you don't have a basement water seeps in sometimes you get little cracks and you get water in your basement well that's not a big problem if you don't if you don't do anything about it but eventually you're gonna get a whole bath your own basement full of water well in this case you get slight little cracks occurring in the cartilage and over time that damage starts to grow most patients won't feel this problem until it causes the damage to the bone and that's the third stage of osteoarthritis when it cracks through this level what then happens is we get angiogenesis or blood vessels creeping in in the body's attempt to repair this the body wants to repair this the blood vessels grow inflammatory mediators are released and we start to get neovascularization what does this look like in a real person rather than a cartoon and this is an arthroscopy this is the two layers of cartilage you can see the great big crack that goes down and into the bone this is the probe the cameras obviously vacuum and you can see how afraid it looks it's an unraveling piece of yarn same on here and this is the damage to the cartilage is missing and this is the damage to the bone and this is ultimately what it looks like on pathology this is very deformed bone and what's left of the cartilage and lots of fibrin and this plug that you see right here the fibrocartilage plug this isn't it right here the body's attempt to create a scar a scab that to bridge the cartilage unfortunately it doesn't work very well and this crack will continue to widen over time now I told you that osteoarthritis was not an inflammatory condition well I kind of lied it is but it's a very low level inflammation it's a variety of risk factors that come together to cause this crack and the hardening in the bone and then the immune system and inflammation get in there that's where things really go awry you can see that there's changes that are taking place from the synovium these are these awful T&F things again the joint space itself and some more cytokines and then there's the cartilage and a number of enzymes that all result in destruction they all come together in a very low very concentrated rate in just the area of injury so osteoarthritis is not a systemic inflammatory illness is a localized interplay of a number factors that then gets finished off when the immune system and degradation get in there and they break it down so how do we treat osteoarthritis well a first thing is thinking about education and lifestyle modification knowing that we're all going to get it should provide reason enough for us to potentially alter our lifestyle for some of you it may mean losing weight for others it may be exercising for some of you it may be stopping running for others we have to modify our activities to our own strengths right you have to play to your own strengths other important things are physical and occupational therapy so if you're weak in certain muscle groups strengthening them having occupational therapy and ergonomic environments that help us put less strain on the joints is important the reality is that these can only go so far and the inevitable reality of osteoarthritis will creep into our lives at some point in which case pharmacologic therapy has to play some role whether you like it or not so for many of us it's acetaminophen up to 4 milligrams a day for some of you may call it paracetamol it is a fairly safe medication when taken over a lifetime and a reasonable dose certainly not to exceed three to four grams depending on a given individual many of you have taken tylenol in the past show of hands have you ever taken Tylenol ok everybody's awake is it has anyone ever felt better in their joints because they took Tylenol oh that's like one hand why is that we always the doctor says did you take tylenol and you go it doesn't work well many of us don't take tylenol correctly it can't be taken as a here and there if you take tylenol to help with your joint pain you have to take it religiously you have to take it three or four times a day every day if you want any symptomatology relief so many of us can't live by that type of algorithm so for many of us we try to hone all it just doesn't help us and we divert instead to what we think of as non-steroidal anti-inflammatory drugs nonsteroidals IV proof IV profile Naprosyn Aleve celebrex Vioxx we start thinking about medications that have proven effective at treating our inflammation and joint pain but the reality is they often come with some side effects they can increase our risk of developing ulcers in our stomach so when we use drugs like nonsteroidals we should protect our stomachs they cause high blood pressure you need to be aware of that it can alter our kidney function you need to be aware of that as well yes I would say for symptomatic relief most individuals would suggest nonsteroidals work better I'm suggesting to you that's probably a good place to start is with Tylenol because it is invariably a safer medication if it's helping with your symptoms well I didn't say take a lot three or four times a day with a dose not to exceed four grams a day and if you have any concurrent intake of other liver hepatic toxins like alcohol or other medications you may need to take less than three grams per day so intra-articular steroid injections what about dr. piezo he comes in two sees me he says gosh mark that I'm getting arthritis in my knee I've done my physical and occupational therapy I've taken my Tylenol I've tried nonsteroidals but I still have some inflammation in my joint can you put something in there that'll help well steroids will often be very effective and temporarily reducing inflammation in the joint it can be very effective of that sometimes for a few weeks sometimes for a few months keep in mind that we don't do this religiously or every week or even every month because steroids have a catabolic effect they can break down some of the connective tissue so intermittent interarticularis steroids are fine but repetitive steroid injections too frequently or right after an injury or a bad thing there are other medications that can be used such as inter articular hyaluronic acid our joints are composed of fluid one of the fluids that's in our joint is called hyaluronic acid it is a very thick material that in some regards is like motor oil if you put it between your fingers it's really slippery if you try and pull it apart though it's really sticky like molasses it takes on characteristics that we think are viscoelastic it's the ideal lubricant within a joint well for some individuals we can give a shot of additional hyaluronic acid into the joint sometimes it makes patients feel better it's a temporizing fix though it doesn't change the underlying process there are also alternative approaches such as nutraceuticals and I will mention the most extraordinary advance in the treatment of osteoarthritis in the last hundred years has been the development of joint replacement surgery that can restore a deform joint into a normal joint our goal in the next century is to prevent joint replacement surgery from being necessary so the question is how do we do that and I would suggest today we've been fairly ineffective at changing the rate or progression of osteoarthritis in fact we have no medications that will change the direction or trajectory of the disease it's unlike inflammatory arthritis where I showed you a patient who had bad arthritis and we gave a medication prevented it from getting worse I cannot do that with osteoarthritis and the tylenol and the nonsteroidals and the steroid injections and the hyaluronic acid do not change the progression of trajectory of this disease have anyone tried anti-angiogenic drugs well in general anti-angiogenic drugs probably aren't thought to be going to be very effective in this disease because the predominant changes within the cartilage and the bone itself and the Neo angiogenesis are neovascularization are a late manifestation but there are medications that are being studied now to help regrow cartilage we and others are experimenting with both systemic and intra-articular medications that may be beneficial to try and repair or certainly prevent further destruction so let's talk for a second about alternative approaches because I know it's on everyone's mind so what do we think about as far as alternative approaches we think of things like joint juice high potency extra strength high strength they come with all kinds of fancy names and it's even available for our animals animal care so what about glucosamine and chondroitin right I mean why not it probably works right probably safe there's a whole aisle dedicated to it in Costco right never been in Costco and somebody's got their cart and I'm talking a big cart full with joint use I mean I just think about it was like how did I not come up with that idea I mean so let's talk about studies I haven't shown you frankly I've been here for an hour and a half and haven't shown you need data is that means anybody I talk for an hour and a half I've showed no dad I was swimming I've purely talked about anecdote and eminent space medicine so what about data so most of the studies that are done about new medications are mostly funded by industry they'll look at their new novel medication and they'll study it for sustaining of care but osteoarthritis and nutraceuticals are a little different this has been one opportunity where the NIH your tax dollars have actually gone to fund a fairly large trial this is almost unheard of in my specialty that money just doesn't exist but the question about nutriceuticals and glucosamine and chondroitin was such were so vehement and there was such a an outcry that this had to be studied that the US government paid for it they did a study that cost more than ten million dollars and it was called the gait study and it looked at glucosamine and chondroitin in the Arthritis intervention trial they took over 3200 patients patients were over the age of 40 and they had to have x-ray evidence of osteoarthritis patients were randomized like a flip of a coin into one of five treatment groups glucosamine three times a day chondroitin three times a day Luco sameen and chondroitin celecoxib celebrex or placebo anyone want to vote for category one glucosamine winner all right we got one taker - chondroitin well come on there we go there's glucosamine and chondroitin ah we got a few more hands celebrex few more how many of you like the SIBO 24-week study and the primary endpoint was 20% improvement in pain we're just looking for a marginal improvement here 20% improvement right you can get 20% off at Macy's any day right so we're looking for just a 20% improvement in your pain so here the results you can make sixty percent of patients better with placebo 60% where else can you get a 60 percent return on your investment where even the Stanford endowment can't get a 60 percent return on invested I mean placebo who knew celebrex you get a 70 percent return on your investment better than placebo how about glucosamine oh it beat placebo chondroitin it be placebo and how about when you put the two together beat placebo well not statistically not out of three thousand patients this was the best and most conclusive study that's ever been done and the x-rays didn't show any improvement in your cartilage so it doesn't change your joints so I'm gonna get you in one second so there's two ways of looking at this the glass is half-empty if you're a glass half empty kind of guy or woman these nutriceuticals do not work any better than placebo they don't work if you're a glass-half-full kind of guy you'd say doesn't matter to me because I'm getting 60% better it's a very nice placebo no one gets hurt on Glucosamine Chondroitin you can make yourself are any given individual may have 60% likelihood of improvement there's no reason not to consider it you give it a try for four weeks for six weeks if you feel better be all me by all means keep thinking if you do not feel better use your well-earned disposable income elsewhere you had a question so the question really had to do with placebo and with Tylenol so have studies ever been done that compare drugs like celebrex to Tylenol yes they have it's been done with Volterra and it's been done with Naprosyn there's no question that nonsteroidals have a better better anti-inflammatory and better analgesic effect than tylenol when both are dosed appropriately there's no question about it that nonsteroidals carry a far worse safety profile when compared with tylenol and you know that with just a sugar pill you can get 60% better if you're not diabetic sugars good a question had to do with well what about glucosamine and chondroitin as a preventative well x-rays were in fact done in this study to see whether or not people progressed in a subset of these and in the subset the x-rays were done there was no difference in the x-ray progression between those who were got any of these groups so as a preventive there really isn't any strong evidence that any of these halted progression they all came up at the same time so what about just using ice if your knee is warm or swollen ice is good I'm not trying to be funny if if your knee is warm or swollen I like ice if your knee is not warm or swollen but it just hurts sometimes heat can be better it helps bring blood and more nutrients to the area so we often apply ice in the first few days 48 hours after an injury or if it's swollen we often use heat at other times but any given individual come to their own conclusions try ice high he decide which one works better yes so with nonsteroidals there are variety of ways to help protect your stomach so anybody who's going to be taking non-steroidal anti-inflammatory drugs with any chronicity trying to protect your stomach with things such as a proton pump inhibitor like over-the-counter omeprazole that's good a second alternative would be something like an h2 blocker it may not work as well as a proton pump inhibitor a t2 blockers or drugs like tagamet zantac pepsi another prescription drug which can work well is it going called misoprostol it can be used in concert it's a prescription to begin used in concert with nonsteroidals when used to high doses it causes diarrhea so if you're constipated it's a great option um shy of that there may be other choices what about platinum so Plaquenil is a anti-malarial why would we use an anti-malarial in arthritis well back when oils are really bad anti malarial it doesn't work very well but it turns out it is a weak immunomodulator you can down regulate inflammation and so for certain types of arthritis Plaquenil can in fact be a useful drug it is not for the majority of the folks out there with osteoarthritis though it has often more benefiting folks that have active inflammation yes how does the my descriptions of arthritis pertain to arthritis of the back well inflammatory arthritis won't cause erosion or destruction in the bones of the back inflammatory arthritis sometimes caused the exact opposite to occur in the spine it'll cause the bones to grow together in the back with some rare exception in that sometimes the very top of the cervical spine osteoarthritis on the other hand will cause the bone to look just like I showed you in the hands for the knees where you'll lose the cartilage lose the protective layers in between that and you'll get these bone spurs quite devastating quite proliferative and certainly symptomatic okay so there are a number of expressing our thright Asst we've been experimenting over decades we as a field a profession at conjurer protective agents such as matrix metalloproteinase inhibitors inhibitors that will stop enzymes from destroying a joint common ones that you might think about food things like minocycline or doxycycline an antibiotic oh my so I know the wheels are turning now how is an antibiotic preventing arthritis well it's not because it's affecting micro it's not because it's killing organisms but it's because it's interfering with some of the destructive enzymes that cause cartilage and bone to break down so these drugs work not as there if I macro Bowl activity would make you believe but because they prevent enzymes from causing degradation there are a number of other therapies such as bisphosphonates that have been tried what if I were to down regulate those osteo class those nasty little buggers that are crawling around breaking down bone what if I down regulate them well preliminary data suggested they might be effective but they ultimately turned out not to be so affecting the way the bone breaks down and builds back doesn't seem to really affect osteoarthritis in any clear way there are a number of growth factors that we're currently experimenting with now we're injecting things like bone morphogenetic proteins into the knees and fiber last by growth factor 18 to see if we can cause cartilage to stop breaking down even potentially grow back and we've tried anti cytokine therapies including here at Stanford where we've tried using these newfangled biologic approaches to see if we can stop that low-grade inflammation in the joint but we couldn't find any significant benefit for doing it yes it's a good question well would it do anything if we tried taking tetracycline mono cycling or doxycycline prophylactically to try and prevent this from occurring and the answer so far is no we don't have conclusive evidence that it does a great job and certainly carries with it far more potential side effects than would be warranted yes what about acupuncture I liken it a lot to glucosamine and chondroitin there's no convincing data that it works but there's a really little downside and if it makes people feel better and a lot people do feel better I have no qualms with it so I think a combination of eastern and western approaches are absolutely legitimate yes so the question is bisphosphonates those are those like drugs like Fosamax al and renee hacked now the answer is yes and when the largest always thought he's ever done took a drug called actonel and looked at it in osteoarthritis more than 300 patients couldn't find a benefit so the last approach i want to talk to you is stem cells you're gonna get a nice lecture later in this quarter about regenerative medicine keep in mind that regenerative approaches are really what we want to talk about we want to try and restore the balance between joint damage and reparative processes in the joint the majority of our efforts right now have been towards reducing inflammation but the reality is once damage has occurred repair becomes far more important and for most of us we don't present saying doc I think I'm gonna get osteoarthritis in 20 years high percent because my knee hurts damage has already happened that's where I really need a reparative regimen and so a goal here is to promote repair through tissue regeneration and how could that be done well it can be done through stem cells or through progenitor cell differentiation it can be done through tissue patterning and creating an orchestra textual organization to allow cartilage to regrow so are any of our engineers out there you can come up with a good bioactive scaffold our cartilage could probably regrow itself particularly if we have just a few little things to stimulate it and it can restore function these are our real goals and so where does stem cells come from well they can come from a lot of places they can come from embryos I'm not going there I will suggest to you that they can come from a lot of other places fetal and the onic umbilical and even adult stem cells we all have stem cells they can be a found in adult tissues and they often will actively participate in our own homeostasis within our joints so how do we specifically try differentiate and how do we pursue processes that will store normal homeostasis and make use of our own tissue stem cells well here our stem cells and here's how they help us regenerate or divide and turn into a variety of components in our body I'm going to ignore the ectoderm and I'm going to ignore a lot of the other things and hone down on just one side of that slide it's missing Komal stem cells they derive and and divide into things like bone cartilage muscle fat and neuronal cells it's entirely possible in the future that we'll be manipulating missing Komal stem cells to our advantage to try and ops to improve and our reparative processes so our hope for the future here certainly for anyone with a damaged joint irrespective of the cause of that damage whether it was an inflammatory or traumatic or age-related process would be to restore joint tissue through a recruitment of resident endogenous stem cells to the site of damage and then activate them this can be achieved through the control of released morphogens or bioactive scaffolds or to expand a population outside the body in a tissue culture lab and then return it to the body and through pharmacologic manipulation help our body return these joints to normal that should be our goal how quickly that's achievable is another issue and certainly the discussion for another lecture so I'm gonna leave you there and see if there are any final questions before I give the podium back to dr. Pease oh yes are there people with rheumatoid and osteoarthritis yes there are most patients that develop rheumatoid will incur joint damage and then a degenerative process will ensue even if they haven't developed that much joint damage from the rheumatoid eventually as they get older they will encounter other features of osteoarthritis as well the question was about companies that are doing research in osteoarthritis and even companies that might be offering it as a treatment I would say right now that that there is no FDA approved process for cartilage regeneration that all of these things are experimental in nature if anyone offers it to you outside the context of a clinical trial don't believe it is bone cancer related to arthritis in general no bone cancer comes from an aberrant process that takes place within certain cells where they take on a life of their own that's a little different than a degenerative process like osteoarthritis itself yes question is what are the phases of trying to improve this because one component here is an overactive bone growth and spurs in others the process the cartilage is going we don't have it worked out yet the reality is we don't really know what's gonna be necessary to restore this but we're taking baby steps right now we'd be just happy to regrow some cartilage in a normal pattern I can regrow cartilage now but cartilage doesn't grow back in a nice pattern we can regrow cartilage in like every different direction but that doesn't help dr. P so if the cartilage grows north and south and he needs it to grow east and west that's a problem yes I took that slide out but the question was what about Micro fracture procedures well a Micro fracture procedure is where you basically drill down and you fracture that tied mark that level between the bone and you cause blood to repopulate in there and you get fibrocartilage growing fibrocartilage is a temporary answer it's like a body's scab it regrows a little bit of tissue right there but it's not normal cartilage so microfracture procedure is a temporizing procedure think of it as a bridge to nowhere or a bridge to a knee replacement it will make you feel better for a few years until the fibrocartilage is obliterated again and you could go back and do a mother micro friend Micro fracture procedure but it's a temporary solution I'm gonna give it to dr. piso because it's a 20 and if he wants to entertain more questions on well mark I think the audience is spoken for itself that was a wonderful indeed spectacular presentation so I'm hope you enjoyed it I think you'll agree that last week's discussion about the immune system set you up for this but I think the really practical way that you approach the issue of their authorities it was really quite wonderful and I still will be running no matter what but I now know I'm gonna get arthritis but placebos are cheaper than anything else and life will go on so there we are so next week we'll be onto vaccines and mark I think if you're willing some folks would love to come down and just ask a few questions thank you I'll see you next time for more please visit us at stanford.edu
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Medical_Lectures
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Immunology_Lecture_MiniCourse_10_of_14_Adaptive_Immunity_to_Infection.txt
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so in this now we're going to go next to part two in understanding T cell immunity and this is really going to start addressing the subpopulations of T cells that a lot of you familiar with th1 th2 th17 and t rex cells and put it into context so the first question is again there are a very wide array of pathogens and for each pathogen you're going to want to have a different type of immune response that's generated how does the T cells know which responds to orchestrate that's appropriate for that specific pathogen how did T cells cd4 positive T cells differentiated appropriately I wrote th one versus T h2 but will again expand the th17 T regs and how to diverse cytokines transmitted signals that mediate divergent responses and again people were so enamored with the t cell signal transduction lecture yesterday that people came up to me again and said we want to see more signal transduction lectures so one no more factors we want to know more kinase is it just wasn't enough you know people can't get enough with signal transduction so therefore I put in you know a whole unit on cytokine signal transduction to satisfy those people who I notice aren't here today okay okay okay so this is to underline the fact that the response to the version pathogens require different immune responses and if you want to use a simplistic analogy it's kind of like if you have an Army and Air Force and Navy or Coast Guard or whatever you want to mobilize the appropriate limb of the Armed Forces for the appropriate type of attack or defense so if you have an attack coming in by sea you want to have your Navy go out if you have an attack coming in by land you want to have your army go out so and clearly if you pick the wrong Armed Forces approach you're going to really have a devastating impact so you're not gonna want to mobilize your Navy if you're going to have an attack from by land and so to the immune system has to marshal the appropriate immune response to the appropriate pathogen that's most effective for the book for that pathogen so if you look at this cartoon this is kind of outlining which are the immune responses that are most effective against specific pathogens and again you know this is done very black-and-white but clearly it's not black and white immunology is a lot of gray zones and just take that with a grain of salt so we're just saying what is felt to be the best response but clearly other approaches play a role so again here it's showing cytotoxic t-cells play a critical role in killing virus infected cells and the pathogens that are targeted viruses like the flu rabies as you know HIV and some intracellular bacteria so an example would be T V if T V escapes the vacuole and now is in the cytoplasm bacterial proteins would process presented in 2 class 1 MHC molecule and then that cell will be killed but obviously antibodies also play an important role in viral infections as well but in terms of T cell responds it's not a toxic T cells play a critical role cd4 helper T cells play a critical role as you can recall in activating macrophages that have been ingested to be TB is inside a vacuole in the absence of activation that back row flies won't be able to efficiently kill the T V and therefore you need cd4 help it's going to secrete as you know gamma interferon that's going to activate it to kill particularly intracellular bacteria it's also again plays a role in activating cytotoxic t-cells so cytotoxic t-cells are far more effective when they receive cd4 positive T cell help and they become even more effective killers of virus virally infected cells because of for example interleukin 2 and other cytokine secreted by th1 cells so in fact one reason why as HIV progresses and individuals lose their cd4 positive T cells the CTL response is diminished is because they don't have those cd4 T cells to keep revving them up and making them better better killers so again you can think of cells in isolation the immune system is kind of like I guess the soccer team that it's a team and even if you don't have your goalie clearly you can have one issue if you don't have your striker or for other plays you can have other issues because everything has to work together this is just showing what the major role of the individual player is but again keep in mind that it also interacts with other players cd4 positive T H two cells their major role is in stimulating humoral antibody response and in particular in particular for the one that again take to the bank really is very highly demonstrated is this ability to stimulate IgE production and again as I'll discuss th2 cells making interleukin 4 which drives as you we learned yesterday these cells to differentiate and recombine to produce IgE and again as we discussed yesterday IgE binds to an FC receptor that's specific for IgE expressed on the surface of mast cells and then when there's an infection with the parasite cross-links those IgE antibodies in the MassHealth surface release all those mediators and then you get sloughing off of the parasite by that very robust inflammatory response cd4 th17 cells again a recently discovered subtype of T cells those seem to play an important role in extracellular bacteria they're very important in mucosal immunity we'll discuss this this afternoon in mucosal immunity and they pretty much drive in neutrophil response and so far in immunology we haven't talked a lot about neutrophils but clearly they were an important media cells have very strong phagocytic activity and th17 are the particular subgroup of t-cells that recruit that response and finally T reg cells and T reg cells mentioned yesterday are the brakes of the immune system if there's a very robust immune response T reg cells come in and basically suppress t-cell responses again because of the too robust immune system may be destructive to tissue so again the same way you drive the car have an accelerator you have a break these cells are the accelerators and the T reg cell is the brake and you can imagine if you don't have T reg cells it's like driving a car without a break or I mean I'm from the Bronx and that's baseline driving in the bronz people drive like they don't have any brakes but that's another story yeah cd4 positive T cells this is start very simply you basically start out with a naive cd4 T cell that is antigen specific it's extremely naive it has no idea what's ahead of them or her in terms of killing bacteria secrete secreting cytokines it now encounters antigen MHC peptide TCR gets co similar signals gets introduced interleukin 2 and now undergoes proliferation so now you have millions of t-cells there's an antigen that they've that's out there an infection they have to respond to but it's still immature it hasn't yet committed in terms of which way it's going to mature so one possibility is that it may mature into a th one cell and again I'll discuss in a few minutes what factors determine whether it matures along the th1 lineage and in that case it's going to be able to again activate macrophages and again it'll talk about how it also doesn't induce optimizing antibody an alternate route that it could take is to mature into a th2 cell a th2 cell is going to activate these cells to make neutralizing antibodies in addition I GE and we'll discuss again the functions in more detail and also could differentiate th17 as well as t-rex cell but again that I just want to give you the caveat there's a very simplistic way that I'm presenting and clearly a lot of things are happening at once but again in immunology you try to teach at sim bully you understand the simple concept and then you can now start applying in the more complex interactions that occur so cd4 and cd8 cells differentiate along distinct pathways have specific effects so for example cytotoxic T cell MHC + peptide class one it'll kill the virally infected cell that's what a CTL is going to do th one cell now illustrating what I showed in the previous slide it this is a macrophage it has its ingested intracellular bacteria again to reiterate if the bacteria is still in the vacuole this is not considered an infected cell it's taken it out it can still cause fusion of Baga lysosomes kill the bacteria and therefore everything is fine in order to augment that process this macro five likes will get T cell help from the th one cell and therefore it'll will kill it again to reiterate if this bacteria escapes from the vacuole into the cytoplasm obviously you can't take a lysosome and slam it against the bacteria anymore because then you destroy the cytoplasm the vacuole is kind of like a stomach for the cell you could throw all sorts of enzymes into it and yet the cytoplasm is protected and as you know some bacteria TV for example Listeria have factors that allowed to escape the vacuole but once the escapes the vacuole again peptides derived from the bacteria can be presented in class one and now it can be seen by a cytotoxic T cell and it'll get killed th1 and th2 cells but th2 cells basically activate these cells here you see the antigen presenting cell to the th2 cell is a b cell itself is trying to recruit help into itself and that's going to allow to secrete in this case showing neutralizing antibodies and later again also say I GE th17 cells can get activated by fire bless and epithelial cells and they are induced therefore to secrete will turn out interleukins 17 that now causes recruitment of neutrophil into the area again mucosal tissue a good example salmonella infection and finally tear egg cells they basically can be activated by immature dendritic cells and again I'll show you that in greater detail and the effective t-rex cells is to turn off cd4 T cell function okay any questions so why what defines what functions of these cells are and the simplistic way of saying it is you are what you secrete if you see you the functional activity of cells is determined by the summation of the factors that it secretes a question so all of these cells it's a very good question all the question is our T 17 T reg cells th1 th2 are they cd4 T cells and the answer is yes they're all subpopulations of cd4 and you're thinking wait a minute T regs are inhibiting the immune response and we've learned cd4 cells are helper it's a little counterintuitive that at quote helper cell is actually having suppressive activity and the answer is yes that is a sub population of T cells that does inhibit and one could play that you know what could spin around and saying you know as a parent you all know that sometimes you have to inhibit your kids as a way of actually in a big picture helping them so made a t-rex cell even though it's inhibiting the response the big picture it's actually helping to have a balanced immune response that doesn't destroy itself okay so that you are what you secrete so in this case for th one cells the factors that it secretes are interleukin 2 and again this just shows all the different things that interleukin 2 can do but the major factor that want to focus on is the fact that a stimulus T cell growth at the last column is actually very informative because it illustrates what the phenotype of the knockout is in mice if you knock out that particular cytokine and what actually was very surprising is that if you not got interleukin 2 it's really not that dramatic you have some decreased t-cell response you have some autoimmune inflammatory bowel disease but it's not as dramatic as one would have thought considering how important we thought il-2 was for t-cell function again it illustrates why you need to do the experiment to see because there are other cytokines that can pick up the slab gamma interferon is probably the critical factor made by th1 t-cells it has multiple effects it does stimulate B cells to make IgE in this case mouse to a it inhibits t-cell growth it activates the cytotoxic T cells as I mentioned before and interleukin 2 also stimulates CTL proliferation but the major activity of the gamma interferon is on macro flash function it activates the macrophages makes them better killers and also up regulates class 1 and class 2 expression on the surface of macrophages making them better antigen presenting cells it also activates post inventory molecules if you knock out gamma interferon these mice are susceptible to mycobacteria which makes sense because their macrophages can't kill them as efficiently and also to some viruses because it relax some of the antibody responses that are required to facilitate immunity to these viruses and finally lympho toxin which is is is an era but it's not as well described as the other cytokines but its major activity as it activates nitrous oxide production and this plays a critical role in bacterial killing if we now go to th two cells the major side effects that th2 cells produced is interleukin 4 in terms of the proactive function of th2 cells its major impact on b-cells is production of IgG one which is a neutralizing antibody that will bind to viruses and neutralize it IgE again as I mentioned before which plays a critical role in parasite immunity and also up regulates class 2 expression to increase antigen presentation and it actually functions as a growth factor for t-cells and I'll discuss it actually plays a role in polarizing t-cells to th two polarity it enhances the growth of mast cells which action makes sense if you're going to make ite which is body to meth cells you also want to help make math cells grow and if you knock out interleukin 4 there are no th2 responses and you have a minimal I GE production so it's very dramatic phenotype very susceptible to parasitic infections interleukin 5 is also a th2 cytokine it's basically stimulates IgE synthesis it also enhances the instrument for growth and differentiation and it also probably plays a critical role both in mucosal immunity as well as in parasitic infections and finally interleukin 2 it's it does have proactive effects in terms of stimulating class 2 MHC molecules but if major effects seems to be inhibiting th1 differentiation and i'll discuss that in greater detail in a few minutes t red cells the major cytokine they produce is tgf-beta and tgf-beta inhibits growth of b-cells again it's inhibitory so it's going to do a lot of inhibition and inhibits t-cell growth inhibits activation of macrophages it does activate neutrophils and inhibits other cells and if you knock out the about tier egg phenotype the the mice actually die within several weeks of age from these overwhelming autoimmune diseases because again you don't have the break you can accelerate the car just ultimately bang into something and totally th17 again is an illustration of how immunologists want to be your friends so they very easily could called it you have th1 cells th2 cells are actually our t3 cells they could call it CH 4 cells now that would have been not very helpful to you because website of Heinz do you think th17 makes aisle 17 I mean how much easier can it get than that so therefore they named it th 17 to really reinforce that it's making interleukin 17 and the source are cd4 positive T cells in the major effect and again you have to take this with a grain of salt because the knowledge of th said cells are still evolving and this is a figure from 2008 textbook so it's a little bit out of date but I guarantee you that aisle 17 also has impact on B cells and T cells and macrophages may not yet have been discovered and maybe some of the people in this room may be the people that will discover that recently it's been shown that th17 play an important role in HIV infection again may be very susceptible to being affected by HIV but the major impact it has in stimulation of neutrophil recruitment and also stimulates fireblast and epithelial cells to secrete other chemo clients also further enhance inflammation one thing that also is known about interleukin 17 is it plays an important role in inflammatory bowel diseases and Crohn's disease and in fact that's why it's a subject of a lot of research to block this through potentially reverse those inflammatory bowel diseases okay now as I mentioned before the function again is is due to their differential cytokine production so cd8 t-cells in addition to having fact a cytotoxic of that dudas secretion of perforin and granzymes and expression of fast ligand also secretes of the cytokines that play a role in its process now people who experimental II want to analyze for cytotoxic T cell function do an LS body assay anyone here do elispot assay okay what cytokine are you looking in your elispot assay for interferon gamma because activated cytotoxic T cells also secrete interferon gamma the same way that th1 cells do so in fact that's why the reason you're not picking up th one cell reactivity in your assays is what are you're stimulating your cd8 cells with the peptide being presented by one well what you have to add a self antigen presenting cell and you're adding usually a cell line that's expressed in class one MHC so that's why you're only focusing the response to cd8 cells you're not adding a whole protein that can potentially get digested by a macro fiber and presented class two MHC molecules okay so now cd4 T cells th1 cells again as I mentioned it doesn't make multiple cytokines but the one really want to focus on is gamma interferon because that's the one I'm gonna be discussing in most detail and probably plays the most important role Jeeves two cells again makes us so host oversee defines the ones I'm going to focus on predominately going to be interleukin 4 and in interleukin 10 just to make you aware interleukin 13 has very very similar overlapping activities with interleukin 4 and th17 cells makes aisle 17 also turns out mixing aisle 6 and and some TNF and T reg cells the major player is going to be tgf-beta which has significant suppressive activity on cytokines it also makes interleukin 2 the same way th to sell to an interleukin 2 again as we'll discuss in a few minutes is a inhibitory cytokine now again you have to ask the question who cares you know why does it really matter whether you mount the appropriate immune response to a pathogen can it have implications if your immune system makes a mistake and mounts the wrong immune response and what's very illustrative of this process is leprosy so leprosy has anyone here ever seen clinical luck recei you've seen it what what time did you see tubercular time okay so as we'll discuss in a minute the same exact bacteria can cause to very dramatically different phenotypic diseases and this is an example of lepromatous leprosy very dramatic skin nodules disseminated mycobacteria devastating disease individuals lose fingers limbs it's a really horrible horrible disease when you think of leprosy this is the kind of disease that you're thinking about however the exact same bacteria can cause a very different much more mild disease like the tuberculoid leprosy you seen you have some your patients have skin lesions mostly mostly skin lesions clearly this doesn't look very dramatic and this is to Burke tuberculoid leprosy so the question is why is it that exactly the same pathogen can cause two very distinct B no typical diseases you think the same pathogen should cause the same disease it turns out that this is due to what kind of immune response the individual generates towards those that pathogen if it makes the right one it does well can contain it and here is a pathological section from from tuberculoid leprosy and these that you could almost very few detectable levels of bacteria do not very infectious they have a lot of granuloma and local inflammation skin lesions they have some peripheral nerve damage that's toxicity the levels of antibody are normal and they have normal t-cell responsiveness in contrast individuals that have lepromatous leprosy have large numbers of organisms growing through throughout and especially in macrophages they're incredibly infectious but there's such a high bacterial load they have disseminated infection bone cartilage and damage they have hyper gamma globin amia they have high levels of antibody and have low t-cell responses so who would want to wager if you're infected with Mycobacterium leprae it's an intracellular bacteria it gets taken up by macrophages who here would want to mount a th1 response against michael bacteria leprosy raise your hand go ahead th one okay who would want to mount a th2 response against leprosy raise your hand and now we've got to get revved up got a vote you know it's not like it's gonna kill you if you vote wrong like it will for a patient okay again mycobacteria intracellular pathogen taken up by macrophages now think if something's taken up by macrophage what cytokine would you want to secrete rev WAP macrophage killing ability what side of mine say it louder just yell it out you know give me some gather interferon come on wake up right right so I'm good side of come when you want to make come on yell it out beagles get it out here wishing which cytokine okay the Front's good the back is still you know embarrassed okay so what subtype makes interferon gamma th one or th two so who votes for th one if you're infected with Mycobacterium leopard raise your hand and who votes th to raise your hands see you're not voting you gotta vote lay I'll accept late votes th one late load okay so now the body has to devote the body can't sit on the sidelines and say you know I can't decide so I won't Mountain immune response that doesn't work it turns out now if you look at what cytokines are made in response to micro bacteria elaborate individuals that have tuberculoid leprosy are making I'll to jam interferon TNF beta which are th1 cytokines not a lot and and the lepromatous leprosy patients they're not making a lot of those th1 cytokines most importantly they're not making gamma interferon however patients that are have lepromatous leprosy they're making large amounts of out of 405 and aisle 10 so in essence they've made the wrong choice because now they're making high levels of antibody they're not making gather interferon is antibody can be very helpful for an intracellular bacteria know so to make it an immune response but it's the wrong one that's why the bacterial growth is not controlled and that's why they have such devastating disease if you mount the appropriate response make a large amount of gamma to fear on activating macrophages making them more effective killer cells then you're able to control very well is that clear does that make sense so therefore if you had a patient that had lepromatous leprosy what cytokine would you consider giving that patient to rev up your immune system gamma interferon again patient made a mistake let's help this patient okay so now to put it together now th1 cells naive t-cell it becomes a th one cell it's going to make gamete of fear on which is first going to activate macrophages make them better killers it's also going to make cytotoxic t-cells more effective and also a kind of a simplistic view that that some people may have is while th one is cellular immunity and th two is humoral immunity however th1 cells also stimulate immunoglobulin production predominantly IgG one why because as we learned yesterday IgG one can be very important in allowing optimization of bacteria to induce an increased phagocytosis because now this antibody will bind to the bacteria the the complex of antibody bacteria binds to the FC receptor on the surface of the macrophage and now the macrophage ingests and Farr efficiently than in the absence of the antibody and this now synergizes with gamma interferon ability to now regulate macrophage digestion so again it's this one-two punch of antibody plus activation of macropods digestion again makes macrophages that much more effective so you can imagine that if you're infected with Mycobacterium leprosy now we have antibodies that bind to it and again stimulate optimization and elimination before it can infect other cells okay is that clear and cd4 T cells make a broad range of cytokines this is basically a list of them and they have a broad range of activities dominantly activating macrophages increasing t-cell proliferation and also increasing macro fires differentiation in the bone marrow gm-csf is a critical macro flies growth factor that stimulates macrophage differentiation in the bone marrow well you have an infection you need macrophages to eliminate the infectious agent maybe you're going to run out of them in the periphery so you also want to go the bone marrow and recruit more macropods in the periphery so again it's a very parallel approach of again enhancing the immune response it also stimulates ccl2 secretion CCL 2 is a chemo kind whenever you see either c c or c XC at the beginning of a of a compound that refers to a chemo kind and again it recruits macrophages to the side of infection again because that's what's going to be best in terms of eliminating the infectious agent th2 cells naive T cell also the French chasing through th2 cell it basically makes interleukin 4 and into the looking 13 which are going to drive these cells to either make IgE which is going to bind to mast cells FC receptors and service of mast cells where cross-linking will allow deep granulation and is important for parasitic infections particularly worms and it also makes neutralizing antibodies IgG one which can just directly bind for example to viruses or toxins to neutralize them and also stimulates l5 which again up regulates iga production and also stimulates the esta no fill differentiation and activation as I'll discuss in a minute it also makes interleukin 2 which has an inhibitory effect on th1 maturation now why what tells the cell whether to become a th one cell or th 2 so you have this naive lymphocyte it can go either direction what's way is it to go one way or the other and it turns out it depends upon the cytokines that are secreted in the milieu in which this t-cell is located and if it's interleukin 12 and interleukin 12 can be made by dendritic cells or can be made by macrophages this stimulates T cells to differentiate into the th one pathway in contrast if the T cells exposed to interleukin 4 either made by NK t cells or actually made by other th2 cells or made by mast cells this now drives the self is a French into a th2 cells so aisle 12 made by macrophages as well as dendritic cells drive cells become th1 and interleukin 4 drives itself to become th - okay that's pretty straightforward but in addition it also may depend upon what cell is actually presenting antigen if the antigen is being presented by macrophage well likely it's because the macrophage has taken up a bacteria it's digesting and throwing up some peptides it needs help digesting so it's gonna want to drive that t-cell to MIT to help it and it's gonna want to drive it to make gamma interferon and therefore it's going to drive it to become th one cell by also secreting interleukin 12 a b-cell for example one net wants the class switch into a IgE producing b-cell if it presents antigen it will can secrete interleukin 4 and drive the helper T cell into becoming at th - helper T cell now T reg versus T a 17 are mediated by different cytokines and if there's no infection going on so that means that it's peacetime in the body if it's peacetime in the body do you want to be revving up immune responses no absolutely not so if all the ER dendritic cells are immature they're immature because there's no pathogens out there that are activating them so in this case the dendritic cells are making high levels of tgf-beta and they're making very very low levels with either Aisle 6 or out of 23 and therefore when a naive cd4 t-cell is exposed to high tgf-beta it's now stimulated to up Gregg you late its expression of Fox p3 which as you may know is characteristic for T red cells it's a it's it's present inside the nucleus transcription factor and this now drives this T cell to become a tier egg cell and you could detect here egg cells by their expression both of the cd25 al 2 receptor as well as Fox p3 and this t-rex cell now has the capacity to inhibit both th1 and th2 cells okay is that clear and it makes sense if there's no infection no activation you want to have cells that can damp down any residual inflammatory processes that are going on however if there is infection and now the dendritic cell is activated by those infectious agents now this dendritic cell is not only going to be making high levels of tgf-beta but it's also going to be making high levels of interleukin 6 and interleukin 23 and now this is going to drive this naive cd4 T cell in a different direction if there's infection do you want t-rex cells to be generated no because you don't want to basically damp down an appropriate immune response that would be devastating and therefore now you drive the cd4 positive T cell to differentiate into a th17 cell and the characteristic transcription factor that's expressed by th17 cells is ror gamma t plus and this now drives that this cell to make interleukin 17 which ultimately is going to stimulate in this case neutrophil neutrophilic influx now this balance of T reg and T a 17 is particularly important in mucosal tissue because in the absence of infection you basically don't want to have inflammatory processes in the gut you need to have T reg cells if there is infection though now you want to mount a inflammatory response and then you shift over to a th17 response is that clear now a question so apparently it seems he has a great question is that yourself always producing high tea Japan same time I can't give you you know a definitive answer that every dendritic cell is making hi TGIF all the time but apparently there are some populations of resting dendritic cells that make high levels of tgf-beta enough to drive this differentiation to go on excellent question okay so now again you you asked for more signal transduction and I try to be responsive to students so this is just kind of review signal transduction but if you recall one of the factors is in fat and when n fat is phosphorylated it's stuck in the nucleus when M fat is d phosphorylated and now can go into the new I'm sorry phosphorylate and that is stuck in the cytoplasm you can't get into the nucleus if you D phosphorylate it can now move into the nucleus bind to its appropriate transcription site and activate the appropriate genes it turns out that there's more than one flavor of nfm there actually at least four different types of n fat each one of those different types bodies to a different sub population of genes turning on a different population of cytokines and it turns out that n fat a atc one and these are the ones just want to focus on this particular one mediates the interleukin 4 th to response and this particular one mediates gamma interferon CH 1 responses so in addition to cider 5 - 2 - the extraneous aisle 12 versus aisle 4 there's also a subpopulation of n fat if this is generated then the t-cell will be differentiated to aisle 4 th - if this n fat is stimulated after antigen presentation then this T cell will differentiate into th one responses now a question is how does the t-cell know whoof and that to make it turns out that it depends the antigenic load that's out there if you have a high antigenic load which means you're going to crosslink a large number of t-cell receptors that seems to drive a th1 response lightly because that strong signal probably turns on this particular n fat however if it's a lowering to genetic load not a lot of the antigen out there that seems you can only personally for very few number of t-cell receptors a weaker signal and that seems particularly to draw pile for th one so again another paradigm that you'll hear about why you get a th one and why you get a th two is depend upon the antigenic love high antigenic load makes sense think of a bacterial infection large number bacteria a lot of the antigen out there that's going to give you a to eighty th1 response parasites tend to have a much lower shedding of antigen think about allergies allergens like pollen not a lot of pollen out there that will drive a th2 response okay now I will say that the immune response is like a political campaign and I know here in South Africa you also have a politics of an election coming up soon right about five years okay well there's always elections it's probably local people running for office right so whenever you have an election the first thing that the candidates always do is they say how great they are right you know and they say the same thing everywhere they're going to help the economy they're going to increase jobs they're gonna you know the usual campaign thing right what else do they say here in South Africa how decrease housing and what else increase what the decrease crime right I mean what who could be against that right but but what's the problem with that approach the problem with that approach is that everyone says it so you can't decide who to vote for based on what they say so of course you say well this person really is honest this person is very competent and maybe he'll do or she'll do what they say and the other person is lying through their teeth but you talk for you to tell that so ready candidates do to convince you to vote for them what else do they tell you what the others bad exactly like the other person is a liar a thief he cheats on his wife you know a list of the usual the other sold pictures of that person like they never shut it up to any meetings or the person is lazy or they'll show they'll be taking on junk it's outside of the country all the time you know then we may not be true but now you see that and say I'm not voting for that person because this you know has only bad things about them and in fact whenever a campaign gets close that's what candidates tend to do they shift from being very positive to be negative on their opponent and things have really nested right so unfortunately the immune system does exactly the same thing it sinks to that same level so initially it tries to have a positive message th1 cells will say I'm secreting interleukin 12 join the th1 team that's the appropriate immune response th2 cells are saying all secrete interleukin 4 come the MTA's - so that's the appropriate response but then and things get desperate and the naive t-cells who don't know much may not be taken a positive message or th17 cells may not take the positive message now they go to negative and the way they go negative is by secreting cytokines that actually inhibit differentiation in alternative pathways so here in this case th2 cells secrete interleukin 10 and interleukin 10 are basically going to prevent interleukin 12 production and therefore prevent ch1 differentiation and th and in this case here I interleukin 4 also inhibits th17 differentiation whereas th1 cells make gamma interferon and gamma interferon blocks th - differentiation and also the blocks th17 the French and tea red cells make tgf-beta and teaching at bay that basically bought th one versus th two differentiation so in addition to having a positive message you also have a negative message to particularly polarize T cell differentiation to their own particular pathway okay is that clear does that make sense okay and it turns out that this is important because you have to visualize the immune response in the context of a tissue so let's say you have a person who's infected with the parasite what if I have an immune response are they going to be generating who here sells th - raise your hand just go for it okay who says th one raise your hand no it's gonna be too used to it right so now that same person gets infected with Mycobacterium leprae what's going to happen well they're having a th to polarize response they may be making large amounts of interleukin 10 which is suppressing th one so that may be a reason why individuals may make the wrong immune response because another pathogen that they're infected was made part of the resident the one way is hard for th one differentiation to occur in environment we have large amounts of interleukin 10 be made so if you're infected with multiple parasites that may polarize you to the wrong way and it's not that the immune system is stupid and doesn't know that mycobacterial library requires th one it may be that just a victim of its environment and it can't overcome them okay so no matter how much positive signals it provides to go th one is story by the pre-existing polarization - th - and the same thing can happen the opposite if a very strong th1 response you're infected with Mycobacterium leprae you pull around th one now you get infected with the parasite you can't ship to th to the appropriate response 7 there and that's why : faction could have very important effects on altering the immune response and again some of you are familiar with now HIV Iko infection and the question is what impact that has on your ability to fight either pathogen okay so t-cell respond to go stimulation proliferation and the effector so the critical role is played by cytokines and cytokines order for them to activate a cell has to bind to a receptor and the basic receptor that we're going to be focusing on are henry of america receptors that basically consists of either two or three chains that are combined when the cytokine binds to that particular receptor and as I mentioned previously interleukin 2 rapidly turns out to have three receptors a gamma beta heterodimer can give you monitoring and induced signal transduction by having the third alpha chain now you switch to a high affinity receptor and again as I showed you previously the monitor infinite interleukin-2 receptor requires high amounts of interleukin 2 to be activated and one could have imagined in a very vigorous infection you have a lot of interleukin 2 being made and that's one way of recruiting t-cells in the face of a really severe infection it's not not in time to go through this very careful checks of activation and the antigen presentation every see - this is like an all-hands emergency for the immune system you want yourself to get activated and that's what high out - we'll do a normal situation you you want to have the appropriate checks so therefore only low-level interleukin 2 will activate a t-cell if the infection is relatively cannot that dramatically severe okay so signal transduction by satellites while I have good news for you because instead of this complicated process that T cells use for T cell receptor transduction cytokine transduction is actually quite straightforward and simple it only requires you to know two families of proteins and the word family is the first number as you recall it being receptor molecule itself doesn't have an intrinsic cytokine as a single translation molecule attention one bound to it and for cyber coins it turns out it's a family of molecules called jacks and these jacks are located on the header demerit chains again same motif as yesterday when the chains are separate these can't interact with each other nothing happens or whatever the cytokine comes it binds to both of these chains brings them together and now the two jacks can interact and as you can tell these two jacks are slightly different color which means that it's not just one jam is a family of different jack molecules and they could have different jacks on one chain different ones on different chain now they interact and what do you think they're doing to each other and to the chain what kind of what's happening what's process + correlation which is indicated by this little purple dots now that this phosphorylated it can be proved from the cytoplasm another family of transduction molecules called stands and now these stats get recruited to the cytokine receptor now these stats phosphorylate each other now that stazon phosphorylated they can form heterodimers and single chain stats are limited to being in the nucleus once you form a heterodimer they now can migrate into the nucleus where they combine to the appropriate target genes and turn on genes that's it Jack's nap fine now I mean chili logically why do you think cytokine signal transduction is so simple whereas a t-cell is so complicated well a quick response could also think about i how much cytokines do you think you have compared to how many mhz plus peptides you're going to see make cytokines is a lot of cytokine out there which is probably going to interact with a lot of receptors at the same time so that individual receptor problem doesn't have to do a lot of amplification but a t-cell receptor maling be activating a handful of molecules therefore it has to have a high level of amplification and that's why you got much more complicated system to give you that high level of amplification okay any questions okay question that seems to be anyone that doesn't yes one is I I suspect that there is one I actually don't know the answer and in je way at this point they know the answer either but I suspect if you looked an alert right now you would find the definitive answer because one thing you have to know is always a negative okay so now a question that you should have asked is how many side of Mines do you have you know to give me an exact answer do you have a few cytokines or a lot of cytokines a lot of cytokines well the question is you know how does it sell no I mean you have two molecules here families how do you I to give you one if that out for another effect aisle 12 another thing how do you get that specificity with this system of basically just two families of signal transduction molecules right is that a reasonable question well it turns out that there's a lot of stats and there's a lot of jabs and this is what provides the specificity for the immune response and it turns out that for for interleukin 12 its effect is driven by staff for and interleukin 4 in stat its effect is driven by staff because each one of these stats is going to recruit a different job and that jack is going to turn on a different family of genes so bound to the receptor for interleukin 12 is going to be staffed for down to the receptor for aisle 5 ISM of these stat 6 and that's what gives you the specificity the different Jack's different stats associated with a different side of contraceptive and that these jacks different jacquelina are going to bond the different of transcription factors and this transcript decides to turn on different so now if you put it into context of the th of the th1 response you basically have either looking 12 being produced and we in a pinch looking 12 of those as it turns on staff for and that's now going to drive th one differentiation entry looking for is going to bind to its receptor turn on status 6 and drive th to differentiation and years ago a student in my class took told me that an easy way of remembering it is he calls it the rule of 48 and what you basically do is do th one cell times 12 times 4 is 48 teens to sell times 4 times 6 so 4 times 2 is 8 8 times 6 is 48 so that's an easy way of remembering we're stat is th one and which that is is th to just remember rule of 48 and again the way Americans can remember that is up until Hawaii and Alaska join the Union I guess in the 50s there are only 48 states so that's how you kind of maybe remember the 48 ok the other way of remembering is just multiplied 12 times 4 that's 48 ok so so the immune response the temporal course basically shows you you can establish infection you induce the adaptive response and this indicates how much antigen is required to activate the immune system nobody antigen no infection no activation of the immune system the bacteria start replicating it comes in starts replicating greater energetic load now you induce the adaptive immune response you have the adaptive immune response get rid of the infection you clear it and ultimately the pathogens clear and now at this stage when you want to generate the muna logical memory you want to remember that you have been infected with this pathogen because the second third fourth times you want to mount a much more vigorous and wrap it in the response because you don't want to get infected and therefore the important in this system some effector T cells become memory t-cells and the way this happens is you have entered presenting cell t cor naive t-cells and Panthers antigen most activated T cells become the effector cells make sense and want to clear the infection doesn't do you any good to remember you've had the infection if you if you're killed but didn't amount to good immune response so you most of them become a factor but a small fraction of those become long-lived memory cells and I was seven out of fifteen apparently play a critical role in terms of allowing these memory cells to survive and they also continuously would like to be exposed to MHC plus peptide but it could get little mental stimulation from some peptide and these memory t-cells can live for decades they if you're infected a second time not only are they there but they're also hardwired to be a much more rapidly responsive cell where it's naive t-cells to antigen it takes a while for them to undergo of proliferation activation memory T cells have like this hair trigger they rapidly respond because their transduction pathways of rewire again explaining now why when you get infected the first time you found an immune response but you have a bad infection the second time you basically mount the rapid immune response you may get a very modeled infection and perhaps the third time you basically don't even notice that you're infected because you mount this very very rapid focused significant immune response that's why you pretty much get sick once from most infections and you don't ever get sick again okay any questions yeah good city for salaries ones that you still have a good immune system yeah it's a great question but the question is well the implications hard for example the cd4 cells go up well that that's great but on the weekend the question you have to ask is where are those cd4 T cells and is it if as you pointed out HIV is knocked out these memory T cells we all know that memories are not earned very easily you have to go through the whole process experientially and if HIV seems there are reports that have potentially targets memory T cells one reason may be that memory T cells may express higher levels of ccr5 for example and be more susceptible to being eliminated by HIV but now that means that if you're exposed to a pathogen that previously you would have mounted a very robust response because the memory t-cells are there you may now almost have to relearn immunity to a degree and therefore I'll take a while to get reconstituted so it's an excellent point just because the cd4 T cells go back up to previous levels does not mean qualitatively the immune response is equivalent to the way it was before for the sick point that you're making because you're wiping out these memory T cells so it turns out that there are two types of memory cells and again this is relevant to HIV because this has been addressed in terms of what memory cells may be preferentially wiped out an HIV infection and they're basically either central memory cells or affective memory cells and you basically have dissemination of the effector lymphocytes throughout the lymphoid system into God and so some of the memory t-cells may be directly generated after the infection occurs from naive t-cells another fraction of them may be derived from effector T cells that then can differentiate into memory cells for the two types of memory cells that you'll hear about are central memory cells and effector memory sells and the difference between the two depends upon their expression of ccr7 remember if you express ccr7 where are you going to migrated to ccr7 remember dendritic cells in the periphery and Langerhans cells see antigen now Express ccr7 where they migrated to lymph nodes so - if these memory if these memory cells Express ccr7 where are they going to migrate to - lymph nodes and that's why central memory cells Express ccr7 and they remain in the lymphoid tissue why because that's what antigen is going to come the next time you're infected that's what you want you memory cells to be better in addition we also want to have frontline memory cells that potentially can be rapid responders you don't have to worry about these cells being self reactive because they've already been appropriately activated so you don't like having these cells in the periphery because they're not gonna start attacking self because they're and injected specificities to a pathogen and therefore these cells lacks ccr7 they will therefore go home to the lymph node they're migrated to the tissues where they can therefore be frontline t-cells the next time you're infected another reason why you mount a much more rapid response to the second time your third time you're infected because you have these peripheral memory t-cells okay is that clear okay original antigenic sin is a very cool concept whenever you want to you know give something a good name right originally in the next sin so the idea is is that the first time that a person can be infected with the virus and this is really known very well for influenza the this flu virus has a large number of antigens and each antigen is indicated by a different color as you know the flu mutates frequently and therefore expresses different proteins so the next time a person is infected they'll be infected with a different strain of influenza and that different strain has different some different proteins and some have proteins as the original flu virus now you would think that the second time that you're infected you would make antibodies to all of the flu proteins right would that make sense it turns out that that's not the case your immune system basically says well I recognize that protein I recognize that protein from the previous infection and therefore only makes antibodies to the older four antigens that it's seen before it ignores the new antigens that's called the original antigenic sin namely immune response tends to respond to the antigen that it's seen previously and it ignores new antigens when they're presented at the same time well now it actually makes sense because what cells are responding to the old antigen what kind of cells what type memory cells they're fast they're rapid they're there so therefore it's going to happen quickly they're going to bind clear the antigen and probably prevent the whole long process of making new immune responses against the anthon is that you've never seen again you're infected twenty years later with another strain of flu your memory is so strong that you'll ignore the new antigens and you only make an Tobias against the one that you may have seen twenty years ago and the reason that's pretending that you were immunized with and this is a critical limitation of the immune system in terms of its inability to diversify beyond what it's seen earlier okay and this is just to show you M cells and again I mean the sons discusses this afternoon and mucosal lecture so I'm just going to just skip that and now basically summarize how do cd4 T cells or orchestrate responses appropriate to the infective pathogen by differentiating into the appropriate subtype making the appropriate cytokines how does cd4 positive T cell differentiation properly th1 th2 or th17 t reg depending upon positive cytokine signals that is specific for and negative signals that prevent differentiation and alter pathways and finally how do they versus attic lines transmit signals that mediate the inversion responses by using Jack's stats and different families of Jack's interacting with different families of stats that bind to different gene families okay so then have a good lunch I'll come back and we'll start talking about mucosal immunity yeah thanks a lot for your attention
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Medical_Lectures
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The_3_Rs_of_DNA_Molecules_to_Medicine.txt
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stanford university now i'd like to start this evening by introducing someone who couldn't be here with us last weekend who is the co-director for this course and that's dr sherry wren who's right here dr wren is well deserving of your applause she is a currently a professor of surgery an expert in oncologic surgery and chief of surgery at the va hospital of course which is affiliated with stanford medical school but beyond that she is a person of really quite significant renown in education she's won virtually every teaching award that we've given she's served as the director of the medical senate which is really quite an esteemed role and she's also nationally recognized for her work and surgery as a governor of the american college of surgeons and this year became an associate dean in academic affairs so we're really pleased to have sherry with us for the course thank you sherry and because this course has become so popular our speaker for next week jill helms jill why don't you stand up she couldn't wait to be here in fact she asked whether she could go first tonight but i told her no she'd have to wait till next week because our we're going to drill down tonight even to smaller entities than which she's going to speak about next week she'll talk about the fascinating area of stem cell biology and regenerative medicine but tonight we're going right down to the molecular basis of life and this is a really important topic one that has certainly changed as i indicated to you last week in just such remarkable ways over the last 40 to 50 years really with some of the most fundamental work taking place here at stanford just to remind you of what i said last week that began when the school was really founded here in 1959 with arthur kornberg's seminal investigations he won the nobel prize for his work and just two years ago his son roger kornberg also won the nobel prize to kind of book in that family and really catalog some of the extraordinary work in uh in the study of dna and its transcription uh and um as we think about uh the future of our topics tonight's session will be really i think quite essential for your knowledge base now our speaker tonight professor gilchu has a interesting history in his entree into medicine he started out at princeton university where he was a extraordinary student in physics actually went on and got his phd in physics at mit and then a light went off and he said i think i'd like to be a doctor and so he then went to harvard medical school where he completed his md degree uh trained in internal medicine at the mass general one of the most distinguished hospitals in the nation and then elected to move into the area of oncology came to stanford to do his fellowship in medical oncology join the faculty and has had a remarkably distinguished career since then both as a professor of medicine and as a professor of biochemistry he works in the area of dna repair damage related to both ionizer radiation ultraviolet light which may sound ethereal but is actually related to some disorders that fall into my own world of pediatric oncology where risks for uv light damage can actually lead to certain malignancies so he is someone who's leading the edge if you will in fundamental new knowledge and beyond that has also been one of our most distinguished teachers he is the person who leads the learning curve of our new medical students in the foundations of molecular medicine so i'm really pleased to have uh this evening for you and with you dr gilchu so thank you phil for that very nice introduction um we have a lot to cover the three r's we're going to do tonight usually that takes six years in elementary school and i'm going to try to do it in less than an hour so that i can get into applications of the three 3rs because you don't want to just learn about the 3rs and we're going to go all the way from molecules to medicine this is in braun auditorium that's the periodic table and i've got my molecules here in my periodic table here so if you forget i'm wearing the tie so the way this talk is organized and i don't think there's a laser pointer but i may not really need it but it's okay three r's are replication recombination and repair and i'll tell you about them and then i'll tell you about a couple of very short applications using recombinant dna and new drugs and then implications implications of the three r's the application of the technology to to new drugs and implications that has for our healthcare system which you've probably heard a lot about lately so we're going to go right into the three r's replication copies dna recombination rearranges dna and repair fixes dna and i'm going to start talking about some complicated things but i just want you to know what replication does is it like on the starship enterprise that replicated food and what replication does is it makes an exact copy of dna just as on the starship enterprise now um when we start getting more complicated don't hesitate to raise your hand and interrupt me because if i've skipped a step i i don't want to keep moving on i want you to interrupt me and and ask a question okay if you have a philosophical question we can wait till the end all right now phil mentioned arthur kornberg so here's a picture of arthur he he actually founded one of my two departments the biochemistry department by bringing it lock stock and barrel from st louis and in 1957 he was able to reconstitute the process of dna replication in a test tube the headlines at the time says said stan professor reproduces life in the test tube not quite right but he reproduced a really key step in life and he found that the re reaction only required a few components which i've listed here in red a dna polymerase which copies the dna a template dna which is the dna to be copied primer dna to get the dna polymerase started just the way you prime a pump that's why it's called primer dna and nucleotide bases which are incorporated into the dna and you all know that there are four bases which are basically the code for life and how they're organized in the dna tells the instructs the cell on what proteins to make and how to make them all right so basically what you have is you start out with a dna molecule and if the cell wants to divide it has to make a new copy and this is an electron microscopic image of the replication of the e coli genome e coli there are more genomes of e coli in you than your own genome uh because they're gazillion e coli sitting in your gut and so they're very handy we don't have to dig very far to find them and that that dna is being replicated and these things are replication forks which are marked by those arrows and those are the areas in which new dna is being synthesized and those forks are moving outward until you finally end up with two circles and then the two circles will go into two daughter e coli bacterium bacteria when they divide when it divides okay same thing with your cells so here's where it gets a little complicated but i need to give you a little bit of a picture because i have to show you the replication fork in order to show you some of the applications later so replication differs there are two strands of dna okay it forms a double helix and the dna actually has a directionality to it and we call it five prime to three prime so you'll notice that the dna molecules are anti-parallel to each other they each have a five prime to three prime direction but the two dna strands are going in opposite directions so the five prime and three prime ends of the two molecules are are next to each other up there all right now it turns out dna polymerase is kind of dumb it only goes in one direction five prime to three prime it doesn't go backwards there are actually very deep reasons probably that why it doesn't go backwards but it's very important for it not to go backwards okay now since the two dna strands are anti-parallel if you're going to copy that dna at a replication fork one of the strands is very easy because the the on the upper strand you're you're copying the yellow dna into new red dna and because you're copying it in an anti-parallel direction the dna that's being synthesized is going five prime to three prime opposite to the template dna strand the old dna strand so that that polymerase keeps going in a continuous way and then displaces the replication fork to the left on the other hand so from the left-hand picture to the right-hand picture you can see the replication fork moving to the left my left your right okay going that way all right now because the replication fork is going that away and because the lagging strand the other strand of dna is in the antiparallel direction and because dna polymerase can only go five prime to three prime it can only go backwards away from the replication fork so what it does is it goes backwards for a little bit and then it hops towards the replication fork and goes backwards again creating very short fragments which were discovered by a molecular biologist in japan named okazaki so they're called short okazaki fragments and because it's going backwards each of those okazaki fragments can then be synthesized to meet so on the lagging strand an okazaki fragment gets synthesized until it bumps into the previous okazaki fragment and then a ligase a dna ligase joins the two dna ends are you with me so far okay so yeah go ahead if it has two sides a five prime and a three prime if it's taking a one that's going from three prime to five prime and making one that's five prime to three prime it's making it backwards that's right how is it replicating things that hasn't gotten you yet it's not it's gonna it's only gonna get to so in the top strand that one's easy right that's going five prime to three prime and you notice that the new dna that's being synthesized will therefore be anti-parallel to the dna it's copying so the red strand on the upper fork is going in the opposite direction from the yellow strand on the upper fork and that's going to keep going displacing the fork so the fork is going to get displaced and unwound it's the same facing the other way okay so the question is is it the same basis but facing the up other way it's quote copying the dna but it's not copying it as an exact copy what it's doing is it's it's every time it sees the base a it puts in a t every time it sees the base g it puts in a c so it puts in a what's called a complementary base a's and t's are complementary to each other g's and c's are complementary to each other so the code is actually got a code of g c a t but the other strand has a code that's kind of the bizarro mirror image of it it's not the exact image but it's a complimentary image okay did you ever read superman comics so superman was you know the night for all good but there was a bizarro world which was a copy of superman's world and there was a bizarro superman who kind of looked like superman but wasn't quite the same it's that's what this is you're putting in complementary bases of dna so so so the con since you're always putting in the complementary bases of dna when you copy that other strand again you just put in the complement to the complement which restores the old strand so as long as you're putting in the complement each time you've always got all the information there oh thank you is that is are you okay with that the superman thing worked okay no no actually there's a big there's a big machine called the replisome and actually you know many how many people here sew mostly women i'm looking not not a whole they're not a whole lot of men are raising their hands but the way a sewing machine works is this strand is working like a sewing machine so it's it's going backwards and then it goes this way and then goes this way right just like a sewing machine so there's a polymerase that's doing this job and there's another polymerase that's doing that job but they're all part of a big machine so the whole thing is being held together right at this fork and then this fork is moving is that okay okay yeah i forgot to repeat the question but let me try to remember the fragments that that it works in what's the length of the fragments in which it works yeah no this leading strand the question was how long does it do the replication yeah on the leading strand on the lagging strand okay so i'll start on the leading screen the leading strand is really long it just goes and goes and goes okay in fact there's a sliding clamp that keeps it on and the sliding clamp looks like a clamp it's like a circle on the lagging strand what it'll do is it'll depending on the what organism you are if you're a bacterium or a human it's slightly different so it's a few hundred to many hundreds of of bases so like a sewing machine the length of the stitch depends on the species but it's fairly short a few hundred just to give you an idea the human genome has six billion base base pairs so it's just little snippets uh-huh we're okay uh-huh how long will it be before it actually becomes just like two molecules yeah so in e coli in the bacteria in our guts they can divide in about 20 minutes takes us a little longer it takes us a little longer so the question was how long does this process take and and e coli can do it their their dna genome is only five million base pairs they start at one spot and they go we've got 23 chromosomes so we we obviously are you know 46 but we we uh we every chromosome starts independently but because our genome is so large we actually initiate replication at many spots on each chromosome so we can get the job done in a reasonable amount of time okay so let's move on that was pretty tough actually i think i'm hoping that's the toughest um okay now i've tried to make it simple we're moving on to recombination that rearranges the dna so those of you who don't know what rearrangement is you remember mr potato head and this is mr potato head optimash prime do you re do you watch those kind of movies my teenage son does but anyway transformers okay okay so anyway this is now optimash prime you've got a mustache here i don't know what this is some kind of a nose a different different thing out for the arms i don't know may maybe a beard i know i don't know but you can make all kinds of mr potato heads and dna recombination is a way of rearranging the dna to make new pieces okay it turns out that that kind of stuff is going on all the time you probably thought that you were born with a fixed complement of dna and your future was set but i want to disabuse you of that notion you are actually being bombarded by changes all the time your dna is constantly in flux by this mr potato head-like system so you need to hear about recombination okay now there are two types of recombination that and one is called homologous recombination no this is going to be a hard one homologous recombination but we'll get you through it and the other one is site-specific recombination so here goes watch homologous recombination it's a mess first of all i have to tell you what the word homologous means dna molecules are homologous if they have similar dna sequences two identical twins when they're born before mutations occur and before all kinds of recombination occurs are 100 homologous that's a night that's what an identical twin is certainly you know in utero there are 100 homologous now men and women are 97 homologous chimps and humans male chimp and a male human are 97 homologous so that explains a lot but i think you got the concept they're pretty similar but there's some important differences okay all right now this is the hard thing ha what a mess that is homologous recombination repairs dna double strand breaks so i started out with two dna molecules that were intact one was a double-stranded dna molecule that was yellow and one was a double-stranded dna molecule that was red okay now the yellow molecule developed a double strand break now a double strand break is a disaster because that's a broken chromosome and the cell will die unless it can fix it so what homologous recombination does is the yellow dna double-stranded dna molecule was the broken one right and so what it does is when it's broken what often happens is some of the stuff at the ends gets all screwed up okay and in fact the information gets messed up so what it does is it says aha i can find if i can find a nearby dna molecule that's homologous i will use that to do my repair okay so what it does is it does something called strand invasion it takes the three prime remember i told you about five prime and three prime ends it takes the three prime end of this dna molecule and it invades the homologous dna now remember the dna molecules that the two dna strands were complementary to each other so they will actually hybridize to each other by hydrogen bonding okay hybridized hybridized means that the two dna molecules will stick together because it turns out that there are nice hydrogen bonds between the g nucleotide and the t c nucleotide that cause them to hydrogen bond and stick together and there are hydrogen bonds between a and t nucleotides that cause them to stick so the dna molecules will actually zip up because the complementary bases want to stick together via hydrogen bonds that guy i can't teach you all of chemistry but i think you remember some of that okay these are not covalent bonds but but there are they are bonds that nevertheless will cause two molecules to stick together so these each each double-stranded dna molecule wants to stick to itself but under the conditions of homologous recombination you displace one of the strands to allow this broken dna molecule to strand invade once it strands invades then dna polymerase remember that guy can actually copy this dna so if the double strand break was created where you lost a few bases at the ends okay you can recover them by copying the missing bases off of the other dna strand now you know i'm not real good at fixing things unless i have a copy to work from so when i start doing plumbing and i forgot how i messed things up i go to the other bathroom and i check it out and then i copy what happened i've got a bathroom in the house with two sinks right next to each other that's very handy so this is what's going on here okay then after you do this the other side of the brake the other side of the break actually can copy off of the display strand all right this then what you can do is you can hop back and you get something called a double holiday junction which you'll notice has two l's in it so it's not a vacation it's it's named after a british biologist robin holliday and what happens is these junctions actually get resolved by nicking the dna in either one of two ways and the dna strands are cut at the arrows and then you end up with two intact chromosomes chromosomes with something called crossover and something without now you don't have to remember that for later but you end up now either depe depending on how you cut the dna it doesn't really matter you end up with two intact chromosomes now some some information quote has been exchanged so if you're copying a dna molecule that's 99 the same you'll copy the mistakes the one percent of the mistakes okay you okay with this i think this is the second hardest slide questions yeah you did i think getting started is the hardest what causes the strand break in the first place is the question just being alive all the time you're making oxygen free radicals while you're metabolizing you're at body temperature right now and oxygen free radicals are coming in and they are they are making strand breaks in your dna and when they make two strand breaks that are near each other your dna falls apart the other thing is cosmic rays are coming in and getting you okay or you went to the dentist i went to the dentist yesterday and he wanted 14 x-rays bang x-rays are especially good at making double-strand breaks so just being around causes this to happen so this better be working or you're in trouble uh-huh is it only ionizing the ring rf energy yeah no well you know in terms of the electromagnetic spectrum you actually need enough energy high enough energy and tuned enough to break this to break the strands of dna ultraviolet radiation for example will not do it ionizing radiation is tuned just right to break it but there are other things that do it plants for example make a lot of poisons and they have drugs that will actually create double strand breaks in oncology we harvest those drugs and give them to our patients not because we don't like our patients but because they happen to kill cancer cells just a little bit before they kill the patient and so they have some benefit uh for cancer chemotherapy and some of our most important cancer chemotherapy drugs make double strand breaks so lots of ways of making double strand breaks it turns out your immune system in order to to generate immunological diversity makes double strand breaks on purpose to generate huge huge numbers of different kinds of antibodies so this is just happening at it uh-huh where's the spare dna come from this process only operates only operates during replication where did the spare dna come from all i have to do is go back to the replication fork because we spent so much time on this if there's a double strand break over here there's the spare dna right on the other side of the replication fork now you're going to ask me another question what's the other question remember this is always happening when replication ain't going on right when replication isn't going on there is another machinery that actually repairs the brakes which i'm not going to tell you about it happens to be what i'm i i dearly love because it's one of the things i i spend all my time thinking about in the laboratory but i'll i'll spare you the pain it's actually quite simple it just jams the dna ends together um and it's re if it doesn't work you don't have an immune system so great question okay any other questions i think i i think most of you are coming along do you understand the questions okay yeah no yeah well what happens is this only happens during the replication right and the strand invasion only occurs between homologous dna strands ooh what percentage homology do you have to have it's got to be at least 95 percent or so otherwise it gets it it's it's shades of gray if it's 100 homology the strand invasion is very efficient if it's 98 it's pretty efficient i think you know somewhere around 90 percent it just doesn't happen very often but you're asking a really good question because maybe there's someplace else on this newly replicated dna that it could hop over to and sometimes it does and when it does you could have a deletion so think mistakes happen mistakes happen that's why you're he's worried about errors he's asking you know how homologous do they have to be right very good now replication occurs replication occurs with very high fidelity one error in 10 to the seven bases so if you do homogenous recombination right here at the replication fork very few errors but it turns out that homologous recombination also operates during meiosis when you're making egg and sperm and there it occurs between different chromosomes and there's where we get all the diversity i see in front of me although i don't think you're as diverse as our medical school class no you're you're definitely not but we'll work on that okay but but but but this thing is can operate when it's not precise and there's a special machinery for making it work during meiosis okay so we're we're good with this now yes the replication machine when it has an a it needs to make a c is it on the other end uh it happens to be a t but it's not important yeah yeah so the question was what about this replication machine um it's coming along and it sees an a and it needs to copy the a and the a is going to be copied into a t but remember what arthur cornberg did he found out that you have to have a's t's g's and c's into in the reaction so those things are just floating around in the cell and so when it hits an a at this point it's going to stick in a t there because they're floating around there are t's floating around is that okay yeah all four nucleotides have to be there so so it'll just grab one and bring it in so that part was easy for it to do it it doesn't have to search okay good aha you mentioned an a or a t there's no indication on the graph where you wonder what about the c and the g are they yeah yeah so just imagine you know there's some sequence here and the sequence could be g c a t whatever and they're all in there yeah and then when it hits a g when it sees a g here it goes ah i have to put in a c and then oh i didn't repeat the question he says what did did you get the question from my answer so he's he's saying what's going on here basically what's in this sequence i don't see a gca that's because i left it out um but there are g c a t g g t t t whatever okay and when the polymerase hits the g here it'll put a c opposite and then it'll move a little bit forward and stick in another nucleotide and so on great aha what is the speed at which the breaking and joining is happening and does the speed uh change your age this the speed of the repair or the oh oh how fast does how fast does this repair occur i actually i i don't know somebody may know i don't that's not a well-defined number i know for the pathway that i'm interested in non-homologous and joining it's over the span of hours so for homologous recombination my guess is it's probably of that order so it takes it the question was how long does it take to do this repair it takes a couple hours sometimes and so there have to be all kinds of signals to tell a cell hold it don't do anything yet i've got to fix things okay how long does the break how long does it take to make a break well you know how long does it take to break something they know the answer you know cosmic rig bang it's done right so it's always easy to break things yeah yeah so so i think i think your real question was how often do the breaks occur in my body that's your real question what's the frequency of breaks it's happening all the time yeah yeah it's it's happening all the time i'll i'll i'll get it some really hard numbers it'll really scare you uh is the repair as vital in a young person as it is in an old person well i'm older so it's more vital to me but um i think it's equally vital really it turns out that if you don't repair things properly mutations occur and if you build up enough mutations you get cancer and on average you have to mutate about six very important genes in order to get cancer and as you age you sort of built each one of your cells builds up a few mutations and one of them if one of them happens to get the magical six it can turn into a cancer cell so if you look at the incidence of cancer as a function of time it goes up as the sixth power of age whoa t to the sixth power that's why old people get cancer and young people don't but the cancers that young people get are the cancers that only need one hit so there are a few rare cancers that just need that one hit but most of the common ones breast colon prostate cancer pancreatic cancer those need on average about six hits uh-huh how long can it be broken for this mechanism to still work how long how long can the dna be broken for this mechanism to work well if the dna is broken and the replication fork gets too far away then you have to pull in the backup pathway the the non-homologous and joining pathway that it's not homologous because it isn't homologous recombination so we call it non-homologous and joining it'll try to do it yeah it'll detect it'll see the break all kinds of signals will go out ah broken right and then and then it'll try to bring in all the machinery to repair the break lucky for you yeah now here's here's some more machinery this is going to be important later several proteins promote strand exchange remember i told you that in order to have this strand exchange you have to take a dna molecule that's already zipped up and somehow unzip it to let that broken piece of dna squeeze in right remember that so what happens is that broken dna actually this is one of those things where you take a few steps back in order to go forwards the broken dna actually gets resected leaving a three prime overhang you actually remove some pretty good material and then that leaves a single stranded piece of dna and that's a single binding protein called rpa replication protein a and that coats the single strand of dna that's really to protect it temporarily so so uh that just coats it to protect it and then two proteins rad 51 and brica ii brica 2 might have a familiar ring to it because brica 2 stands for breast cancer 2 gene so we're very interested in that rad51 stands for radiation radiation because it was first discovered in yeast and yeast that were mutated in that gene were especially sensitive to ionizing radiation which makes double strand breaks so that's why it's named rad51 now those two proteins make it to this junction between double-stranded and single-stranded dna and what they do is they see this protected piece of single-stranded dna and they load on and and and they cooperate so that more and more rad51 molecules get recruited to make a filament on the single strand of dna and it is that filament that actually makes this especially good at doing strand invasion to create what we call this d loop it's called a d loop because biologists okay are really cockeyed and sometimes they walk around like that and there's a d there okay so so these filaments allow that d loop to form and then because that's a reaction that doesn't want to happen all by itself okay and then now you can copy and repair your break and so on and then you're free free and clear all right now it turns out that brica 2 is loading rad51 on okay so it's just ushering the rad51 onto the single strand of dna but brica two has to have some help to get where it needs to go and guess what brca1 helps brica 2 get there brc a1 has a familiar ring it's also involved in breast cancer okay and brica 1 guides bricka 2 to the dna end some of this we just discovered last year but it's so neat you have to hear about it because it all makes sense now right okay not not me but they discovered last year okay now i want to show you a filament because i promised you molecules to medicine here are the filaments this is what it looks like the rad 51 molecules are wrapped around the dna and now you can see the wrapping can you see the wrapping with it you know the lights are shining on me but i think you should see the wrapping which looks like that wrapping around the dna so that's the active ingredient that causes single strand invasion all right now bricawan and brica-2 mutations cause breast and ovarian cancer affected individuals carry one mutant and one normal allele for brick one or bricka two so one of their two chromosomes has a mutation the other one doesn't but if you get a mutation of the second allele in bret in a breast cell it disrupts homogeneous recombination because you can't do the you can't load rad51 on anymore and you can't do the strand invasion so homologous recombination gets gets shut down now because you don't do homogenous recombination you don't repair double strand breaks as efficiently as you once did and as a result you develop more and more mutations and it's it's it's easier to get those six mutations that you need to become a cancer cell because you have a repair system that's been knocked out so women who carry these mutations will get breast cancer in their twenties instead of in their 50s and 60s it takes much less time because they've knocked out this repair machinery you okay with this yeah allele what does the word allele mean i should kick myself that i i apologize allele just means that i have i have two chromosome sixes chromosome six chromosome six therefore i have i forget which chromosome brica brca1 is on but br there's one gene on this chromosome brca1 gene on this chromosome and one brca1 gene on this chromosome okay those are the two alleles so allele is a fancy word for saying i've got two copies of every gene with a few exceptions right we men lack two copies of some very important genes women on the other hand this may be an explanation for their superiority have two copies because they have two x chromosomes right we only have one copy but except for the x and y chromosomes we have two copies of every gene the unfortunate nature of the fact that we have only one x chromosome means we're susceptible to a lot of diseases because we don't have a backup copy so that's why that's basically the root cause of our inferiority on the other hand that's why we have to subjugate women for centuries in order you know but they're getting even now so so so on the other hand we have a y chromosome and the y chromosome is what causes us to become men so that encodes genes for pro football and things like that so you know those testosterone related things okay any other questions but thank you for allele that was very important uh-huh so if somebody gets tested for brca they already have it in their body but they don't have the mute if they're negative oh yeah so what does it mean to be for a woman to be tested for brca1 or brca2 right what it means is that you're going to look at her normal cells you really don't know if she has a breast cancer brewing right you're just going to take her normal cells and you're going to see if one of the two alleles has a mutation in either one of those genes now those genes are inherited so that woman got one of those mutant genes from either mom or dad and sister or brother also could be carrying one copy of those mutant genes yes and so as a result if you find one of those mutant genes it tells you that the process of becoming a breast cancer has been foreshortened and you're at very high risk for getting breast cancer so if you turn out to be a carrier for a mutation in brick of one or bricka two you either are going to say to yourself i have to be very careful and have very careful surveillance for breast cancer or i might decide to have radical surgery to remove the offending organs either the breasts and or the ovaries so i can't get breast or ovarian cancer so that's a very difficult choice some women have to make okay now how do you how do you do surveillance this all comes around right how do you do surveillance for breast cancer what do you use a mammogram what does a mammogram use ionizing radiation what do ionizing radiation gamma rays do they make double strand breaks which are repaired by the homogeneous recombination which is encoded by brick of one and brick you know which ricoh and briquette are involved in so some of the machinery or some of the devices we have boy you guys have learned fast you now realize what a problem we have so we're sort of on occam's razor here uh-huh so if this is the way all humans repair their cells why does breast cancer seem to be more prevalent in women oh what a great question so all the cells repair dna the same way with this machinery all of our cells are using this machinery so how come men women get stuck with breast cancer that seems unfair now that that question is actually a really deep question it almost falls into the realm of philosophical discussion because the answer is we don't know we do know that you can get ovarian cancer so it's not just breast cancer but then why yeah that's just women too why not every other cancer why not we just don't know we don't really understand that's all i can tell you now it turns out men who have mutations in brico 1 or bricka 2 can sometimes get breast cancer they can't get ovarian cancer obviously but they can't get breast cancer so that does happen but we really don't know why why is disruption of this particular dna repair pathway so important for the breast and relatively less important for other tissues i know no one knows the answer to that uh-huh something is actually protecting them from breast cancer um uh what's yeah the question is are men being protected from breast cancer well i think we men are being protected from breast cancer because we have a lot less at risk we don't carry around breasts and also breast cancers actually are estrogen dependent and we're not making estrogen so i think it's not got to do with the y chromosome it's got more to do with the fact that we we have less breast tissue and we have we don't have estrogen hanging around that are stimulating the breast cancer cells to divide but but there's enough around so that men who have this mutation can actually get breast cancer breast cancer does occur in in men but it's mostly because we don't make estrogen i think and we treat breast cancer with anti-estrogens so all right now i'm going to tell you about site-specific recombination retroviral integration is one example and i'm going to tell you that because retroviruses oh what's the word retrovirus mean retrovirus retrovirus refers to the fact that these viruses have don't have dna in their genome they have rna in their genome and in order to copy their rna they actually copy their rna into dna which is kind of backwards because usually dna gets transcribed into rna to make proteins so retroviruses do it backwards so they're called retroviruses okay they cause human disease hiv causes aids you all know that hepatitis b i bet you didn't know was a retrovirus but it actually goes through an rna to dna copying mechanism and it causes hepatitis and liver cancer and retroviruses undergo recombination with the dna of the human genome you may not have known that retroviruses are hopping into your dna so this is what happens this is the retroviral dna it's already copied itself it started out as an rna molecule but it made itself into a double-stranded dna molecule by doing its reverse transcription and this blue here is the target the human chromosome and this viral dna the viru the virus carries with it an enzyme called integrase it's the name integrase stands for its ability to integrate the viral dna into our genome so that's why it's called integrase and it basically attacks us by sticking itself right smack into the middle of our genomes basically at random sites and then from there it hijacks our machinery to make more virus particles and to move on with its life at our expense okay just an example of site-specific recombination that is mr potato head except it does it's an unattractive mustache that's being stuck on yes yes they do it's a special dna polymerase that instead of copying dna the question was do do the retroviruses carry their own special machinery to accomplish this strange maneuver and the answer is yes they carry the enzymes in the little virion that a site particles say a pharmaceutical maker might target and yeah and because the retrovirus is somewhat different he's saying because it carries its own special enzymes it's integrase and its own special dna polymerase is that something that a pharmaceutical manufacturer would target and the answer is yes the first hiv drug that was ever developed was called azito thymidine azt and it is a nucleotide a fake nucleotide that actually fools the retroviral polymerase much more than it fools our polymerase so that's why it worked it didn't work to it has side effects because it's not perfect because our polymerases also use it to some extent but it's the other primaries the ritual viral polymerase likes it much more so exactly right if you understand the differences at the molecular level you could be an entrepreneur just like our friend here who could think of a way to cure aids so that's why we need to know this stuff okay finally repair okay that's what fixes dna we already talked about one process of repair because it's basically a recombination is recombinational repair i am going to tell you that dna repair is very important because dna damage is unavoidable the human genome is six billion base pairs counting all the chromosomes they're 0.3 for a third of a nanometer per base pair now if you multiply six billion times a third of a nanometer you get two meters so every cell in your body has dna that would basically be more than most of your height okay now your body on the other hand has 10 to the 13 cells so you have to multiply 2 meters times 10 to the 13. that's 2 times 10 to the 13 meters which means nothing to you so i recalculated it in terms of solar system diameters 50 solar system diameters of dna in in your little body and damage in any one part of that can eventually lead to cancer that's a lot to protect so this is very important and also you want to know because dna is damaged by many anti-cancer drugs and of course no surprise to you many anti-cancer drugs can cause something what do they cause cancer so we take a patient with hodgkin's disease treat them with these wonderful drugs and we say you're cured and then five years later they get another cancer sometimes at a rate of about five percent it's usually leukemia and so actually i think phil ended up taking care of them so i basically gave him my business five percent of the time so that's a very unfortunate thing we do need better things then than this okay so there are three major classes of dna damage and each class has many mechanisms and i don't have time to tell you about all of them i am only going to tell you about one and i'm going to tell you about this one because it's going to really scare you it's called base loss these bases are attached to the dna these are the bases that hold the code they're the g's c's a's and t's they do all the coding okay now these things believe it or not fall off about 5 000 of them fall off of the dna in a given cell per day so you have all this dna it's pretty long it's 50 solar system diameters so from that point of view 5 000 isn't a whole lot but 5000 is a pretty high number and so there has to be a repair system to deal with those five thousand okay and i'm going to just tell you about that repair system because then you'll be really caught up about an article that was published last month in the journal of medicine if you know about this so base excision repair deals with this see here's a piece of dna look at that gctatc so this is something somewhere in your genome right and look at that all the bases are little colored blocks and there's one colored block there that's missing dna helix with a missing base 5000 of those bases fell off that one cell in one day so that spontaneous loss of a base creates an apurinic or a pyrimidinic site that's an ap site now the bases fall into two classes purines or pyrimidines so biologists came up with a fancy name which is very it should be just an a basic site but i i'm going to call it an ap site because that's what biologists call it just you know pretend you can pretend i never said it but base excision repair restores the loss basis so there's an ap endonuclease that recognizes that ap site the site without the base okay and it places a nick there and so you have a dna helix with a little gap now if you have a dna helix with a little gap you know how to repair gaps because you've got a template strand to copy from and dna polymerase can come in and copy and now opposite this g goes a c you okay so this whole course the whole lecture is starting to come together right okay okay and now you're repaired so you're safe now there is a protein called parp i'll tell you in a minute what parp stands for it binds to the dna gap to make this process more efficient you can imagine that you notice in brico 1 and bricka 2 you had you had a protein that did the job but then you had another protein to help it do it faster that's because this stuff is so important you got to go chop chop as fast as possible so parp accelerates this and this has got to go really fast because this is happening at the rate of 5000 per day now double strand breaks only you're only getting a few in your cell per day but this is several thousand per day okay so what's parp parp recruits repair proteins to the dna single strand breaks and it stands for poly adp ribose polymerase adp is like the brother of atp that's that's the energy molecule well it turns out adp isn't just for energy what it can do is it can be constructed to make big polymers of itself so parp actually binds to the dna gap and then it becomes activated and then it attaches it adp ribose polymers to itself and to other proteins it makes a cloud of adp polymers and we don't exactly know why it's doing this but we do know that its ability to do this is very important for recruiting the dna polymerase and ligase to the single strand break to do the repair so think of it as slime these adp polymers are just sort of hanging out there they're very unstructured and i think what's going on is it's like velcro just their slime just stuck out there so that these very important enzymes find their way there and do the job very quickly okay i mainly had to tell you about that so you knew what part meant but it's kind of cool anyway now homologous recombination saves part 1 deficient cells people were trying to figure out what part 1 did so they made what's called a knockout mouse you can do genetics on a mouse and take a gene and wipe it out and say if i wipe out both copies or both alleles of the part one gene just make it totally part one deficient what happens to the mouse it turned out the mouse was perfectly healthy so i didn't pay attention to the parp literature i said to myself how important could it be a mouse can live just fine without it and then some important things started happening i started having to learn about it but it turns out that the reason the part 1 mouse is perfectly happy is because homologous recombination is a backup for part deficiency so now we're going back to the replication fork see it's only bigger now and suppose i have defective base excision repair there's a gap there that isn't going to get filled very quickly because i've got a part deficient mouse for example look at this replication fork it's moving to your right it's headed right for there okay so these are railroad tracks and one of the two tracks has a gap and this is the train this is a big machine here which is like a locomotive going to the right very quickly okay so what happens the migration of the replication fork converts the single strand break into a double strand break see now if i if this break is here now if i start and do lagging strand synthesis i can't copy this thing right there's going to be a gap polymerase can't copy across the gap but since this is a new replication this is newly synthesized dna this is exactly the scenario for which homologous recombination was designed because you have a double strand break here you can do homologous recombination on the other strand all right to repair this break remember homologous recombination where we had the d loop and then we copied the missing material and then we restored the dna and then we're fine so that's why the part deficient mouse was perfectly fine part deficient mouse was okay not because parp was not important but because parp is so important there's a backup system stupid me i should have realized right how could such a cool enzyme that was doing such cool things not be important it was because it was so important are you okay with me on this this is pretty hard okay i'm gonna have to give you a break soon so homologous recombination repairs the double strand break now here's the kicker this is why you want to learn about this people have developed drugs that inhibit parp1 and it turns out they kill bricawan and brica-2 breast cancer cells july 9th 2009 whoa now you can imagine part 1 inhibitors how about parb1 and hip now brick one and brick of two breast cancers only form five percent of all breast cancers what about the other 95 percent what if i treated them with parp1 inhibitors what do you think would happen i don't hear nothing remember the purple and deficient mouse perfectly happy because he had a homologous recombination actually she had a homologous recombination so the other breast cancers this isn't going to help right but for the five percent of breast cancers that have homologous recombination knocked out this is going to be really great will this drug have side effects well it could it could but it works only when brica one is knocked out completely both alleles both copies of brick brica1 are knocked out then you don't have homologous recombination what if a cell has one copy knocked out and one copy good that cell still does homologous recombination so you've got a drug that knocks off the cancer cell where both copies are knocked out but for a normal cell where one of the two copies is still good it's not going to be very harsh yes so the question is if somebody takes this drug what's happening to that person's repair while the drug is being taken and the answer is if you took that drug you would be back here and you would all your normal cells would have little gaps in them in their dna right but not to worry your normal cells do homologous recombination and your normal cells would repair that when the replication fork moves forward right and then it repairs that break so you're taking it and you are shutting down basic excision repair but lucky for you you've got homologous recombination even if you happen to be i don't know what your genotype is but even if you happen to be brick one or bricka two mutant you've got one normal copy and that's enough to make homologous recombination go so that's a great question that's asking the question about side effects a different way uh-huh so that's a really cool basic science result but that's a really cool basic science result right so you know okay so now you've got maybe some presumably there's some carbon inhibitor that they use to do the study but you know suppose somebody says okay well let's you know let's move down the road to make this into a treatment yes what kind of time frame oh no it's about to get approved by the fda it this drug this is oh let me let me show you when something gets into here then this is called the new england journal of medicine this was a clinical trial where they treated women with breast cancer this is coming very soon i don't know if it's does anybody know if it's fda approved already huh this was a phase two trial that was reported that's cheap yeah this is so so anyway this is going to be approved very quickly because in the phase 2 trial the women who got treated had hardly any side effects how long ago did we know the science how long did it take to get to this point you know i was ignoring parp literature i'm a dna repair guy and i was ignoring it but you know there were people working on i think we knew about parp and basic excision repair maybe five or ten ten years ago it's been kind of perking along for the last ten years yeah yeah but you know we didn't fully realize what was going on for a while and some of this machinery was getting put together from different angles people were studying homologous recombination other people were studying basic scission repair so and all of it came together i think this is beautiful and i think the reason i need to tell you this is you just saw how the three r's come together to make a drug that has no side effects that can take these very unfortunate women with a terrible cancer and treat it all but you know that's pretty good so what company should we buy i forget i forget i i actually do forget i i but you could look it up you know you could look you could look it up because these these slides will be online and all you have to do is do part one or you could do new english journal events in july 9 2009 in part one and on google it and you'll get the article instantly okay yeah it's too late to buy the stock obviously okay applications recombinant dna and new drugs do you want a five minute break no i you know i'm wearing a foley catheter so i not really but but if somebody okay i a lot of women were saying no but you know some of the men might want to go if anybody if you want to go i won't i won't be hurt where are the bathrooms up here to the right or left just outside so if you want to go i won't be hurt uh-huh it'll be good it'll be fine i always give the medical students a break after 50 minutes and then if if one of my co-lecturers don't give a break i go in and yell at them okay so we're going to do applications recombinant dna and new drugs okay so history i showed you arthur kornberg 1957 1973 the invention of recombinant dna and i'm showing you these characters because you might bump into one of them if you're on campus here this is stan cohen who has a lab just across campus drive in the main right two floors above dean pisa okay so uh he's he's still there uh stan falco he's he's emeritus now but i see him on campus all the time her boyer i don't see him because he was at ucsf and jenna texts her boy or he left ucsf for genentech the boom in genetic engineering prince charles picks a bride so this this gives you a time frame for where we were and paul berg still comes to work every day so you'll you might bump into him he's this is the beckman center right across the way so these three guys with the red arrows had a midnight brainstorming session at a scientific meeting uh at around midnight and they were hungry and they were looking for corned beef so they found on maui a new york style deli and they started talking about plasmid dna these are little bits of dna that bacteria have in them that actually carry antibiotic resistance for example and they had a brainstorming session where they thought we could maybe use these plasmids which are the bane of antibiotics and manipulate them to do some good for ourselves make them into a tool so they they started having ideas about that paul berg meanwhile was working hard at stanford not on maui doing experiments with the sv-40 virus and he in building m which is also at the time one floor above dean piezo's office and what happened was they came up with the idea of recombinant dna and so the outcome was stan and herb patented the idea that turned out to be the biggest patent that stanford and ucsf i think ever the most the most lucrative patent they ever had uh so stanford and stan and herb also shared in the wealth paul won the nobel prize and you know someone had to pay for the dinner on maui stan paid for the dinner on maui so i'll tell you an interesting story when i was a first appointed to be a professor here they had this idea that they would hook us up with a senior faculty member who could be our mentor and i i was really lucky i got assigned stan and so stan took me out to lunch and i asked him if he would please pick up the check i can't remember what happened but i it didn't work so anyway he actually he actually in many ways deserves one of those independently of and so the alaska award was actually recently awarded to him and the lasker award is designed to try to make up for oversights that stockholm had but you know and stockholm can always make up make up for the problem because stan has done so many other things that they may choose to recognize them anyway for other things that he's done anyway so that's that story the other part of the story is these are congressional hearings because people feared what recombinant dna could do they feared that some bioterrorist could create an organism that could wipe out the world and so what ended up happening was paul was the leader in recombinant dna and paul decided that it was be a politically good thing if scientists put a halt onto on their own research before congress got into the act because one thing you don't want congress to do is to try to figure out how to regulate something so paul called a conference at a cilamar down the coast near monterey and all of the leaders in this field decided that they would have a self-imposed moratorium until they could figure out what would be safe okay and then congress didn't have to act because everything came to a halt and since paul was ahead of the game the fact that he was stopping meant that everybody else felt that they had to stop too and so we actually emerged from that with the ability to make recombinant dna molecules and you've never heard of an accident and so on and so forth so things have come out okay so in 1973 invention of recombinant dna and what you do is you open a plasmid which is circular piece of dna and you take into it and you put into it insert into it uh a foreign dna molecule say for something you want to make say growth hormone which is one of the first things that genentech wanted to make and and what you do is you can make now this dna by polymerase chain reaction which is basically replication in a test tube and then you introduce the recombinant dna into cells and the the construction of this recombinant dna is just dna recombination in a test tube right so i taught you all about that so you could do this so now you could make some some of your favorite protein which could be a drug now you might be interested in growth hormone if you have a you know if you have a child who's very small but you can also you can also make some very important growth factors one is called epo or erythropoietin and it's a growth factor for red blood cell precursors and it stimulates the production of red cells and it the way it does that is i'm showing you a molecule of ipo in red here and it binds to what's called the epo receptor on the surface of cells in the bone marrow and these cells in the bone marrow put together this receptor as a dimer so there are two subunits that are the same as each other and epo binds in this pocket here and the cell gets the signal aha it's time for me to divide that's a red cell precursor and it starts dividing and make making more young red cells to go out into the circulation to help us breathe okay very neat now the reason this is important is epo is made by our kidneys and so who are the people who are going to get anemic who do you think people who have cancer i heard but since it's made in the kidneys who else dialysis patients or anybody with kidney disease whose kidneys aren't working exactly and so the good of this is it treats the anemia associated with either renal failure patients on dialysis or with cancer chemotherapy and it replaces transfusion which often takes a few hours with a single intramuscular injection boom it's in there okay the problem of course is it takes two weeks to work for the red cells to emerge out of the bone marrow but it's very convenient and very quick instead of a transfusion it conserves the supply of donated blood and this is a really big problem because in the bay area especially there's a shortage of blood but the bay area imports blood from the rest of the country so we get blood from the rest of the country to make up for our shortage and it eliminates the risk of blood-borne diseases so we do have a shortage of donated blood especially here in the midwest it isn't as bad they seem to have different values there it isn't because of hiv by the way the incidence of hiv in the bay area of course you know is is only going to be a few percent right so it's mostly because people don't have time to donate so my wife and i donated three of the 10 units of blood the blood bank stanford blood bank collected this weekend so so those the guys here you could actually donate two units which i just did which is kind of fun you can go swimming and see what happens things like that so that's in it that's an advertisement for donating blood if you if you think you're eligible um because there really is a short it's very sad to go in there because when i offered two units they were getting close to closing but they said we'll take you and and i discovered it's because up to that point they had only collected seven units and it was the end of the day so that was it so the bad is that that picture of ippo that i showed you is part of this advertisement that might be optimus prime i don't know so and ippo is really good for the tour de france and other athletic pursuits and the reason it's good is it is a recombinant human epo it has precisely the same protein amino acid sequence as human ipo it's very hard to detect it's a mimic of the human protein so you can actually give yourself a shot of epo and raise your hematocrit so that you now can carry oxygen much better to your muscles and become a better cyclist and get away with it so so that's the bad and the ugly of it okay now i'm going to tell you some more things that are even uglier because this is really ugly as far as i'm concerned this is this is one of the most shameful things on the planet right now it's just as shameful as people starving in other parts of the world because this is a country that's actually wealthy enough to take care of its own citizens so the view from below the clinic i'm going to give you two examples of how spending more money can lead to a worse outcome because one of the things you've heard about is how are we going to ensure all the uninsured how can we afford to do it and i'm going to show you two examples that amount to billions of dollars or we ended up spending billions of dollars to get worse outcomes and so this gives me hope i mean this is this was shameful that we did it but it gives me great hope that if we structure the health care reform properly now we're getting into things i think you're really interested in we will be able to give better health care to everybody for less money how much less about two-fold less i'll show you why okay so from the ground pharma educates patients this is an ad that they give to patients are you ready to start chemotherapy low white cell blood counts how neulasta nulasta is another growth factor just like ippo and it it stimulates the production of white cells help protect against infections questions to ask your doctor the question is hey doctor can i have new lasta okay the reason they're advertising is because it cost two thousand five hundred dollars for each injection so the company makes a lot of money every time we prescribe this that's a lot of money that's the administration fee now i'm being a little unfair that's the administration fee if you don't have insurance if you do have insurance like blue cross they've negotiated the price downward to six thousand five hundred dollars it's a lot of money so the hospital loves us oncologists medicare actually medicare only pays a little bit more than that so doctors don't like medicare that's why the the uh ama is so scared that a public option would be just like medicare in fact i think the public option just died today okay so yeah so it may come back they're going to give it another try i don't know anyway um it i don't really care what option they have as long as they design it properly i'm deathly afraid they won't because too many of the wrong people are lining up behind us with too much enthusiasm and they have a lot of money to to make i wish they were a little bit more you know some of the big industries are i really love what's happening now so this is erenist a long-acting erythropoietin same company same kind of advertising amgen makes ipogen and aaronest johnson and johnson makes procrit and eprix they divided the market amgen does the cancer patients johnson johnson does the dialysis patients 2004 sales of epo 8.6 billion dollars now amgen stock this isn't a real graph it goes from 70 to 50 it doesn't go to zero but the price in us dollars in january was here and then in the course of a couple months dropped precipitously something happened what happened what's called the dahanka 10 trial november 2006 just before the stock drop it was prematurely terminated what was it it was to determine if aaronos improves the outcome in head and neck cancer patients who are treated with radiotherapy the reason for doing this trial was there were theoretical reasons for thinking that it might help because cancer cells when they're hypoxic when they don't get enough oxygen are actually resistant to radiotherapy they don't die as easily so the so the it had a good theoretical idea if the cancer cells get enough oxygen they'll die more readily from radiotherapy right so the reason the company was really interested is there was a good theoretical idea but what they were going to do is they're going to treat patients with hemoglobin's less than 14. now 14 is a really high number so they were going to get the hematocrit up there to where the cyclists want to be if you go to the blood bank they'll take you if your hemoglobin is 12.5 or above and they'll take blood out of you so 14 is really high okay the result five year disease disease-free survival was worse on aaronis than the control whoa p equal 0.02 journal of clinical oncology 15s supplement that was a supplement because it's only an abstract it hasn't been published yet but it was reported at the american society for clinical oncology meeting last june let me show you now that was last june so 2006 okay the re the report came out last june 2000 2009 look at this date 2003 they were selling ippo since you know right along 2003 456 789 all the way right through this is a paper published in lancet not of the invisible journal at all it is it is up there with the new england journal of medicine in prestige this is a kaplan-meier survival curve this is months time and months this on this axis is what's called progression-free survival at the beginning of the trial 100 percent of the patients have not progressed yet by definition but as time goes on more and more patients start the cancer starts to progress so as the curve goes downward that means you're losing more and more patients because the cancer has progressed so the lower the curve the worse the worse you are right so as time goes on more and more patients get picked off by the cancer now there are two curves here one is a placebo control where you just give sort of normal saline boom you just inject normal saline those patients aren't doing great head and neck cancer is a very bad cancer so after 24 months actually only 40 of them are still cancer free but you gave some patients epo and look where that is the curve is below cost thirteen thousand shot dollars a shot with insurance blue cross maybe six thousand five hundred is a lot of money 8.6 billion dollars going into this i think we oncologists were spending more than the dialysis doctors what's going on there well i told you there is this receptor called the epo receptor that when it's stimulated causes the red cell precursors to grow now when you name something you usually name it because that's how you discovered it it's called the ipo receptor because it was found on red cells there was no law that said cancer cells couldn't have ebola receptors and indeed cancer cells do have hyporeceptors they have two kinds they have the normal epo receptor on the surface of red cells and they have a heterodimeric epo receptor as well not all cancer cells have this but many of them do and those cancer cells that have these receptors will respond to ipo by growing so that's what that diagram means and we were doing this for years even after we knew about this it's disgraceful 8.6 billion a year was that was that report done pursuant to a study yeah this is head neck cancer this this was published in lansing it was reported to the world right the company have an obligation to report that well the fda knew about it is the indication for the drug head cancer yes yes you one of the indications for this drug is anemia of chemotherapy was that your question it was being administered to head and neck cancer patients was how did it hap okay right your congressman we have a dysfunctional fda as the answer the question was how did that happen the answer is we have a dysfunctional fda uh-huh that's my opinion i mean so something comes out of atlanta does everybody sit down and read this you know if you're practicing i haven't been giving ippo for years so let me tell you why some people were giving ippo the company you know was engaged in a scientific discussion maybe this result is real maybe it isn't real maybe there are reasons for this result maybe there are shortcomings in the study maybe not right but but let me tell you a lot of a lot of people in private practice were giving ippo even when it probably wasn't the best thing to do when the patient really wasn't that anemic why do you think they would do that well the patient was tired was getting tired and you know chemotherapy is tough and they say well maybe i'll give you some people because maybe it'll help you feel better and oh by the way so there's a huge conflict of interest among physicians and so so one of the problems with medical care here is our system is based on fee for service and this is how fee-for-service can be a problem but i'm preaching now let's face it i'm editorializing how would the public option uh fix that how would the public option fix that let's wait let's wait let's let's stick with questions until the end because i want to give you enough time to ask questions but we can talk about how the how would i fix fix the whole system but you know that's philosophical how much of that difference um oh to me nothing you know because i happen to be on salary here huh i am a salaried physician oh private practice no in private practice you're running a private practice so that goes to support your practice yeah oh yeah yeah we can talk about those things later i'm going to give you a short history of vioxx you all heard about it fewer gastrointestinal side effects published in the journal of medicine in 2000 between 1999 and 2003 there were eight trials with independent endpoint committees that showed an almost four-fold increase in heart attack risk of vioxx versus naproxen it turns out there were eight trials without in independent eight point committees and it turned out vox was actually protective all these trials were sponsored by merck but the independent endpoint committees can't have merck employees on them 2004 viax was withdrawn if you want to look it up current new england journal of medicine 2007 the victor trial was to see if vioxx improved outcome in colon cancer patients because it turns out that there was some data that suggested vioxx would prevent colon cancer because aspirin does and aspirin probably does so through the receptor that vioxx works on something called the cox-2 receptor okay so this trial had a good scientific basis to it but the problem was that in the background these heart attacks were going on so i'm gonna i'm gonna run real fast through this but there's viax biology that actually predicted the problem there's the cox-2 enzyme here that that actually stimulates inflammation and this is what you want to stop okay but in addition to that there's an arm of this pathway which goes through the cox1 enzyme that does very important things for platelets to make blood clots and protects the stomach okay from ulcers aspirin and napperson block both pathways so what you do is you get relief from arthritis but you also have stomach problems so what merck did was they had the the best drug for knocking out cox 2 and leaving cox 1 alone so you could protect the stomach that paper in the new england journal of medicine in the year 2000 was trumpeting their success in protecting the stomach it was wonderful and in fact i think vioxx if we only had it now we really ought to have it because there are people with arthritis who simply can't take anything like this because they have too many stomach problems but the problem is this was a very profitable drug and they it to everybody even people without stomach problems they also found out that this arm of the pathway stimulates growth of polyps in the colon therefore if you knocked it out you could prevent colon cancer from recurring for example or maybe even have it as a preventive so if it were a colon cancer preventive everybody over 50 would be taking it who knows how much money that would be so that's why they like that trial it really had an upside but the problem is it could also affect this arm of the pathway and remember it could end up disrupting the clotting mechanism so that you could have unwanted clots in the blood vessels of the heart and this is something they did know about and when they designed the first trial that was published in the journal of 2000 they specifically excluded people with a risk for heart attack but that's okay as long as they market it to people without a risk for heart attack but that's not how it happened so the number of people who died in vietnam was 58 000 just to give you a rough idea so has pharma bias the literature yeah 26 percent of all the papers in the new england journal of medicine are written by ghost writers somebody not listed as an author wrote the paper and i can tell you that if a scientist has an article in the new internal medicine they can't afford to hire a ghostwriter it's probably and i know that almost all of the pharmaceutical industry-sponsored papers are written by ghost writers i've talked to a ghostwriter about this randomized myeloma trials this is a fatal disease and when you do a randomized trial one arm gets the best possible treatment that's known today and the other arm gets the experimental treatment normally you would expect experiments like that if they're if you really don't understand if you don't know what the outcome is you would expect them to favor the new treatment about half the time when these trials myeloma trials were not sponsored by industry it came out in favor of the new drug about what you would expect about half the time when they were sponsored by industry they came out in favor of the new industry drug 75 three-quarters of the time so i think the literature is biased what about oncologists where do they fit this was our convention center uh orange county convention center in orlando stop by the booth 3324. and down here this is what i got in my email from cam stark i think those are hundred dollar bills benjamin franklin who's he 50 100 fistful they send that to us and i responded i took it and put it in my talk but you know other um high drug prices yield high administration fees for oncologists so these are pleased to announce a practice enhancement service line practice enhancement well in case you don't know former editors of the newland journal of medicine are really angry the truth about the drug companies on the take it's easy to see why they have books like that uh-huh why do they have to yeah i yeah why why do why the question is why do we how do we get away with charging these high administration fees these things are just a shot how difficult is that boom right in the arm i don't know why did the the insurance companies decide how much to pay there's there's it's complicated but you know it's got to change it has to change i think it really needs to change so the view from above i'm going to go through real fast this is the whole outcome this is california this is a graph of annual medicare spending per beneficiary this is overall quality ranking this is where we are you can see the line actually gets worse the more you spend now on the one hand that's disgusting but on the other hand you can understand why i'm so optimistic because if we spend less we'll get better outcomes okay this is health care spending in the us as a fraction of gdp so the gdp is growing and we are spending more but we're outpacing gdp back in 65 we were six percent of gdp we are now about 17 or 18 because the rest of the economy collapsed and we're headed towards 20 percent well it can only go so far okay this is a graph the richer a country is the more it spends on health care that makes sense we want our pa we want our citizens to be healthier but uh so we've got turkey poland mexico down here gdp per capita this is health care expenditure per capita there's the u.s even when accounting for the fact that we're one of the richest countries we are about 2x over the line what about quality of care this is a double graph the red dots are per capita spending this is where we are in terms of spending the red dots and all these are now these countries include japan sweden but they also include look at this costa rica cuba chile look who's down there with costa rica and cuba tied in life expectancy in who rankings i think i think we're we're right there with costa rica and cuba as well in in who rankings of quality of care not just in average life expectancy so how can we fix the cost problem this is the pie drug costs are rising uh fastest of all but physicians earn money for treatment not prevention and hospitals succeed when patients fill their beds so i have mixed feelings about expanding the hospital because i think the ideal system we should be hoping that the hospital shrinks so that used to say five cents but anyway for five cents yeah i think i went on a little too long but we so um so you're expected to stay at the molecular level but we've gone way up to dealing with the health care system and i you know i think as gill knows we share many of the same views about this and i think this is a extraordinary time that we're facing today for all the reasons that were articulated so because we still have um just a couple of minutes let's see if we have any questions uh that relate to any of the topics that have come up so far the question is can you access the shots that i found in my presentation on the web yeah i found those all on the web and i'm going to send kathy my powerpoint presentation so that if you want to look at them you can and also uh you should be able to find them by doing a google search so we're gonna we're gonna be posting yeah we're gonna i'm gonna post a whole slide show on the web offer yeah and subsequent presentations we'll be posting them online okay yeah oh when the retrovirus infects the cell and and integrates itself into our dna correct so now you have millions of these cells affected how does the body be sorry that cell is is done for yeah because what happened is the retrovirus the question was what happens when a retrovirus invades one of our lymphocytes that part of our immune system that's what hiv does it invades our immune cells that cell now has a copy of the retrovirus in its genome well it's only going to recognize that dna that cell is going to look at that dna and say that's mine so the retrovirus is so clever so let me just tie this back for a second because i know we're going to be running out of time shortly so i think that what you've heard from dr gill is really the extraordinary story of not only how dna works but how this whole process of recombination takes place and i think the illustration that you gave about how that in some ways relate to the whole recombinant dna technology was an extraordinary one and that actually is what spawned most of biotechnology here in silicon valley and elsewhere but he also told the other side of that story which was great discoveries can also be greatly abused and it really does depend upon whether or not we match um utilization to need and sadly that is where um our system is really out of whack so we we're a nation uh that has by far the greatest bioscience discovery engine in the world today and as you've heard and i stated last week and we'll continue to reiterate in other settings we spend more on health care than any other nation in the world with nothing to show in the population basis other than that we're number one in administrative overhead and you've heard some of the reasons for that tonight so i think we've had a rich exciting discussion tonight i hope you've enjoyed it next week i will actually be missing because i'm giving testimony to a congressional committee on health care reform i've been back to washington many times over the last several months it's hard to know where things are heading as it stands but i think with the senate finance committee's decision today i doubt they'll want to hear from me next week but we'll do our best but next week will be an opportunity for regeneration and jill helms will be speaking about stem cells and regeneration so please come back and we'll look forward to your comments and questions thank you for being here tonight for more please visit us at stanford.edu
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Medical_Lectures
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19_Biochemistry_Signaling_II_Lecture_for_Kevin_Aherns_BB_450550.txt
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Captioning provided by Disability Access Services at Oregon State University. [classroom chatter] Ahern: Okay, folks, let's get started. Student: Let's get started! Ahern: I like that attitude. [class laughing] Ahern: I looked at the calendar today and realized that next Friday we have an exam. Also, that's dad's weekend so it's good to get this out of the way, huh? Maybe have dad come take your exam for you? Maybe not have dad come take your exam for you? [Ahern laughs] Okay, today I'm going to finish up signaling and I will get talking a little bit about the considerations for metabolic controls and this involves Gibbs free energy and I'll give you some things about that. The TAs have been going through and probably gotten through with you in recitations, the considerations and problem solving for Gibbs free energy and so, as always, if you have questions or problems or concerns, come see me and I'll be happy to work with you as well. Last time, I spent some time getting ready to talk about how it is that, how it is that the beta adrenergic receptor and epinephrine play very important roles in increasing blood glucose. And this is very important. We have an emergency, when we need to escape, we need to do something, or we need to have muscular contraction. Having a supply of glucose, excuse me, in our blood stream is very important. As I also referred to in class last time, glucose in our bodies is essentially a poison that when we have too much glucose in our blood stream, we have very severe side effects. People who have diabetes for example have an insulin response system that is either absent, in which case they have type 1 diabetes, or, and there's other manifestations besides what I'm going to tell you, or they have a cellular system in their body that is not responding properly to glucose. I'm sorry, not responding properly to insulin. So the normal response of the body to insulin is that binding of the insulin to the insulin receptor will cause cells to take in glucose. We'll see at the molecular level today how that happens and why that happens, or why it happens is because glucose is a poison. And so if we don't decrease our blood glucose levels after we've had a meal, then they go very high and as I mentioned last time, what this can cause is severe problems that people who have diabetes experience. May involve kidney failure, it may involve blindness, it may involve the longer you have this amputation. People who have diabetes over a long period of time not uncommonly have amputated limbs. So it's very, very severe consequence of having blood glucose level go high. So it's important then that we spend some time talking about how it is that insulin causes cells to take up glucose. And so not surprisingly, there is a signaling pathway that's involved. The signaling pathway, in fact the signaling pathways that I'm going to describe to you today do not, underline not, involve 7TMs. So 7TMs we remember were the 7 transmembrane domain proteins like the beta adrenergic receptor, like the angiotensin receptor that we're involved in causing cells to activate a G protein, that means that the things I'm going to talk to you about today do not involve G proteins. No G proteins involved. Okay, so insulin is a relatively simple molecule. What you see on the screen is a depiction of insulin. It's comprised of two chains that are covalently linked together by disulfite bonds. Disulfite bonds you can see right there and down here. And those disulfite bonds are what hold the two chains together. So first of all we can say that insulin has quaternary structure and interestingly the way that insulin is made is insulin is made as one long chain. Then it folds and the disulfite bonds form, then protease clips off some of the segments so that you're only left with two linear pieces, kind of like what you see on the screen here holding everything together. Now insulin manifests its effects on target cells by binding to a specific insulin receptor. So the insulin receptor is a protein that's located in the membrane of target cells and it has a structure that looks schematically like what you see on the screen. The top part of this image is the outer part of the cell. The bottom part of the image is the inner portion of the cell. The insulin receptor exists as a dimer normally. We'll see the epidermal growth factor receptor that I will show you in a little bit exists as a dimer only when it binds to the epidermal growth factor. The insulin receptor is different. It exists as a dimer but the binding of insulin to this dimer causes some drastic changes to happen to it that cause insulin to ultimately bring glucose into the cell. Now, like the other receptors we saw the other day, insulin as I mentioned is a hormone just like epinephrine is a hormone. Hormones don't make it into, at least the one's we're talking about, don't make it into target cells. So insulin doesn't make it into the cell. It causes all of its effects by causing some changes within the insulin receptor. Now the insulin receptor is a transmembrane protein as you can see here. It has some different components to it here. There's an alpha subunit, there's the beta subunit. And these work together to communicate the information into the cell. So how does this process work? Well, it turns out that insulin receptor is a special kind of kinase. I talked before about a different kinase. I talked about protein kinase A, I talked about protein kinase C. And these were kinases that we found dissolved in the cytoplasm of the cell. The insulin receptor is a kinase as well. You can see it's imbedded in a membrane. And, in addition, this kinase is different than protein kinase A and protein kinase C and it is a tyrosine kinase. It's a tyrosine kinase. So it's a membrane bound tyrosine kinase. Now, a tyrosine kinase, as its name tells you, is a kinase that puts phosphates onto target tyrosine residues. It puts tyrosines onto target residues. Now, what's interesting and odd about the insulin receptor and many receptors that are membrane bound exist like the insulin receptor does, is that the insulin receptor is a tyrosine kinase but it's normally, when you see it in a state like you see it here, it's completely inactive. And this tyrosine kinase ends up activating itself. How does it do that? Well, the binding of insulin on the external part of the receptor causes a shape change like you've seen before. Now, before the binding of that insulin occurs, the tyrosine kinase portions are down here. Each side has a tyrosine kinase activity in it. But each side is unable to function because of the way that these catalytic sites are oriented with respect to each other. They're just sitting there doing nothing. Binding the insulin causes a shape change that allows one of the tyrosine kinases to phosphorylate the other one. So there's a shape change. This now places into the active site of one of the portions of the dimer. It puts the target tyrosine into there. Well, the phosphorylation, let's say we're phosphorylating the right in this case, the phosphorylation of the right one now causes it to become active. And so it turns around and phosphorylates the left one. So now they're both fully active. They're able to do their thing. As a result of that, there's a series of phosphorylations that happen up and down these beta subunits. So several target tyrosines will get phosphorylated on these beta residues. That's an essential component of the insulin signaling. So first of all, we have to jump start everything, we jump start it by putting one phosphate on, then we go back and fourth, back and fourth, back and fourth, and get phosphates all over there. Everybody with me? Now, what happens as a result, here's the tyrosine kinase first of all. There's the side chain of tyrosine, there's the addition of a phosphate, and again like we've seen before, this changes this guy which is largely an OH group into something that has a negative charge. Not surprisingly, that negative charge may change again itself the shape of the protein in some way. And that causes all the other changes to happen that I've been talking about. Now, you can see on this receptor right here that this phosphorylation induces a pretty big change in shape. Here is this guy before phosphorylation and look how far this has moved over here after phosphorylation. So the shape change that's happening as a result of the phosphorylation of those tyrosines is inducing a pretty good size movement inside of this protein. There's a term that we use for this, I haven't given it to you and I should give to you at this point. It's called receptor mediated tyrosine kinase, or RMTK. This is a receptor, the insulin receptor's receptor, meditated tyrosine kinase. And we will see, we won't actually go into them in this class, we'll talk about one other one. But there are many receptor mediated tyrosine kinases that we find in cells. Many, many. And they all play important roles in signaling. Well how does insulin signaling work? So far you've seen how the receptor gets activated. What is involved in signaling through the insulin receptor? Well, now you see this a little bit more clearly, hopefully. You can see there's a lot of the guys, lot of things that are involved here. First of all, we see that this is the receptor that has bound to insulin. And once it is bound to insulin, there's this cross phosphorylation that happens across the beta units of the insulin receptor. One of these phosphotyrosines, as you can see here, is a binding target for a protein known as IRS-1. That's not in internal revenue service. It does better things than the internal revenue service does. There's another one called IRS-2 that will also do this that's not shown here. But this guy, this is a protein, in fact everything you see on here are proteins. This protein binds to phosphotyrosine. It has a domain that we refer to as a SH2 domain. An SH2 domain is a common structure that we find in many proteins that is capable of recognizing and binding to phosphotyrosine. This is a phosphotyrosine. This now is a perfect target for IRS-1. Well, this bringing of IRS-1 in place allows it to become phosphorylated on its tyrosines as well, so again, we have have this phosphorylation picnic that's going on here as it were. And these phosphorylated sites become targets for another protein. It's another enzyme, as you can see it's another kinase, phosphoinositide 3-kinase. So when we had the beta adrenergic receptor, we saw movement. We saw this G protein moving back and fourth to adenylate kinase. And we saw the cyclick AMP moving in the cell. All these things are happening right here in this one site. We'll see right here a little bit of movement, but for our purposes, essentially everything is happening at the same place. Well what happens here? What is this protein? This protein is known as phosphoinositide 3-kinase. It also has a SH2 domain and it binds to a phosphotyrosine on IRS-1. So we're making kind of a big sandwich here if you want to think about it that way. This enzyme, as you can see, catalyzes the formation of a molecule called PIP3. Now PIP2 you've seen before. PIP2 was involved in the cleavage reaction of phospholipase C that I talked about on Monday. If I take PIP2 and instead of cleaving it, I put an additional phosphate on to it, I make PIP3. I've put an additional phosphate onto this molecule. And yes, PIP3 is acting as a second messenger. PIP3 is able to travel in the membrane, as is PIP2. They move in the membrane very readily. And it moves in the membrane and it itself is a target for binding by PDK1. PDK1 is PIP3 dependent protein kinase. So we see kinase, kinase, kinase, kinase. We see this cascade that we've talked about before. This was a tyrosine kinase that got activated. This is a phosphoinositide kinase that got activated. This is a kinase that's getting activated, and we'll see that this PDK1 phosphorylates this important protein known as AKT. Yeah? Student: That catalyzes the reaction of PIP2 to 3? Ahern: The green guy catalyzes the conversion of PIP2 into PIP3, you're exactly right. Yes, sir? Student: Is IRS-1 the only one [inaudible]? Ahern: IRS-1 is simply a bridge in this scheme. It's simply a bridge. Student: It's not important to [inaudible]? Ahern: Nope. Student: Is there an amplification that happens during this process or will it always be together? Ahern: A very good question. Is there any amplification that occurs in this process? The main amplification actually occurs right here where this guy can phosphorylate a lot of PIP2s, but you don't see the same sort of cascading amplification that we've talked about before. That's a very, very good question. Well, we've gone here, here, here, we've got a protein kinase that's active. This protein kinase is going to phosphorylate. This protein known as AKT. AKT plays many roles in the cell and mercifully not going to show you all the roles in the cell, nor am I going to show you the series of proteins that it phosphorylates, that phosphorylates, that phosphorylates, that phosphorylates, that phosphorylates. But, I will tell you what the end result of this phosphorylation is. AKT is a kinase as well. And this enzyme will stimulate ultimately a change in the trafficking of proteins in the cell. What does that mean? Well trafficking, it refers to the movement of proteins. When we talked about the endoplasm reticulum and the Golgi apparatus the other day, and I said that these glycoproteins have various license plates on them that tells the cell where they should go. Should they go to the membrane? Should they get exported out of the cell? That's trafficking. Those guys get moved into the cell according to instructions that are on them. This guy here is altering the trafficking. What does it do? It changes one important protein where it goes. The important protein that it changes is known as glut, G-L-U-T. And as we'll talk later, there are several gluts. Glut stands for glucose transporter. Now, what this pathway is doing is it's taking glut, which is found normally in the cytoplasm, and it's moving it to the membrane. And since glucose, I'm sorry, since glut has the property of transporting glucose, the cell starts taking up glucose. Now, that's a lot of steps that you needed to know. Yes, okay. You need to know the steps. But that's a lot of steps to get glucose inside of the cell. As a result of this, cells start taking glucose out of the blood stream, and when they take glucose out of the blood stream, they are reducing blood glucose, reducing the toxic effects of glucose, and getting it to the cell that might either burn it or store it in the form of glycogen. So insulin ultimately is countering the effects of epinephrine. It's countering. Epinephrine is increasing blood glucose, insulin is reducing blood glucose. We see that they're doing very different mechanisms, but those are the results of the action of those different hormones. And yes, insulin is a hormone. It's a peptide hormone, meaning it's a protein that's a hormone. Okay, so I'll stop and take questions at that point. Or give you a chance to catch your breath. Yes, ma'am? Student: Since the glut goes from the cytoplasm into the membrane, and it takes glucose and with it, it counteracts epinephrine you said? Ahern: Yes, so what her question was, 'Glut, because it's going to membrane, is taking in glucose and that taking in of glucose is countering the actions of epinephrine, the answer to that question was yes. Question? Student: Was it changed by AKT? Ahern: So her question is, "Is glut changed by AKT?" Glut's location is changed by the pathway that's stimulated by AKT. There's several kinases that act before we ever get to that change. And all that's happening is glut is having its location changed from the cytoplasm to the membrane. Question over here, Lawrence? Student: This PT table [inaudible]? Ahern: PDK1 phosphorylates AKT, that's correct. Student: And that of course, affects blood...? Ahern: I'll tell you what, everyone is curious about the steps, maybe I'll make you memorize them. No, I won't make you memorize them, but let me show you the overview of the pathway, okay? Student: No! Ahern: Yeah, so I've taken you down to, oh, they've changed it this time. I've taken you down to here. You can see that there's actually several steps that's involved ultimately in moving the transporter to surface. They used to have a figure in the old book that showed like 20 steps that got us down to there. You wouldn't want to know the 20 steps. Yeah? Student: So what does amplification mean here? Ahern: I'm sorry? Student: What does amplification mean? Ahern: What does amplification mean? Student: Yeah, in this diagram. Ahern: Here? Student: Yeah. Ahern: So amplification is simply, well, I think it's a little misleading here. If we activate the receptor, then we're essentially activating the phosphorylation of many, many things. For the figure I've shown you, we're only looking at one thing, that's why I'm saying there's not really an amplification there. The insulin receptor is involved in phosphorylating many things. We're looking at one at the moment. There's other things that it can phosphorylate and activate. We're not looking at those. So let's leave that amplification out for the moment. Yes, back here? Student: The cell has a way of releasing the insulin and stopping the whole phosphorylation process or? Ahern: Yeah, so how does the cell stop this process? That's a very good question. Just like we saw before, we have to have a way of getting insulin out of the membrane. The cell has to have a way of handling that insulin and yes it does. And that's, again, beyond the scope of what we're going to talk about here. Was there another question? I thought I saw a hand. That's what's involved in the insulin signaling pathway. As I said, the receptor is involved in many things. The insulin receptor is one that, if you take my molecular medicine class in the fall, I'm sorry in the winter term, I'll talk a little more about that. It is a very important receptor that's involved in a lot of things, including phenomena as diverse as aging and cancer. So the insulin receptor has its fingers in a lot of pies, an awful lot of pies. Haha, glucose, you see. Alright, I don't think we need to talk about that. Alright, so that's the insulin receptor and the insulin signaling pathway that we will talk about here. I want to talk about another receptor mediated tyrosine kinase. And this is one that binds to the epidermal growth factor. The epidermal growth factor is a hormone and like insulin, it has a receptor that it binds to. The receptor is membrane bound. And the receptor is a tyrosine kinase. So it binds to insulin, I'm sorry epidermal growth factor, or EGF, binds to the EGF receptor. There's a schematic diagram of it, I don't like the schematic diagram as much as I like this. Now, I earlier pointed out that the insulin receptor exists as a dimer all the time. The epidermal growth factor receptor does not. You see it in the dimer form only when the receptor has bound to epidermal growth factor. So we can see that here's one half of the receptor that's bound to epidermal growth factor. Here's another half the receptor that's bound to epidermal growth factor. And only after both of these guys have bound epidermal growth factor do they dimerize as we see here. Now, there's a figure that's in your book and I don't like the figure as much as I like this little schematic. You see this little red sort of loops that are here? These red loops are the major shape changes that occur upon binding of the epidermal growth factor. So before the epidermal growth factor binds to the receptor, this loop is sort of folded over onto this thing so they can't interact. But the binding of the receptor, I'm sorry, binding of the epiderm growth factor by the receptor causes them to literally stick out and touch with the next one. That's how they dimerize. So the system is set up so that the receptors don't dimerize until they have both bound to an epidermal growth factor. Well what happens with the binding? Upon the binding, very much like what we saw with the insulin receptor, these kinases, which are inactive, become active. One phosphorylates the other, phosphorylates the other, phosphorylates the other, phosphorylates the other, and you see that we get a series of tyrosines with phosphates on them. Those tyrosines with phosphates on them are targets for another protein known as Grb-2. And Grb-2 has a SH2 domain just like we saw before. It's recognizing and binding to a phosphorylated tyrosine. Grb-2, like we saw with IRS-1, serves as a bridge. Excuse me, the other side of Grb-2 binds to this protein known as Sos. Sos now, here's a G protein. It's not really a G protein like we saw before. It's a different kind of a G protein. So the beta adrenergic receptor had what we classify as a pure G protein. This protein called Ras is a very interesting protein. It's like a G protein but technically it's not the same thing. So I wasn't lying to you earlier when I said we don't have G proteins involved at this point. Ras is one of the most interesting proteins in your cells. You see that, like a G protein, it binds to GDP and like a G protein, when it gets activated, drops the GDP and picks up a GTP. So for all apparent purposes out here, it's functioning kind of like a G protein. Now, the G proteins we talked about before either activate phospholipase C or activated adenylate kinase. Ras instead activities a signaling pathway series of events. One of which ultimately stimulates a cell to divide. One of which ultimately stimulates a cell to divide. And Ras has many, many pathways it can affect. But one of those is stimulating the cell to divide. Yes? Student: So did Sos activate Ras? Ahern: Right, so the binding of the Sos to the Grb-2, good question, the binding of the Sos to the Grb-2 cause a shape change the in Sos? The shape change in the Sos caused the change in Ras, which was the dumping of the GDP and the replacement by GTP. And as a result, we have an activated Ras. So we can see in this pathway that here's a growth factor. A growth factor is a hormone, in this case it's a peptide hormone, that's stimulating a cell to divide. That's what growth is all about. Not surprising. Multi cellular organisms need to control their growth. I want my left leg to be at least approximately the length of my right leg. I know there's a little bit of difference in how long legs are but I want them to be approximately the same length. I want to have the control so that I'm determining when cell division in my bones is occurring. If I do that and I control that growth, then I will be reasonably symmetrical in my appearance. Now this protein Ras, as I said is one of the most interesting proteins that we find inside of cells. It is an example of a class of proteins of which there are a few hundred that play very critical roles in this decision to divide or not to divide. They're involved, these proteins that I'm getting ready to describe to you play very critical roles in signaling and usually in some level affect the decision to divide or not to divide. This class of proteins has a name, it's very important, they're called protooncogenes. Proto, P-R-O-T-O dash oncogene, O-N-C-O-G-E-N-E. Well what is a protooncogene? A protooncogene is a protein intimately involved in cellular control. Usually by a signaling pathway. That intimate nature of its action in controlling the cell is essential for the cell to function properly. It's essential for the cell to function properly. If it doesn't function properly, if the protooncogene doesn't function properly, it behaves as what we refer to as an oncogene. An oncogene has another name. It's a gene that causes cancer. Now, how does a protooncogene become an oncogene? The most common way in which that occurs is mutation. If we mutate the coding sequence for Ras, we may convert it so that it no longer performs its normal function. It may stimulate the cell to divide uncontrollably. When I mutate a protooncogene, I can make an oncogene. So the difference between a protooncogene and an oncogene is a mutation. Unmutated equals protooncogene. Mutated equals oncogene. It can lead to uncontrolled division. There are many examples, there are several hundred protooncogenes that are known. And normally, they function exactly as they're supposed to. They're supposed to control whether a cell divides or not divides in response to the signals that it's getting. But when they mutate, we can have real problems. That's why we worry about mutagens. Cigarette smoking, pollution in our air, pollution in our water, junk that we're eating in our food. These things may favor mutation, mutation of DNA in general, you're increasing the chances that you're going to cause a protooncogene to become an oncogene. Now in the case of Ras, I'm going to tell you exactly what happens. There are many examples though of different mutations that can happen. And I'll show you one other one after I finish with Ras. Ras, like the class of G protein, I don't want to say like other proteins, but like the class of G proteins, is a very bad enzyme. Remember I said that the G proteins were bad enzymes, bad in the sense that they're very inefficient at breaking down GTP. Ras is the same way. Ras will cleave GTP, and as we can see in the scheme, when GTP gets cleaved, Ras is no longer active, it goes back to here. As long as Ras is active, it's going to stimulate the cell to divide. One of the mutations in Ras that converts it from a protooncogene into an oncogene affects the ability of Ras to break down GTP. It affects the ability of Ras to break down GTP. Now in the case of Ras, it's a fairly small protein. There are two, it's actually three, but two that we focus on, two critical amino acids at the active site of Ras. Positions 11 and 12. You don't need to know those numbers. Mutations at either one of those amino acids that converts that into any other amino acid causes Ras to be unable to cleave GTP. Yowza. Any mutation can do that. That can involve a single base pair change in the coding sequence of Ras at that position. Now, if you want to think about why you want clean water and clean air and good food, and you don't want to smoke, and all of these various things, Ras is a really good thing to think about. There are animal systems that have been shown that they can induce a tumor by making a single base change in the coding of Ras. Now the formation of the tumor is a complex process. I'm not going to say in a human being that's necessarily what's going to happen. I can tell you that making Ras mutated is not a good career move. In general, mutating protooncogenes are not good career moves at all. You're asking for trouble if you start doing that. So be careful what you eat, be careful what you drink, think about the environment, think about your health, because these things really are very important in your survival. Yes, sir? Student: [inaudible] require 3 or 4 separate mutations that would disable like apoptosis and induce constitutive cell division? Ahern: So his question is, doesn't the formation of a tumor require several independent, separate mutations? And there are thousands, tens of thousands of mechanisms that can lead to a tumor. You are correct. That's why I say I'm not talking about necessarily in one sense, but at least in some animal systems, that has been shown to be possible to do. So you got to be careful. You don't know. I mean how many, is it 2, is it 3, is it 20? If there are some systems that you could do where you might take 2 or 3 of the right type of mutation, or maybe the wrong type of mutation, you don't want to mess with that. Student: But if a single cellular signal just activated Ras constitutively, wouldn't you still add a regular active like a P51 that would initiate apoptosis and... Ahern: Okay, so, let's talk about apoptosis later. What he's asking about is a phenomenon where cells commit suicide. And you are right, there are checking mechanisms in cells that will help prevent cells from becoming out of control growth. So the mutation of proto-oncogenes is a necessary step for formation of a tumor. So I'm only telling you one way by doing this. Apoptosis is one way of preventing that, but again, let's save that until we talk about apoptosis, okay? Because there's many factors to consider. But I want you to be left with the gravity of this, which is that mutating your protooncogenes is not the best thing to do. Yes, Neil? Student: How does the cell go into uncontrolled division? Ahern: How does a cell go into uncontrolled division? Well, okay, you guys really want to get into this here. So cells control their cell cycle. In multicellular organisms, we see the cell cycle that they go through, there's a synthetic phase, a mitotic phase, and there are resting phases, and there are specific proteins that will allow movement through those phases. So when we have uncontrolled growth, we do not have regulation of those phases. That can involve, again, multiple steps in the process. So I'm just talking about one mutation here, folks. So I'm not going to go through the whole cell cycle, but the point is that the more protooncogenes we mutate, the more likely we're going to have something that we don't want. Yes, sir? Student: So does the GTP play a role in the deactivating, so when it mutates the GTP is broken down...? Ahern: Okay, so I'm not sure I understand the question, but the point is that once it's bound to GTP, it's activated. So there's no role of GTP or GDP because all that we have to have is this activated. If the Ras cannot break it down, then it's always in the activated state. The only shut off mechanism is the breaking down the GTP. I'm sorry, maybe I didn't understand your question, but if I we can't break this down, it's on. It's on. Okay. So that's a pretty important, pretty cool system to understand. There's a long set of steps I didn't take you all the way through. There we activated Ras, Ras activities Raf, activities MEK, activities ERK, and phosphorylates transcription factors. Phosphorylates transcription factors. Transcription factors of proteins that bind to DNA that activate transcription. If we turn on the wrong genes, getting back to Neil's question back over here, if we turn on the wrong genes that are otherwise stopping cell cycle, now they're starting cell cycle, we can have uncontrolled growth. So I know I'm giving you a very sort of black box image of this, but the point is the to we lose control of the system here, everything else that follows can be a really big problem for us. Okay. Ba-da-ba-da. The last things I want to talk about with respect to signaling and then I'm only going to talk about one of these and that's this guy right here, bcr-abl. This one's an interesting one and it's interesting particularly for people who live in Oregon, interestingly enough. And this thing that you see on the screen is a way of making an oncogene from a protooncogene. Now I talked about well, we mutate. Maybe the DNA polymerase doesn't copy something properly. Another way of having changes happen that are the equivalent of mutation are to have recombination. You guys have learned about recombination in biology I'm sure. This happens when two DNAs that were not originally together get linked together by a cross over phenomenon. A very common, I shouldn't say very common, but a relatively common cross over that can occur that is a recombinational event that can occur, occurs between two genes known as bcr and abl. Abl is a receptor, I'm sorry, abl is a tyrosine kinase involved in signaling. It's a tyrosine kinase involved in signaling. Bcr is another gene that's up here on chromosome 22, abl is on chromosome 9. Cross over events that bring these two guys together happen as I say relatively commonly, not every day, but relatively commonly to make something that we call bcr-abl. What happens in this case is that the abl gene gets linked to a portion of bcr gene. So the bcr genes here, we see the bcr gene in red. We see this portion of the abl that gets linked to it. And we make essentially a new protein. Now if we completely alter the function of the protein, it probably wouldn't cause too much of a problem. However, this fusion keeps the tyrosine kinase activity of abl in the active form. This guy is still a tyrosine kinase and abl is involved in telling cells to divide or not to divide. The result of this fusion gives a phenomenon that's very interesting. When we talk next term about gene expression, we'll talk about how much transcription of a gene occurs. We can imagine that some genes might have on average, let's say 1,000 copies of its messenger RNA made. Another gene that's used a lot might have 20,000 copies of its messenger RNA made. Bcr, it turns out, has a lot more copies of its self made than abl does. Abl only has a few copies made normally. So what's happening as a result of this fusion is abl is being brought under the transcriptional control of the bcr gene. So now instead of having just a few messenger RNAs for abl, the cell is flooded with them. Well you've got, if you have thousands and thousands more than you would normally have, each one of those has more opportunity to get activated and to activate cellular division. So here's a case where the amount of a protein that we're making, the amount of the protein that we're making is affecting the cell's ability to control itself. Now we've got an awful lot of this stuff here. That's the bad news. This mutation happens in a type of leukemia. It happens in a type of leukemia known as CML. The good news is that there's a pretty darn good treatment for it. And the pretty darn good treatment was actually invented at OHSU. Now, it involves a drug that inhibits this enzyme. It is a tyrosine kinase inhibitor. In the back of your minds I hope you were thinking, Do tyrosine kinase inhibitors have effects on cells? And the answer is they can. Inhibiting this tyrosine kinase is one way of keeping this tyrosine kinase under control. Because if this guy doesn't have the ability to phosphorylate tyrosines, it's going to in fact not be stimulating that cell to divide. We have a better way of handling this mutation in this cell. The tyrosine kinase inhibitor that was invented at OHSU was known as Gleevec, G-L-E-E-V-E-C. It's very effective against this type of mutation, or this type of alteration, and interestingly enough, this Gleevec doesn't have many side effects. Why? Well, it turns out that it really binds to this fused protein very well and this fused protein isn't found in regular cells. So when we think about an anti-cancer drug and we think about something that we want few side effects, we would really like to be able to target something that occurs in cancer cells but doesn't occur in other cells and Gleevec actually does this quite well on this particular fusion. So in this case, the fusion actually gave us a unique target that a regular cell doesn't have. It's something we think of a magic bullet or a silver bullet that is targeted at a cell that is in trouble. Questions about that? I brought you guys to silence. Wow. Yes? Student: Will cellular systems still recognize like in this case, a new protein, that it will recognize it as foreign? Ahern: Are their cellular systems that recognize this as foreign? The cell would have no way of recognizing it's a foreign thing. When we think about recognizing foreign vs. natural, we're talking about the immune system which is working outside of cells. So no, there's not a way of recognizing this. Good question, though. Okay, so we're getting late. Maybe we should sing a song and call it a day. I've got a signaling song. Anybody here like Simon and Garfunkel? Alright. This is one of my favorite Simon and Garfunkel songs. I'm an old guy. Come on here. Oh, wrong one. It's called "the Tao of Hormones." It's to the tune of "the Sound of Silence." Lyrics: Biochemistry my friend It's time to study you again Mechanisms that I need to know Are the things that really stress me so Get these pathways planted firmly in your head Ahern said let's start with epinephrine. Membrane proteins are well known Changed on binding this hormone Rearranging selves without protest Stimulating a G alpha S To go open up and displace its GDP With GTP, got too high there Because of epinephrine Active G then moves a ways Stimulating ad cyclase So a bunch of cyclic AMP Binds to kinase and then sets it free All the active sites of the kinases await Triphosphate Because of epinephrine. Muscles are affected then Breaking down their glycogen So they get wad of energy In the form of lots of G-1-P And the synthases that could make a glucose chain All refrain Because of epinephrine. Now I've reached the pathway end Going from adrenaline Here's a trick I learned to get it right Linking memory to flight or fright So the mechanism that's the source of anxious fears reappears When I make epinephrine. I had a little bit of that fear at the end there. Alright, take care guys. [class clapping] [classroom chatter] [END]
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Medical_Lectures
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Hepatocytes_Liver_Histology_Part_47.txt
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now we will go in details detail of the liver lobular structure the liver parenchima and its arrangement of the hepatocytes can be explained in three ways we should explain it in reference to Classic lobule which I mentioned previously number two we will explain it in reference to portal lobule another way to look at the liver architecture and still one more way to look at the architectural and functional aspect of the liver is when you talk about hepatic asinus right I will first draw the classic lobule right three or four of them then I will explain other two types as well when we talk about the classic lobal okay I think I need to make it a little bit smaller it has gone beyond my board how many lules are there in your liver many many mean how many yes too many okay I think you're millionaire as far as these lules are concerned right your millions of these hepatic lobules these structures which I'm drawing these are basically classic lobules right now you will tell me the structures this is a branch from herpetic arter yes and the one branch should be present at every corner of this hexagonal structure which is classic lobal and this is right hepatic artery and and what is this Branch at every corner portal ve is it right from the portal way and here we have bile drainage right so B Du yeah I've changed the color scheme a little bit and this is now your bile d right and of course you remember there were lymphatics also here these are lymphatics oh I don't know this lul is under sered what's wrong with it we should put here portal and yes please what else we need to put bile duct so this is Portal Triad but actually there's always lymphatics also in this area now you will tell me what is this in the center Central wi right and now we will come back to our Arrangement what was from here yes Soo sides right I will just draw two sinites now exactly how is the structure of sinites cides are made of endothelial cells but these endothelial cells are not making a continuous lining they're discontinuous and these endothelial cells are having fenestrations these endothal cells are having fenestrations so these are cides are wide diameter capillaries which are extending from the outer part of the classic luule to the inner Central vein and they're lined by endothelial cells but endothelial cells are not completely lining as they are very very big gaps and even within the endothal cells there are very large fenestrations and these fenestrations are diaphragmed plus there what is this basil lamina even basil lamina is discontinuous and it has very very big gaps and it is not complete then another very important aspect of this is that there are very special type of cells here which are called cuffer cells what are these cuffer cells these are derived from circulating monocytes or macrophases right so these are fixed cuffer cells are in fact the fixed macres right present into liver cdes and along with the endothelial cells cuffer cells are also lining the cides so that when blood is moving from the periphery to the center uh components of the blood are freely exposed to the cuffer cells so cuffer cells can you know blood is also coming from the suine if they are fragments of the broken rbcs they will be removed by the cuffer cells am I clear so cuffer cells are present filtering whatever toxins or bacteria or fragments of the rbcs are passing from Portal blood right or through the cdal blood any question up to then here were your hepatocytes right hepatocytes are large cells they are about 20 to 30 microns right 20 to 30 Micron meter these are polyhedral cells are with multiple faces the you have to remember one face of the hocy is towards the blood and other faces towards the B canalicular system if you really want to know exactly how hytes are and what is their relationship with the bile drainage system let me make two hytes here now this is a nucleus right usually there's a large nucleus in the center of the hepatocytes but as your age increases number of nucleus in the hepatocytes May increase right and this was the cocal system so naturally this part of the this face of The Hite is exposed to the components of plasma is that right this face which is towards the face space of Des this has multiple micro will light to increase its surface area right surface area right another thing which is very important that b nuli are basically a gap between the cell they're not suppose this is one cell membrane going this is another cell membrane going and these cell membranes become separate from each other and now they go down so this is B canaliculi actually B canaliculi is not lined by any special cells these are just gaps grooves present in between the two heyes it's worth repeating bery drainage system does not start as very special duct the bery CL start as a gap or space or the grooves present in between the two hocy membranes is that right and the then what is there a special what is this tight junctions here is that right so basically bile is drained from these cells up to this side and then this tunnel is going in between many many cells is that right so here I've shown it in a very wide way but actually the cells should be touching each other and in between them a very fine T tunnel is moving through is that right am I clear no problem right now these cell membranes are specialized in a different way for example the membrane of the cell which is facing to the space of desay right it has different transporter systems and it has different type of secretory system and the membrane which is facing toward the B canaliculi it has also transporter membranes and proteins but they are different than the part of the membrane which is facing to the space of DY for example you know blue Rubin when rbcs break down and hemoglobin break down one of the breakdown product of the hemoglobin is blur Rubin blur Rubin is from the you know a lot of rbcs break down into suine from there BL Rubin come into blood right and blur Rubin is reaching suppose I make this B Rubin is coming here this is a b Rubin molecule through the cocal system BL Rubin molecule will come to the space of D then this surface of the cell this surface of the cell has the transporter which will take up the bluin to the cell I'm going to tell you how the pite work just one example then blur Rubin will go in right when blur Rubin is brought here it is now after that this is conjugated with glucuronic acid now this gluconic acid this is gluconic acid it is added to the BL Rubin right by the hocy Machinery then conjugated blue Rubin is actively secreted into where belri nuli you understanding so it means that hocy membrane which is facing to the space of Dy it should have the transporter to transport the B Rubin from space of dis to the cell and once buurin gets conjugated with glucuronic acid then it cannot go back most of it will go to the B Ali because those transporter protein which can transfer the bobin from the aptoite to the build Channel or build are only present in this part of the membrane right and there are so many other examples right that for example from here we can take up many drugs in from the blood and then drugs are modified and then some in some cases of the drugs their metabolites are actively secreted into where bcal system and in this case these metabolites will eventually go to git and lost into feal matter but there are other drugs those drugs are taken from the space of dis into the cell they are conjugated and thrown back to the blood so they will be lost into urine you are understanding the point by these examples I'm trying to put in your mind is that different faces of the parasytes are specialized in doing different functions that is why when you will study pharmacology you will learn the few drugs or their metabolites are lost in feal matter the other drugs their metabolites are lost into urine because liver is one of the major uh you can say organ which deals with drugs and uh conjugating the drug or changing the drugs we call it biotransformation right and it depends on the liver cell the throwing that modified metabolites of the drug into b or throwing it back to the space of Deion going to the blood am I clear no problem now this is the exact position of the B canalicular system right another thing which is very important that hpy cells are having lot of mitochondria 800 to 1,000 mitochondria present per cell they're having lot of lomes they have having lot of peroxone you know they have to destroy all some bacterias which may be wrongly coming from git to the portal blood so pyes should take them and cuffer cells should take them and Destroy them then from the G if the toxic substances are coming right they should be catabolized in hepatocytes so they have lot of lomes they have lot of peroxisomes then they have lot of lot of rough endoplasmic reticulum and smooth endoplasmic reticulum with smooth endoplasmic reticulum they are producing the lipids they are catabolizing the estrogens progesterone testosterone uh substances like that and with the rough endoplasmic reticulum these cells are synthesizing lot of plasma proteins right from the this side they will be taking up the amino acids then they will synthesize the plasma proteins like albumins and then they'll throw them back to the space of D so that they become the part part of the blood am I clear so and another very very rather extremely important thing in the space of Dy here there's a very special type of cell this cell is extremely important especially in pathological conditions physiologically this cell is acting as a fat fat storing cell or vitamin A storing cell you know vitamin A is lipid soluble am I right or wrong
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Medical_Lectures
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25_Biochemistry_Glycogen_Metabolism_I_Lecture_for_Kevin_Aherns_BB_450550.txt
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Captioning provided by Disability Access Services at Oregon State University. Kevin Ahern: Okay, folks, let's get started! How was your Thanksgiving break? Student: Good. Kevin Ahern: I bet everybody was really dying to get back and get into biochemistry? Student: No. Kevin Ahern: No? Student: I was, a little bit. Kevin Ahern: Look at this way, you go from one turkey to another. Student: Oh. Ha, ha, ha. Kevin Ahern: The old lead balloon, that's as good of a joke as I have for you today, folks. Believe it or not, we're in the home stretch. We have, counting today, three lectures. Whoo! The end is near. I did not bring note cards with me today. I said I was going to. I did not bring note cards, so I apologize for that, number one. I've got number two and number three, I think, here, also. I will bring them on Wednesday, so on Wednesday you will have note cards, for sure. If you're really desperate and you really have to have your note card, you can come by my office and pick one up, but realistically it's not going to change things an awful lot if you go 48 hours different without your note card. But it's a 5-by-8, I can tell you. If you want to get a 5-by-8 and practice on it, you can see it's a pretty good-sized note card. Student: Wow! Kevin Ahern: But you have to get the note card from me, remember that. So the note cards have to be gotten from me, and you don't have to use it, but you have to turn in a note card, with your name on it, that you got from me, with your final exam. If you don't, you will lose points. So make sure that you get a note card from me and you turn it in with your final exam. Are we clear? Okay, that's number one. Number two, I had a big presentation this morning and I was working my presentation last night and forgot to send you guys an email saying the exams are graded! So the exams are graded, and let me just say a few words about the exam. I was totally delighted with the exam. The performance on the exam was one of the highest averages I've ever had. The average was 76.5. I have a curve. I will post it as soon as class is over today. I will post the curve on the website so you can see it for the overall sum of your grades for the first two exams, so you can see exactly where you stand. The low score on the exam was 15. Student: Ugh. Kevin Ahern: The high score was 103. I had, I think, two or three people who had a perfect 103, including the extra credit questions. So I was very impressed and I was looking, as I looked at the grades, themselves, I saw that that average went up because a lot of people in the low-to-mid jumped quite a bit, and so it was very, very satisfying to me. I just felt very, very good about that and I really respect what people did with that, so that's kind of cool. So it was worth the wait, hopefully, for you to get your exam back. As always, if you have questions, let me know and we'll go with that. Number three is we have a final exam coming up. That final exam is in here, on Monday, at 9:30. I will do a review session for it. In fact, I have put in for, I believe it's Friday evening at 6:30, I have put in a request for a room. I will announce that when I get the room for sure, but the review session will almost certainly be Friday at 6:30. That gives you a chance to get dinner and then get review session. As before, I will videotape that. Now, let's see. What else do I want to say here? I want to say we finished almost everything about gluconeogenesis. The last thing I did not talk about on Wednesday of last week, which seems like a long time ago, by the way... Student: I know, it does. Kevin Ahern: The last thing I did not talk about there I wanted to save for today because it's kind of involved and so I wanted to make sure everyone had the same opportunity to see it and ask questions and so forth about it, and it's the combined regulation of glycolysis and gluconeogenesis. I'm going to start out by showing you a complicated figure. Actually, no, I'm not going to start with that. I'm going to start by telling you the sort of philosophy of glycolysis and gluconeogenesis regulation. The philosophy is that glycolysis is a catabolic pathway. Gluconeogenesis is an anabolic pathway. These pathways, for the most part, occur in the same place, which is the cytoplasm. Gluconeogenesis only has two reactions that aren't in the cytoplasm. One's in the endoplasmic reticulum and one is in the mitochondrion. All the other enzymes of both pathways are in the cytoplasm. Moreover, many of the enzymes are the same enzymes in both pathways, which means that many reactions are driven by concentration, which side of the equation has necessarily large enough amounts to drive a reaction one way or the other. That means that we have to be careful to regulate these pathways. If we don't control these pathways, we're going to have that futile cycle that I talked about before, where, imagine, let's imagine the following scenario. Let's imagine I had glycolysis and gluconeogenesis going at the same time. What would happen? I would start with pyruvate. I would put in six triphosphates to get to glucose. I would burn glucose and get two triphosphates and be right back at pyruvate. I wouldn't have gained anything, but I would have lost four triphosphates. And then I start it again and I go up, I go down, and each time I turn that cycle I lose four triphosphates. That's futile because it doesn't give the cell anything but heat. So it's important the cell not waste its energies, and the cell doesn't waste its energies by controlling pathways like that in what we call a "reciprocal" fashion. Reciprocal regulation is something you're going to hear a lot about today and you're going to hear a lot about it on Wednesday, also. Reciprocal regulation. Well, we start to see it right here. Here's a schematic, going down for glycolysis on the left, going up for gluconeogenesis on the right. Some of these things we've talked about already. Let's look at the regulators of this pathway. What you see on the screen are the allosteric regulators, the allosteric effectors of the important enzymes. In glycolysis, we know there are hexokinase, which is not shown, PFK, and pyruvate kinase. In gluconeogenesis, there are these two enzymes. They are this guy and also glucose-1,6-phosphatase, which is also not shown. So we just sort of throw out the first one up here. We just throw it out. These guys and these guys, we're very interested in. As I said when I talked about glycolysis earlier, the most important pair are these two, right here. PFK and FBPase-1, or fructose 1,6-bisphosphatase, if you want to call it that, PFK and FBPase-1 are regulated reciprocally. F2,6BP we talked about before. Notice that in very tiny amounts it turns this enzyme on. In the same tiny amounts, it turns this enzyme off. It has opposite effects on the two enzymes. Look at AMP. AMP turns this guy on. AMP indicates low energy. With low energy, we want glycolysis to go. PFK is activated. We look over here. PFKóI'm sorry, AMP turns off FBPase-1. Citrate turns off this guy. Citrate turns on this guy. It's reciprocal. It's not perfectly reciprocal. There are things that affect this one that don't affect this one. But when we look at the thing as a whole, F2,6BP is a reciprocal regulator. AMP is a reciprocal regulator. It has opposite effects on catabolic and anabolic enzymes. To a lesser extent, we see some of that down here. ATP turns this guy off. ADP turns this guy off. But we don't see the same kind of reciprocal regulation that we saw with PFK and FBPase-1. Now, reciprocal regulation turns out to be very, very important when we have pathways occurring in the same place, at the same time, or that can occur at the same place, at the same time. Cells generally regulate them so that they don't occur that way, for the most part. Well, it gets even a bit more complicated than that, because the question arises, I told you earlier when I talked about PFK and I said the most important regulatory effector for PFK was fructose 2, 6-bisphosphate, right? And I just showed you that it was a very important regulator for FBPase-1, as well. So, unfortunatelyóand you're going to say this as well as I doóunfortunately, it means we need to understand how do cells make and break down fructose 2,6-bisphosphate. That, you're going to see, it's going to look much more complicated than it is, so I'm kind of conditioning you for what I'm going to show you, and I'm also going to tell you that I can throw a million words at it. It's kind of like the mechanisms of serine protease action. We can throw a million words at it, but until you sit down with it and look at it yourself, it's going to seem like a million words. So let's take a look at the overall pathway by which fructose 2,6-bisphosphate is made and regulated. Remember, this is the reciprocal regulator of PFK and FBPase-1. This looks complóoh, Jesus, yeahóthis looks complicated. That's always the first reaction. It's not as bad as it seems. There's a lot of information on here, and the guts of its right here. All this is showing us is, on the side we're breaking it down. On this side over here, we're making it. There's an enzyme that makes it, and there's an enzyme that breaks it down. An enzyme that makes it, and an enzyme that breaks it down. Now, let's take a look at this enzyme. This enzyme is one of the most fascinating enzymes in biochemistry, because this enzyme is actually two enzymes. The same protein molecule catalyzes the synthesis and the degradation of fructose 2,6-bisphosphate. It's the same protein. This protein has two activities. One activity makes it. It's called PFK2. PFK2 catalyzes the synthesis of fructose 2,6-bisphosphate. FBPase-2 is the other half of it, and it breaks it down. Make it, break it down. Here's the enzyme. Here's the two activities. Well, as we can see, at any given time, only one portion of the enzyme is active. Only one portion of this enzyme is active. Oo-ooh! Good job! Not my day today. What's the difference between these two? The difference is a phosphate. If we put a phosphate onto this enzyme, we flip the activities. That turns FBPase-2 on. That turns PFK2 off. If we take the phosphate off, we favor the reversal of that. Well, that's not surprising. You've seen before how covalent modification of enzymes can affect enzyme activities. We're simply putting a phosphate on, we're taking a phosphate off. It has opposite effects. This causes the PFK2 to become active. Going to the right causes the FBPase-2 to become active. What catalyzes these things? Well, protein kinase Aóthere's our friendóprotein kinase A, when it's activated, catalyzes this enzyme getting a phosphate on it and FBPase-2 being active. Let's think about what that means in terms of the cell. If FBPase-2 is active, not looking at the screen what's going to happen? We're going to break down F2,6BP, right? When we break down F2,6BP, what's going to be the effect on FBPase-1 and PFK1? PFK is activated by this molecule, so if I take the molecule away, what's going to happen? Less active, right? If I go to the right, PFK1 is going to become less active. F2,6BP is an allosteric inhibitor of FBPase-1. If I take it away, what's going to happen to FBPase-1? It's going to be active, right? Well, since those are the critical enzymes controlling whether we're running glycolysis or gluconeogenesis, now you can look at this and say, in general, what's going to happen to glycolysis and gluconeogenesis if I phosphorylate this guy, right here. Well, I'm going to go over here. I'm going to break this guy down. I'm going to favor gluconeogenesis. And look, when glucose is scarce, that's exactly what I want to be doing. I want to be making glucose. Remember the flight or fright? Remember the grizzly bear chasing me and my adrenaline starts flowing, and I said that we had that kinase cascade, and the kinase cascade activated protein kinase A? There's our protein kinase A. And I said that the result of activation of protein kinase A resulted in production of glucose. This is one of the ways in which we make glucose. Not surprisingly, if we're making glucose, we don't want to be breaking down glucose, so we inhibit glycolysis, because we're no longer activating PFK with fructose 2,6-bisphosphate. So in one simple step, depending on how you look at it, of course, but in one simple step, we've reversed those two pathways. Well, what happens now when I've got my glucose stores back up? I've escaped the grizzly bear and I'm sitting around and eating pizza. I've got plenty of glucose around, and glucose is a... poison. So I've got to deal with that glucose. I've got two things I can do with glucose. I can break it down. I can turn it into glycogen. We'll be turning it into glycogen later in the week. Today, we're going to break it down. So when we no longer are activating protein kinase A, we are no longer phosphorylating. Phosphoprotein phosphatase becomes active, and, by the way, phosphoprotein phosphatase is activated by insulin. Insulin is causing this process to go to the left. Why? Glucose is a poison. We've got to do something with that poison. We're going to take phosphates off. We're going to activate PFK2. We're going to inhibit fructose bisphosphatase-2, FBPase-2. What's going to happen? We're going to start making fructose 2,6-bisphosphate, activate PFK1. Glycolysis is going to run. When fructose 2,6-bisphosphate is present, FBPase-1 is inhibited and gluconeogenesis stops. Insulin favors going to the left. Epinephrine favors going to the right. That, in a nutshell is what's happening. Now, I want you to lay this out yourself. I'll be happy to answer any questions, but I want you to just sit down, lay it out, and you'll discover it's really not that complicated. Yes, back there? Student: What about non-strenuous activity, where you happen to have an abundance or scarcity of glucose? Kevin Ahern: So what if you have the in-between situation, basically, is what you're saying. We have an in-between response. The body will generally modulate glucose levels to provide glucose, as needed, as much as possible. So maybe we'll phosphorylate, in this case, we'll burn some of our glucose. We'll phosphorylate some of this, but not all of this. Does that make sense? Student: Yeah. Kevin Ahern: Thanksgiving took all the questions out of you guys. Yes? Student: So the glucose production, that's happening in the liver, only, right? Kevin Ahern: Glucose production, gluconeogenesis, is happening primarily in the liver and a portion of the kidney. That's correct. Okay. So look it over. If you have questions, see me, but that's basically what's up with that. That is the last of what I want to say. Oh, here's the enzyme, by the way. There's the enzyme that's there. There's the part that puts the phosphate on. There's the part that puts the phosphate off, and there's that tiny little ribbon that connects the two of them. It's an amazing enzyme, absolutely amazing enzyme. We turn our attention now to something that is an easy metabolic pathway. It's going to concern us for the rest of this week. So you say, "Well, it's not an easy pathway!" Well, I'm going to convince you, I hope, that glycogen metabolism is actually one of the easiest metabolic pathways to learn. Its regulation is complicated, but the pathway itself is extraordinarily simple. Let's talk about glycogen. We talked about it earlier in the term, and glycogen is a storage form of glucose that animals use. It's a storage form of glucose that animals use. We talked about how plants use amylose and amylopectin. We combine those and we get starch, right? But plants don't have glycogen. What's the difference between glycogen and amylopectin? Anybody remember? Student: The linkages between [unintelligible] Student: There's more branches? Kevin Ahern: There's more branches in the glycogen than there is in the amylopectin. So they're all polymers of glucose. Amylose has only alpha-1,4 bonds, so it's just a long linear chain. Glycogen has alpha-1,4 linkages, but every now and then it has 1,6 branches. There's a 1,6 branch. About every ten residues or so, glycogen has a 1, 6 branch, which means that glycogen, even though it's full of glucose just like amylose is, is structurally very different. It has a lot of ends. The more branching you have, the more free ends we have at the non-reducing end. You remember what the non-reducing end is. Is this a reducing sugar or not a reducing sugar? How many say it's a reducing sugar? How many say it's not? I'm sorry but the person who said it was a reducing sugar was right. The very first one has a free aldehyde. The very first one has a free... it's alpha-1,4 linkages. There's 1,4. That means if this is the end of the molecule that would actually be an OH there and that could become an aldehyde. Student: So the last one on the right is the reducing sugar? Kevin Ahern: In this case, it would be, yeah. Now, that's not important. I'm just throwing that out at you, just to see what you remembered after all that turkey. The difference between glycogen and amylopectin, they're both branched. Amylopectin also has 1,6 branches, but it only has them about every 50 residues or so. I'm going to tell you in a second why that's the case, but that's the structural difference between amylopectin and glycogen. Did you have a question? Student: Yeah. Isn't that initial glucose subunit before all the branching takes place, on the very internal chain, wouldn't it be non-reducing because it's covalently attached to that little seed molecule that starts the whole thing off? Kevin Ahern: That is the seed molecule, right there. So if this is the end, then that's going to be an OH, right there. That OH makes it, a free anomeric carbon on an aldehyde on an aldose will always make it a reducing sugar. I'll show you the structure of that, if you'd like to see it. Come see me. Now, amylopectin's chemically different from glycogen in just the extent of the branching. Why is that important? Well, the reason it's important, and this is why you're able to be an animal, and I'm not talking about in any sense except for walking around, you people... [laughter] I know where your minds are! How are you able to be an animal? One of the most important ways in which you can be an animal is thanks to glycogen. Glycogen is stored in our muscles. It's also stored in our liver. It's in muscles for very quick energy. It's in our liver for providing that buffer to keep our glucose levels balanced, hopefully, over time. The reason that the structure of glycogen is so important to being an animal is because glycogen has so darned many ends. All those branches, all those ends, are important, because, as you will soon see, the way that glycogen is broken down is from the ends. More ends, more breakdown, more quick release of glucose. Animals have to run. They have to escape. They have to catch prey. They have to take notes in biochemistry. All those things require quick energy. Having a system that has a lot of ends allows for a lot of glucose to be released very quickly, when necessary. Plants don't have those needs. Plants don't go running away from their prey. If they could, they might evolve into something different. But they never made anything of themselves. They just kind of sit around like plants, right? "If only we had thought of making glycogen," plants say to themselves, "where would we be now?" But, no. You guys are really quiet today. Student: It's a Monday. Kevin Ahern: It's a Monday. Student: Thanksgiving, we had a four day break. Student: Yeah. Kevin Ahern: So do you see the fundamental difference? That chemical difference really plays out as a very important thing. Well, let's look at the metabolism of glycogen. Actually, this is whatóthere you go. Is that the figure you were referring to? Student: Wasn't the very, very internal molecule not a made-of-glucose molecule, though... Kevin Ahern: It is. It's a glucose, yeah. Everything in it is a glucose. What's that? You want to draw it on the exam, you said? Student: No! Student: What?! Student: You could just draw a bunch of lines. Kevin Ahern: No, you've got to draw it, we'll line it up and we'll put it on top and see. Nope. No partial credit. Sorry. [laughter] Student: Oh, god. Kevin Ahern: Let's look at the breakdown of glycolysis. There's what glycogen looks like. That's these little black guys here. Fates of glycogen. Glycogen turns out to be important as a source of glucose. But, of course, we know glucose is not the end of the story because glucose, by itself, doesn't do anything except poison us. We want to have the energy from glucose, which is why we have glucose around in the first place, and what this is showing you is what happens when we break down glycogen and how it's converted into energy, the glucose in it. I'm going to show you in a second an unusual reaction. It's a really cool reaction. The glycogen isn't broken down directly into glucose, for the most part. Ninety-nine percent of it, or, ninety percent of it is broken down into this guy, right here, glucose 1-phosphate. Where did we see glucose 1-phosphate before? Anybody remember? Student: Glycolysis? Kevin Ahern: Not glycolysis, no. Galactose metabolism. Do you remember when we had the UDP glucose and it got released, and it was released as you don't remember thatóglucose 1-phosphate. I told you, at the time, glucose 1-phosphate would be important in glycogen metabolism because it can readily be converted into glucose 6-phosphate. This enzyme phosphoglucomutase allows this interconversion. It can go up, It can go down. It's pretty much equal in terms of which direction it goes. Student: Is it "phophoglucomutase"? Kevin Ahern: Ha-ha-ha-ha! What are they doing this in this textbook? "Phophoglucomutase." [laughter] That is now an acceptable name for this enzyme. If you want to call it "phosphoglucomutase," you can. If you want to call it "phophoglucomutase" [laughing] or "phuphuglucomutase," I don't care. [laughter] Now, glucose 6-phosphate can go to glycolysis. That's important. Glucose 6-phosphate can get released as glucose and go into the bloodstream, if this happens in the liver. Glucose 6-phosphate can be converted by the pentose phosphate pathwayówe'll briefly talk about that next termóinto ribose, and ribose is very important for making nucleotides. So this molecule is central to a lot of different pathways. How do we get glucose 1-phosphate? Let's take a look at that. Here's the end of a glycogen molecule. One of those ends that we talked about, one of those millions or thousands of ends that are on the end of a glycogen, we're sitting at it right now with an enzyme that breaks it down. The enzyme that breaks this down, that catalyzes this reaction, is known as "glycogen phosphorylase," P-H-O-S-P-H-O-R-Y-L-A-S-E, unless you're a textbook publisher, in which case it's called "phophorylase." [scattered laughter] Now, this is a reaction like you haven't seen before. It looks very straightforward. Here, we've got a glycogen molecule. Here, we've clipped off a glucose 1-phosphate, and here's the glycogen that's lost one of its residues. Very straightforward, right? Well, not quite. Look what's happened. We put a phosphate on there, in the process. How did we put that phosphate on there? We didn't use ATP. When we talked about putting ATP onto glucose before, we said that took energy, right? Where did the energy come to put this phosphate on here? Any thoughts? Wild ideas? Yes, sir? Student: It's energetically favorable? Kevin Ahern: Why is it energetically favorable? It is energetically favorable, but why? Student: Negative Delta G zero prime? Kevin Ahern: Why is the Delta G zero prime negative? Student: Is there energy in breaking that bond? Kevin Ahern: There's energy in breaking this bond. This bond has some energy in it. The energy in breaking this bond is transferred to making glucose 1-phosphate. So it tells us that that alpha-1,4 bond has some energy in it and that we can use that energy to make something. Well, why do we want to do that? Well, it turns out, whenever we can save energy, that's good, just in general, right? Insulate your glycogen, right? So that you don't... no. Alright. You don't waste energy, you see, if you insulate your glycogen. Alright, Anyway. We got a phosphate onto here and we didn't have to invest ATP energy. We just saved a triphosphate. Muscle cells, if I am running and jumping, I don't want to burn my ATP breaking down my glycogen. I want to burn my ATP using the energy from glucose. This allows me to put a phosphate on there without using any ATP energy. This is really cool because now I can isomerizes this guy to make glucose 6-phosphate andóbang! I'm in glycolysis without investing any ATP to start. Very good. So this saves a reaction. The enzyme is called a phosphorylase. The name, again, tells us what it does, meaning it uses a phosphate to break a bond. It uses a phosphate in breaking a bond. It's different than a hydrolase, which uses water to break a bond. So instead of using water, we're using phosphate to break that bond. We're almost done, okay? We're almost done. There's only one other thing I have to tell you, and that is the fact that glycogen phosphorylase is a finicky enzyme. Of course it's a finicky enzyme. It has to be, right? Glycogen phosphorylase will only work to within about four residues of a branch. It gets to that point. It starts up here. It keeps chewing, chewing, chewing. It takes these red guys off here, and it says, "I ain't going any further." It will not work any closer than about four residues to a branch, the branch being a 1,6, right there. Then, something else has to happen. Well, the something else that has to happen is another interesting enzyme that has two activities associated with it, but we branch them into one name. We could memorize that it's called a transferase and we could memorize that it's called an alpha-1,6-glucosidase, but we, being biochemists, are kind of lazy. We like to call both of these activities "debranching enzyme." I'm going to tell you what debranching enzyme does, but these two reactions are catalyzed by the same enzyme known as "debranching enzyme." What happens? Well, let's look to see what this enzyme is doing. Follow the blue guys. Here's the three blue guys here. The three blue guys get transferred from this branch down to this branch. That leaves behind one green guy. They're all glucoses, by the way. So they're all glucoses. They're not different. The difference being this guy is linked by an alpha-1,6. The enzyme, debranching enzyme, uses water to break that guy off and we get free glucose. This is the only place we get free glucose in glycogen metabolism. Student: And then the glycogen phosphorylase will then be able to... Kevin Ahern: Then glycogen phosphorylase now has a new template it can work on and it can go chewing back until it gets back to another branch. Student: The free glucose, the green one, is that [unintelligible]? Kevin Ahern: The green one is the only free glucose that's released in the process. Student: [unintelligible] Kevin Ahern: What's that? Student: [unintelligible] Kevin Ahern: Right. So you might wonder, well, why in the other case does it use glucose 1-phosphate it used glucose to make glucose 1-phosphateówhy, in this case, is it releasing free glucose? It's not being consistent. No, there's something that's different here. What's different here? Student: water Kevin Ahern: It's using water, but why doesn't the other one use water? Why doesn't this one use phosphates, is my question? Student: It's not a high energy Kevin Ahern: It's not a high enough energy bond. An alpha-1,6 does not have as much energy as an alpha-1,4 does. It doesn't have the option. Well, fortunately, there's only one of these per branch that's made, so the cell says, "Okay, I'll take and use some ATP and put you into glycolysis." Bang! You got it. Student: So the debranching enzyme requires ATP? Kevin Ahern: Debranching enzyme? No. There's nothing here that requires ATP. Getting that into glycolysis requires ATP. Okay, questions? Now, believe it or not, with the exception of the phosphoglucomutase that's needed oop, turn that guy off the phosphoglucomutase that's needed to convert the glucose 1-phosphate into glucose 6-phosphate, you've just seen how you break down glycogen. Bang! What enzymes did we see? Phosphoglucomutase interconverts glucose 1-phosphate and glucose 6-phosphate. It's a mutase, so what does that tell you? It has a 1,6 intermediate, and, yes, that can get released as a free molecule. It does get released as a free molecule. The second enzyme was glycogen phosphorylase, that broke 1,4 bonds close to a branch, and the third enzyme was debranching enzyme, which changed the branch and released free glucose. Three enzymes in the entire pathway. Cool! Glycogen breakdown is very simple. I'm going to talk about glycogen synthesis in a second and you're going to see it's almost as simple. Here's the phosphoglucomutase. This is the glucose 1-phosphate. There's the intermediate. There's the product, glucose 6-phosphate. This is a reversible reaction, either direction. If we have excess glucose 1-phosphate, it'll go to the right. If we have excess glucose 6-phosphate, it'll go to the left. When would we have excess glucose 6-phosphate? What conditions would give us excess glucose 6-phosphate? What metabolic pathway...hint, would give us excess glucose 6-phosphate? Student: Gluconeogenesis. Kevin Ahern: Gluconeogenesis, right? So if a cell is building glucose, it's going to be building glycogen, too. We'll see in a second that glucose 1-phosphate is needed to make glycogen. So if we're making things in gluconeogenesis, we're going to the left. If we're breaking things down in glycogen breakdown, in glycolysis, we're going to the right. Yes, ma'am? Student: Which one did you say is reversible? Kevin Ahern: The entire reaction is reversible. Student: Oh. What are the yellow things? Kevin Ahern: That's just part of the enzyme. So there's the active site of the enzyme. There's the rest of the enzyme. There's the serine residue that's involved. That's really all it is. It's just showing you that side chain. Alright. DIPF would to do what to this enzyme? Student: Inactivate it. Kevin Ahern: Inactivate it, right? Okay. I should have asked you what the molecule was that'll do it. Okay. I'm going to jump down to glycogen synthesis, because I think if we talk about the metabolism and then we save the regulation for later we'll be better off. So let's talk about the synthesis of glycogen. It's just about as simple as the breakdown is. There's one extra enzyme, one extra enzyme. So, the enzyme, again, we think "phosphoglucomutase" for interconverting. Now we want to make glucose 1-phosphate, because we want to make glycogen. But it turns out that glucose 1-phosphate can't be added to a growing glycogen chain. Why? Well, remember that alpha-1,4 bond had some energy in it? Right? If it has energy in it, then we have to put some energy into making that bond, and there's not enough energy in water, essentially, to make that bond. So we have to use a high-energy intermediate in order to make that alpha-1,4 linkage. The high-energy intermediate we use is this guy, right here. You saw it before. You saw it when we talked about galactose metabolism. This was a molecule I described as an "activated intermediate." An activated intermediate is a molecule that has a high-energy bond, and there is the high-energy bond. It's a molecule that has a high-energy bond that uses the energy of that bond to transfer a part of itself to something else. So an activated intermediate is a molecule that has a high-energy bond and it uses the energy of that bond to transfer a part of itself to something else. Well, the part of itself it's transferring is this guy, right here, glucose. What it's going to do is attach it to position 4 of a glucose on the end of a growing glycogen chain. If we're going to talk about the enzymes of glycogen synthesis, we have to talk, first of all, about how do we make this molecule. Once we know that, everything else is pretty much like glycogen breakdown. Let's take a look at how we make that. Here's the reaction that makes UDP-glucose. Glucose 1-phosphate, okay, you know how that's made now. Glucose 1-phosphate we combine with UTP. We make UDP-glucose and we make what's called pyrophosphate. Those are two phosphates joined to each other. Let's count the phosphates. One, two, three, four. One, two, three, four. We haven't lost any phosphates, but they've reorganized. Now we have this guy and we have this guy, over here. Student: What did you say the name was? Kevin Ahern: It's called "pyrophosphate," P-Y-R-O-P-H-O-S-P-H-A-T-E. Pyrophosphate means two phosphates covalently linked to each other. Well, we've just made an activated intermediate. What did it take to do it? It took a triphosphate. UTP has the same energy as ATP does. It has the same energy as GTP does. That triphosphate is high energy. The cell is having to invest some energy into making this bigger molecule. That's a fundamental principle of anabolism. Building bigger things takes energy. It took energy to make glucose. It's now taking energy to make glycogen. We're nearing the end, believe it or not. UDP-glucose. What's the next step in the process? Well, the next step in the process is adding that glucose to a growing glycogen chain. This is the reaction that's catalyzed, here. There's the UDP-glucose that we just made. Here's carbon number 4 of the end of a glycogen chain, right there. In this reaction, this glucose gets transferred over there. The energy of this bond is used to make this high-energy bond. We've now made a glycogen that has one more glucose on it. The enzyme that catalyzes this reaction has a very simple name. It's called "glycogen synthase," S-Y-N-T-H-A-S-E. Glycogen synthase catalyzes the addition of glucose to a growing glycogen chain. The product is UDP, of course, and UDP can be converted into UTP and then reused again. Now, we're only missing one thing. What are we missing? How do we get branches? Well, for branches, we've got a really complicated enzyme name that's used to do it, but I prefer to call it "branching enzyme," as I'm sure you will, too. There is, believe me, it's a mouthful of a name. It's about that long, okay? But, in essence, branching enzyme will create alpha-1, 6 branches about every ten residues. Here's an alpha-1,4 linkage. Here's a branching enzyme. Bang! Got it! So branching enzyme is creating the branches. So what enzymes have we seen in glycogen synthesis? Well, we saw phosphoglucomutase, as before. I didn't give you the names of the UDP-glucose synthesizing enzyme, did I? Student: No. Kevin Ahern: Do you really want it? Student: Nope. Kevin Ahern: Should we give it a name? I'll tell you what the real name is and then you can tell me perhaps a more humorous name. The real name is UDP-glucose pyrophosphorylase. Student: Steve! [laughing] Kevin Ahern: Steve. These are all male names. Do we have any female...there's never a female...it's true, every year when I ask for names people always give me male names. Student: Helga. Kevin Ahern: Ursula! Student: Tina. Kevin Ahern: Tina? Student: Amaryllis. Kevin Ahern: Amaryllis? Student: Shaniqua. [laughing] Kevin Ahern: I'm sure they'd like to spell that one. So you may call it either UDP-glucose pyrophosphorylase, which is the real name, or... I'm going to vote on this. I don't know. I think the best names I've heard were Steve... Tina... Student: Lucy. Kevin Ahern:...and Ursula...Lucy! And Lucy. Okay. Steve, Tina, Ursula, Lucy. Steve? Tina? Ursula? Lucy? Lucy is the simplest one, I think. People wanted Lucy. "Lucy in the Sky with Diamonds," right? Student: What was the real name? Kevin Ahern: It's the enzyme that catalyzes this reaction right here. Its real name is UDP-glucose pyrophosphorylase. UDP-glucose pyrophosphorylase. That's the breakdown. That's the synthesis of glycogen. I'm going to cut short early today but I'm not going to finish quite yet. I just want to say one last thing, and that is, on Wednesday I'm going to talk in detail about the regulation. The regulation is reciprocal, but it's also complicated. It involves both covalent modification and allosteric regulation. If you want to look over a lecture material before you come to lecture, next time might be a good one. See you Wednesday. [indistinct conversation] Kevin Ahern: Yes, sir? Student: [unintelligible], why was that a pyrophosphate instead of a bisphosphate? Kevin Ahern: What's that? Student: [unintelligible] Why is that a pyrophosphate instead of a bisphosphate, when it's free floating? Kevin Ahern: I think the term's interchangeable. Student: Okay. Kevin Ahern: Yeah. Student: So "pyro -" means "bond"? Kevin Ahern: Just bond, yeah. Yeah. Student: Okay. Thank you. Student: I didn't catch where you said we could pick up our exams. Kevin Ahern: Yes, they're at the BB office, in ALS-21. Student: Okay. Thank you. Kevin Ahern: Sure. [indistinct conversations] [no audio] [END]
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Medical_Lectures
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Hemostasis_Lesson_5_Antiplatelet_Meds_Part_2_of_2.txt
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[Music] this is part two of anti-platelet medications and I'll be starting off with the P2 y12 Inhibitors the P2 y12 Inhibitors predictably inhibit the P2 y12 ADP receptor on the platelet surface although it seems that most Physicians are not familiar with the formal name of this particular medication class they may instead know them as the theop pirines since most fall into that particular chemical classification these include cigil commonly known by the brand name of Plavix prasugrel which is occasionally known by its brand name of effent and the rarely used tadine theop Pines are all based upon this structure other common features of the theop iines include oral formulations only they are all prod drugs which require invivo biotransformation into the active drug and they all irreversibly inhibit platelet aggregation in addition there is one non theop paradine P2 y12 inhibitor which is called tagore tagore is a nucleoside analog with a chemical structure similar to that of denzine compared to the endopin tagore binds reversibly to the P2 y12 receptor has a quicker onset of action and a shorter duration of action thus it needs to be given twice daily which is its major disadvantage compared to the other P2 y12 Inhibitors what are the common indications for the p212 Inhibitors acute coronary syndrome in addition to aspirin anti-coagulation and plus minus a gp2 b3a inhibitor the specific choice of drug depends on the situation if receiving PCI that is the patient is getting angioplasty and a stent the general preference is for either prasugrel or tagor if the patient is receiving fibrinolytics the preference is for citril and if receiving neither PCI nor fibrinolytics the preference is also for cigil clidr is also used for secondary prevention of non-cardioembolic stroke or Tia but only by itself and not in combination with Aspirin combining cigil and aspirin in this situation results in no improvement in stroke reduction but does result in an increase in clinically relevant bleeding finally for secondary prevention of cardioembolic stroke or Tia citril plus aspirin is used in patients who are not candidates for oral anti-coagulation for reasons other than bleeding risk since the risk of bleeding from cpit plus aspirin is similar to that from anti-coagulation keep in mind that these indications are very fluid based on evolving data and could become obsolete with the next big clinical trial there are also local institutional preferences which may differ slightly and the cost of some of these drugs may be an additional consideration in some areas a common question which comes up regarding use of these drugs is how long should they be used after PCI patients who have received a bare metal stent absolutely must be treated for at least 1 month to prevent instant thrombosis patients who have received a drug eluding stent absolutely must be treated for at least 3 months in the absence of a contraindication such as bleeding or non-elective surgery evidence supports continuing treatment for 12 months irrespective of choice of stent and some cardiologists prefer treating Beyond 12 months but there is currently insufficient evidence to broadly recommend this some additional notes about the p212 Inhibitors clil has a longer onset of action compared to prasil and tagore and may be less effective at preventing cardiovascular morbidity but is associated with less bleeding Paras gril is contraindicated in patients who are 75 years or older and in those with history of stroke or Tia as a consequence of observed higher rates of interradial bleeding in these patients it also needs to be dose reduced or avoided altogether in patients of low body weight do p212 Inhibitors increase the risk of major bleeding in patients undergoing urgent coronary artery bypass surgery the common approach to this issue is to withhold the drug for 5 to S days prior to surgery however there is some variability in this practice and there is insufficient evidence to make any specific recommendation finally I mentioned earlier that copine is rarely used that's because it's associated with trenia and a potentially fatal hematologic condition called thrombotic thrombocytopenic perpa which I will discuss later in this series The Last aspect of P2 y12 Inhibitors to discuss is that of cigal resistance also known as cpil non-responsiveness or high on treatment platelet reactivity this refers to a lack of clinically meaningful inhibition of platelet activity after taking cigil proposed ideologies of this include variations in the metabolism of the drug into its active form as well as drug drug interactions several years ago there was a lot of concern regarding some modest quality evidence that proton pump inhibitors May interfere with the action of cigil but based on the current body of evidence this seems to most likely not be true however remains controversial and some physicians may still show preference away from concurrently prescribing cigil and ppis the generally accept aced clinical presentation of cogal resistance is simple recurrent thrombotic cardiovascular events despite treatment to me this seems to be overly inclusive since a recurrent event is not necessarily indicative of there being no meaningful platelet inhibition after all patients on aspirin still get recurrent events all the time nevertheless that's the definition some of the literature uses despite the fact that this sounds like it should be a big deal routine screening of patients for cpog resistance is surprisingly not currently recommended since it hasn't yet been shown to lead to any benefit the next class of medications is the GP 2b3a Inhibitors recall that the 2b3a receptor serves as part of a bridge along with fibrinogen and Von willbrand factor to attach activated platelets to one another Inhibitors essentially prevent these receptors from doing that job as mentioned at the beginning of the video there are three members in this class eptifibatide and tyrin are relatively small non-antibody Inhibitors while apomab is the Fab fragment of a chimeric human murine monoclonal antibody all three of these drugs are given via continuous IV infusion initially when they were first developed the 2b3a Inhibitors had a larger role in the management of ACS but with the subsequent development of p212 Inhibitors their role has become somewhat reduced in addition the current body of literature on them is quite dense and looks at very specific combinations of drugs in specific situations generally speaking the current indications for 2b3a Inhibitors include use at the time of pcii in an N stemi if either the chosen concurrent anti-coagulant is unfractionated heprin and the patient receives pre-treatment with citrail or if the patient does not receive adequate pre-treatment with any P2 y12 inhibitor irrespective of choice of anti-coagulant in addition they can also be used in unstable angino or n stemi prior to PCI if there is evidence of ongoing es schea irrespective of the choice of anti-coagulation and P2 y12 inhibitor in practice I currently find their use prior to PCI to be quite uncommon the next drug to discuss is dimol it has multiple mechanisms of action first as a phosphodiesterase inhibitor it inhibits pde5 and to a lesser extent pde3 this raises platelet CP and cgmp with a secondary effect of decreasing platelet responsiveness to ADP it also inhibits the re-uptake of adenosine by red blood cells which leads to higher concentrations of plasma adenosine and subsequent platelet inhibition dimol also acts as an antioxidant which scavenges the free radicals which normally inactivate cyc oxygenase thereby enhancing pgi2 synthesis by the endothelium and finally it acts as a coronary vasodilator in effect not directly related to hemostasis but is related to its use in myocardial nuclear Imaging where it is marketed under the tray name pantin as in a pantin stress test despite all of those mechanisms for the purposes of preventing hemostasis Di is not very frequently used in fact I've never personally seen it used outside of a combination with aspirin in a medication marketed under the trade name aggrenox an aronox is only used for the secondary prevention of non-cardioembolic stroke or Tia because it has an equivalent Effectiveness as cigil for this indication and because agox is a twice daily medication with a relatively common side effect of headache most Physicians prefer prescribing cigil in this situation the final medication to discuss is costasol this acts as a selective pd3 inhibitor which increases CM leading to Plate inhibition and Vaso dilation its indication is primarily for symptomatic Improvement in peripheral artery disease that is it reduces claudication there's also limited data supporting possible use for secondary stroke prevention in Asian populations though at least in the US I've never seen it used for this reason costasol may take four weeks for symptomatic benefit in patients with claudication so patient should not write it off if they don't see any Improvement after just a few days and finally it is considered to be contraindicated in congestive heart failure so those are the anti-platelet medications in current use here's a table summarizing the medications their mechanisms and their common indications I'm not to go through the table line by line as you can obviously pause the video If You' like to review it on your own that concludes this two-part video on antiplatelet medications sorry it was on the longer side but it's kind of a big topic similar to the next one on anti-coagulation
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Medical_Lectures
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06_Biochemistry_Protein_Purification_Lecture_for_Kevin_Aherns_BB_450550.txt
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Captioning provided by Disability Access Services at Oregon State University. Kevin Ahern: So we're moving along nicely with the schedule. And though we don't have to stay on it exactly, we've been on it pretty good. I've been pleased with the interactions and also pleased with the questions I'm getting, both in class and out of class. So you guys seem to be engaging in this material and that's a very good indicator of success. So if you have questions, please feel free. Come see me. Come see the TA's. And we're here to help in any way that we can. I only have one tiny little thing to say today regarding the last of protein structure. It's actually sort of an anecdote more than anything else. And then I want to talk about techniques for characterizing and/or purifying proteins. One of the things that biochemists spend a tremendous amount of time doing is just that: isolating, characterizing, understanding proteins, enzymes, etc. And so what you've learned so far about structure of proteins, you will discover will be useful as tools for learning how to isolate them. And so I'll spend some time talking about that today and also on Monday. The anecdotal thing I wanted to mention to you is the very last item on the protein structure page, and it's actually this right here. I've mentioned hydroxyproline to you already and I want to reiterate something here. Now, if you recall, I said that there are 20 amino acids that we find commonly in proteins, but we find modified amino acids in proteins. And the point that I want to emphasize is that those modified amino acids that we see happen post-translationally, meaning that the modifications occur after the amino acid is built into the protein. So, in the case of hydroxyproline, for example, I gave you, I showed you or described to you how Vitamin C was involved in that reaction that modified the proline. That happened after the proline had been built into the protein. The same is true of all the other things there are here. Carboxyglutamate is an important modification, as we will see, that occurs as an important consideration in blood clotting. Carbohydrate-asparagine adduct, where we see, in this case, addition of a carbohydrate to an asparagine residue, this is really imporant in the production synthesis of glycoproteins that we'll talk a little bit about later. Phosphoserine, Phosphorylation is something that you're going to hear a lot about later in the term because phosphorylation is a means of controlling or signaling through proteins. And it's a very, very important mechanism for us to understand. It's specifically phosphorylation that I want to address briefly at the moment. And that is that phosphorylation of amino acids has to occur on side chains and side chains that have hydroxyl groups. So the three amino acid side chains that have hydroxyl groups, of course, are the tyrosine, serine and threonine. These are the three amino acids that get phosphorylated or can be phosphorylated. And we'll see a bit of a pattern to how that phosphorylation occurs. Not surprisingly, you might think, well, why do these have such big effects? You saw a big effect with hydroxyproline because it was a part of that important structural consideration for making a strong collagen. In the case of phosphorylation, what we're doing is we're converting, excuse me, we're converting a side chain from being hydrophilic to actually being ionic. And so, in essence, what we've done is we've changed it from, say, a partial charge to a fully negative charge. In this case, we see two minus groups there. Now, based on what I told you so far about protein structure, you might imagine that changing the charge of a specific location of a protein might have structural considerations for that protein. Imagine that previously we had a negative charge, let's say a glutamic acid residue, that was close to this proline before we put the phosphate on there. When we put the phosphate on there, here's this negative charge before that didn't really have that much interaction with the OH, but now there's two minus charges over here. What's going to happen? Well, of course they're going to repel, and when they repel, that's going to change the configuration. It's going to change the shape of that protein slightly. And, as we will see, and I've mentioned previously, changes in the shape of proteins can have some dramatic effects on the action of those proteins, and we're going to talk more about those as we get further along. So those are some things that are other modifications that can happen to proteins. But I want you to be aware that virtually any time you see a modified amino acid in a protein it is because it has happened after the amino acid has been put into the protein. Okay, so that's the last of what I want to say about general considerations of protein structure. Now I'd like to turn our attention to characterizing proteins. The first part of the characterization I'll talk about is actually purification. And purification isn't a spiritual purification, but it's actually a physical purification. I'll tell you a brief story. When I was working in my very first lab after I had graduated, I worked in a laboratory where we did HPLC, and we had to have very pure solvents. And I was very impressed by this notion of purification that happens in there, the need for purity in all biochemical materials. And so I was very, very impressed with these solvents that we used, and we got them from this company that had purified stuff. So I remember writing a letter to the companyótongue in cheek, of courseósaying that, you know, we found that not only were their solvents very pure, but we had to do a spiritual purification of these solvents before we used them, as well. Of course, I wrote this as if it were completely serious, sent it to the president of the company, and, to my delight, I got this letter back from the president of the company congratulating me on describing for him a new way of purifying his solvents that he could use for HPLC. It was a good exchange. So purification really had a big impact on me as a very young biochemist. Purification is important. When we want to characterize, let's say, a protein or an enzyme, we need to have it isolated away from everything else. When we try to understand an enzymatic reaction, for example, we say, "Okay, well, ìI'm interested in this enzyme. ìI'm interested in the reaction that this enzyme catalyzes." If I only have the soup of the cell, that is, the cytoplasm of the cell that contains this, I not only have that one enzyme that I'm interested in, but I have several thousand other enzymes in there. So it's important for me to understand what this enzyme does that I be able to purify this enzyme away from all those other proteins. And so understanding how to purify one protein apart from others is a very, very important consideration in biochemistry. Well, there are several techniques that we use in order to do this, and I'm going to go through and sort of describe a few of the basic ones to you and then show you some of the applications of these technologies. You can't walk into a biochemistry lab without finding a centrifuge. It's almost impossible to do that and that's because the use of centrifugal force as a means of separating molecules on the basis of their size is a very, very valuable tool. Not surprisingly, different things can be spun down. We talk about "spinning them down." That is, will they precipitate out of solution or will they move to the bottom of the tube? The function by which that, by which they occur, is a function of their size and the speed with which we spin things. So the largest things, of course, as you might imagine, spin most easily to the bottom. So if I take and I'm interested in studying an enzyme in E. coli cells, I can take a batch of E. coli cells and I could use a fairly light centrifugation and spin, and those cells would come to the bottom of that tube. Let's say I took that pellet which we get out of that, and I'm interested in, not just the cells, obviously, because I'm interested in the enzymes that's inside, I can use some techniques to bust 'em open. I might use sonic waves to do that. I might use enzymes to do that. I might use mechanical agitation to do that. It doesn't really matter the means I use. But when I do that, I basically open up the contents of the cell and the insides spill out. Those insides are going to have some things in them. And, in addition, I'm going to have some cell walls that are sitting there, that now are empty of their contents. I could spin those down again. And if I did that, I would basically have done my first separation. I would have, on the one hand, the pellet, which would contain the very big things, like those cell walls, and I would have the liquid component, which would be the cytoplasmic material that I was interested in. I could take that cytoplasmic material. I could do various centrifugations on it, if I chose to. And I could separate them on the basis of the size of those complexes that are in there. Now, we don't need to memorize numbers or anything like that, but I do want you to understand that centrifugation allows us to do a sort of a rough separation based on size, a very rough separation. The stronger the centrifugal force, the more things I'm going to pellet, I'm going to drive to the bottom of the tube. And there's a lot of different techniques involved in centrifugation that allows me to purify things. Now, centrifugation alone will notóunderline "not"ógive me pure material. So it's used mainly as a means of what I would describe as fractionating. When we fractionate things, we break them into smaller pieces and then we work with those pieces to do things of interest to us. Let's imagine, for a moment, we've got two possibilities. I took these E. coli cells that I was describing to you, and I'm interested in understanding a particular protein. The first question I would ask is, "Well, where's this protein?" Is this protein in the cytoplasm? Or is this protein embedded in the cell membrane? Because both of those are possible. The beauty of this is, if I've fractionated it in this way, I've got one fraction that has only things in the cell membrane and I have another fraction that has only things that's in the cytoplasm. Then I can subdivide those further, and that's some of the other things I'm going to be describing to you. So centrifugation, a very rough but powerful tool to allow us to start to separate things in the process of isolating components of cells. Another techniqueóyes, question? Student: Is size like actual physical size, or is it like [unintelligible]? Kevin Ahern: Yes, good question. So is it actually the physical size? Does density or mass play a role? And all of these are variables in how things will separate. So yes, those are factors, especially as we get smaller and smaller, some centrifugation techniques actually work on individual proteins, and what we discover with that is that proteins that are very compact migrate through the centrifugal field very differently than those that are very open. So, yes, those are all considerations, and we're not going to need to dissect those out, but yes, you're correct, they do affect things. A second technique that we would commonly use in a biochem laboratory, it's probably one you've played with in biology laboratories, either in high school or college, and that's dialysis. Dialysis tubing is pretty cool stuff. It is, basically, if you've never played with one, it's basically a tube that is semi-porous. It's semi-porous in the sense that it can allow water molecules and small ions, for example, to move through it, but larger things, like proteins and DNA, can't move through it. And in biology labs we commonly use this as a way of illustrating the concept of concentration and osmosis. If I have a solution, for example, that has a situation hereóhere's my cytoplasmic mix, and let's say it's full of salt, which I want to get rid of as much of the salt as I can, I would put it into a piece of dialysis tubing. The salt ions, the sodium and chloride, are pretty small. They will pass through the tube fairly readily. The larger guys, my proteins and so forth, won't pass through that tubing. And after a period of time what I will see is that the concentration of those salt ions inside the tubing has decreased considerably as a result, and, conversely, some water will actually enter that tubing. And the reason it will enter that tubing is it's trying to basically dilute out the things that won't come out, that is, the proteins and so forth. So I see a pressure that arises as a result of that. Yes, Shannon. Student: Isn't it, would it be impossible to actually get rid of all the salt molecules? Kevin Ahern: Is it impossible to get rid of all the salt that way? In theory, yes it is, because I'm depending upon a differential concentration, and even though I get it lower and lower and lower, in theory, I could never get it completely out. You're correct. But this technique will give us a very nice simple way of getting rid of a lot of small ions very readily. And, in doing this, I have actually increased the concentration of my protein relative to the other things that are there. A technique that is a useful technique that I'd like to describe to you is that called "gel filtration" and it's also called "molecular exclusion." You see "molecular exclusion" over here. "Gel filtration," we use the two terms interchangeably. I actually sort of prefer "molecular exclusion" but either one is acceptable, as far as I'm concerned. Now, to understand this technique, we need to understand the sort of physical nature of the separation. So to use this technique, I have to have something that's pretty cool. So I have what's called a background or matrix material which consists of millions or I shouldn't say "millions," but thousands and thousands of tiny beads. Little beads, maybe a millimeter or so in size, big enough for your eye to see individual beads, but they're still pretty tiny. These beads have a characteristic. The beads have little tunnels through them, little tunnels. And the little tunnels have openings that are pretty uniform in size. That turns out to be important. So I've got a bead, I've got tunnels, and the opening to those tunnels is uniform in size. So to use this technique, what I do is, I take my beads and I suspend them in a buffer. So I suspend them in a buffer, and the reason I want to use a buffer is I don't want the pH to be too high or too low. I want the protein to be stable, because if I change the pH too much, again, I'm going to denature it, unfold it, and cause some problems. So I have it in a buffer. I take that sort of buffer containing these beads and I sort of shake it all up and get it into a nice slurry. Then I carefully pour it into a column. And the beauty of this is that the beads, of course, can't come through the bottom. They get stuck right here. And they form a column of beads. So I've got thousands and thousands of these beads, each with little tunnels through them, each with a hole that's a set size. And, yes, I can get beads with different holes of different sizes. But for any given experiment, I'm doing one size of hole for one bead and I've got thousands and thousands of those beads. Once I have such a column, I might run my buffer through it for a little bit, just to make sure that it's washed all the other junk out and so forth. And then I've got a mixture of proteins that I'm interested in separating on the basis of size. This is a technique that allows us, again, to separate on the basis of size. The exclusion part of the technique goes as follows. I've got in this mixture of proteins some that are very, very large, maybe 200,000 in molecular weight or greater. I've got some that are, let's say, medium size, maybe 50,000 weight or greater. And I've got some that are fairly small, maybe 5,000 molecular weight. And just as an example, I just picked those three ranges, What's going to happen with these three sets of proteins relative to these beads? Well, it turns out that the holes that I've chosen in these little beads that I've got are such that they will only let in things of a certain size. There's a size exclusion. So the great big 200,000 molecular weight proteins won't fit in the holes. They will not enter the beads at all. The 50,000 are borderline, they might be able to enter a few, but they don't really enter very effectively. And the 5,000 molecular weight proteins that I have will basically see a hole and they'll go into it, just because they can. Well, if I apply these three to the top of the column and then I let buffer sort of push everything through, what I see is as follows. The 200,000 molecular weight proteins will not enter the beads and they will travel a very short path through the column. They just go shooting right through. They're the very first thing that comes through the column because they don't get distracted by going through all these little tunnels on the way. The 50,000 molecular weight proteins, that can make it into some of those tunnels, travel a slightly longer distance than the 200,000's do, and consequently follow. These would be the green ones on this display right here. Last, the 5,000 will take the longest path because they can virtually go through every tunnel that they bump into. So they take a much longer path going through the column. So this column allows me to separate them on the basis of size: the 200,000 guys coming out first, the 50,000 molecular weight guys coming out second, and the 5,000 molecular weight guys coming out third. Now, as you can imagine, when I have a mixture in cells, I have all kinds of molecular weights, so I don't just have three there, for example. But you get an idea about the way that we can separate on the basis of size. So molecular exclusion is a very nice way of separating these individual proteins and saying, "Alright, I know my protein is around 50,000 "in molecular weight. I can collect this fraction from around 50,000 and then work with it further to purify it." Yes? Student: How do you know when to... the proteins obviously aren't actually yellow, green and pink? Student: How do you know when to switch, that you're up to the next size? Kevin Ahern: How do you know where they are? They're not necessarily green or red or yellow. It turns out that there's a couple of things that you can do. One is, you can actually put molecular size markers in there that are green or yellow, which will help you. But more importantlyóand your question's a very good oneómore importantly, I need to have a way of determining where my protein is. That means I need to know something about what my protein does. So I know my protein, for example, catalyzes a specific reaction. I could test each one of these and see where is that reaction being catalyzed. And so I say, "Oh!" It appears over here in this tube, so now I know that this is the range where I want to collect my sample." Does that make sense?î Kevin Ahern: And being able to assay what my protein does is essential to purifying a protein. If I don't, if I can't measure what my protein does, I have no way of purifying it. Yes, sir? Student: Won't some of the smallest come out with the biggest because they don't all just go into the tunnels? Some of it will just fall through normally, won't it? Kevin Ahern: His question is, "How pure is this method? Will you get a little bit of the smallest with the largest?" Again, it's kind of like the question Shannon asked about being able to get rid of all of the ions. Yeah, you will have microscopic amounts of things there. This is not absolute purity that we're getting. But, in general, you will see the smallest will come out way, way late. Yes, back there? Student: How long does the process take? Kevin Ahern: How long does the process take? That's a good question. It depends a little bit on the column. Sometimes people really want to get as much purification as they can, and I've actually known people to pour columns that are six feet high. And those could take a few hours to run. If I'm running a shorter one, that might take an hour or two. So it really depends upon what I'm trying to do in terms of my separation. But there are columns that people can pour that are actually quite large. Yes, sir? Student: When you're saying that it's based on the size, are you talking about physical size or the weight? Kevin Ahern: Physical size and weight are related. So, in general, when we talk about globular proteins, even though they have individual shapes and so forth, they, for the most part, have a given size per weight. It's not absolute, but their growth, as they get bigger in molecular weight, their physical size will actually increase, as well. So it's based on their physical size, but since that's related to the molecular weight, there's sort of a one-to-one relationship. But it's not absolute. Yes? Student: Is this used to just primarily [inaudible] process? Or is this ever done sequentially where you would take a narrower range each time to evaluate a broad spectrum of sample contents? Kevin Ahern: I'm not sure I understand the question. Student: Like, for each one of those, if you took the yellow one that resulted from that, and then put it back through another column that had a narrower... Kevin Ahern: That's actually a good question, also. So could I take this guy and run it through a different column that has a different size bead that might be a little bit more selective in the process? And the answer is, I could do that, but there are other techniques that may be more useful to me. And I'm going to show you some of those other ones. But you're right, you could do that, and take it over and say now you've got a smaller bead and so you might be getting rid of some of the other molecular weights that you don't want. But, yes, you could. One of the things that you discoveró just a second, Shannonóone of the things that you discover in purifying proteins is there's no one way to purify a protein. Alright? You have to adapt the methods that you use to the protein itself. And you don't know before you get started what it's going to take to get that protein purified. So there may be several different techniques you'll have to use to get it, and it's going to vary from one protein to the next. Shannon? Student: I was going to ask, how do you know how often to change the tubes out? Kevin Ahern: How do you know how often to change the tubes out? Well, typically what people do with these is they just count drops. So I might say, "Okay, I'm going to get 50 drops." If the drops are coming out at a reasonably even rate, which they typically do, then people will set up fraction collectors so that every minute it will change a tube, and that will have, on average, the same number of drops. Paying somebody just to countóbelieve me, I've done this myselfópaying somebody to count drops before they switch the tube is one of the most mind-numbing things that you can possibly have. [laughter] So this is one of the joys of automation in biochemistry, when you've got a machine that will automatically do that for you. So that's molecular exclusion, gel filtration. Another related techniqueóit's related only in the sense that it uses beadsóis called "ion exchange chromatography." So in this method, we also use beads, as we used in gel exclusion, in gel filtration. However, the beads don't have tunnels or holes in them. Instead, the beads have on their surface chemical forms that have been bonded to them that have specific charge properties. So what you see in this case is a set of beads that have, on their surface, ionized, molecules that when they ionize give negative charge. When they ionize, they give a negative charge. Now, these started outóhow do I take one of these? I take my beads and the beads start out with a counterion. I can't get a bead that has a negative charge on it until I get it into solution and the ion comes off, so typically the counterion might be, in this case, a sodium. I've got sodium ions out here and they're attracted to those negative charges. So I've got sodium ions mixed with these beads and I've got them sitting in a bottle. I take my solution, I take my buffer, and I mix it just as I did before. I pour my column just as I did before. And those sodiums are still sitting there next to those negatively charged beads. Now I've got my proteins. I've got my mixture of proteins. Some of my proteins will have an overall negative charge. Some of them will have an overall positive charge. Some of them will have an overall charge that's pretty close to zero. So it's going to vary with the protein. How many glutamic acids does it have in it? How many lysines does it have in it? And these are going to determine positive and negative charges. Well, if I have beads that are mostly negative, what will happen is, the proteins that are the most positive will actually kick off those sodium ions and replace them. This is the "exchange" part in the name. They're exchanging those counterions, in this case, the sodium ions. So the positively charged proteins will kick off the sodium ions and the positively charged proteins will "stick," quote-unquote, to that bead. What's going to happen to the negatively charged proteins? Well, guess what? They're going to come shooting right through, because they don't want to interact with these beads, at all. So what I've done with this technique is I've separated proteins on the basis of their charge. The most negative ones are going to come racing off. Those zero ones are probably going to follow that. And then the positives are going to follow that. And you might say, "Well, why do the positives even come off at all? Or how do I get the positives off?" That's one of the most common things. If I want the positive ones, you know, I've got them stuck to the beads. How do I get them off? The answer is this. Virtually every kind of interaction we talk about in this class is not a covalent interaction. These are attractive things. So if I can make something else replace those proteins, I can get the proteins to come off. It turns out, if I pour a concentrated sodium chloride solution in there, there's enough sodium there it will displace those positively charged proteins and then I can get the positively charged proteins off. So there's an exchange. First, the protein displaces the sodium. Then high concentrations of the sodium will displace the protein, and I've got what I want. So I've separated my proteins on the basis of charge. This particularly phenomenon I've just described to you, in general terms, is called "ion exchange chromatography," but more pecifically, this is called ìcation exchange.î Cations, of course, refer to the positively charged ions, and what's being exchanged were those first sodiums. They were positively charged. This is cation exchange chromatography. So in cation exchange chromatography the first guys that come off will be the negatively charged proteins. The last ones to come off will be the positively charged proteins. Is there an anion exchange chromatography? You betcha. So if I have anion exchange chromatography, instead of having beads that are negatively charged, I have beads that are positively charged. And exactly the opposite of everything I've just said is the case. Instead of having sodium as a counterion, they'll have chloride as a counterion. And the chlorides get displaced by the negatively charged proteins. The positives, of course, come racing through. So we just flip everything backwards if we have anion versus cation exchange chromatography. Yes, sir? Student: Regardless of whether you're using anion or cation exchange chromatography, wouldn't your initial sample received also include the neutral? Kevin Ahern: So the sample will also include the neutral and it will come out somewhere in between the two. Yes, it will. Remember, we've get a whole, we've got thousands of proteins in here. We've got a lot of different proteins. So we're going to have sort of a spectrum, some with a lot of negative charge, some with a little bit of negative charge, some that are zero, a little bit of positive, etc. And that actually is going to relate to another technique I'm going to talk about in a minute. But you're right. There's a whole spectrum of these that are there. So, again, we're talking about techniques that give us basic, simple ways of separating things. But they're not absolute. I don't get only the one thing I want there. I've got some other components that are there. And there's no technique that I will tell you that is going to give you absolutely one thing. Understand that. That's important. If you wonder what those anion versus cation exchangeóyou don't need to know these structures, I'm just showing it to youóhere's an example of something that would have negative outside. It's got a carboxyl group on there. Here's something that might have a positive thing on the outside. You can see this tert-, uh, quartern-, amine that's out there, actually a tertiary amine that's out there. And these are commonly used, but, again, don't worry about the structures of those. One of the more powerful techniques that's used in a laboratory for purifying proteins is called "affinity chromatography." So, like the other two techniques I just described, it also uses beads. But instead of having tunnels or instead of having charged molecules, this technique uses specific chemicals on the exterior. So to describe this I need to give you an idea about how I might use this technique first. Let's say I'm studying, I'm going back to my E. coli cells and I'm very interested in a protein that I know binds to ATP. I know it binds to ATP because it uses it in a reaction that it does. So I know that this protein will bind to ATP, What I do is I take this naked bead that doesn't have anything else on it, and I treat it so that chemically it is bound to, covalently stuck to, ATP. So I can covalently link ATP to a naked bead, as it were. So now I've got all my beads and they each have hundreds or thousands of ATPs stuck, just out here, facing the solution, in the bead. Well, now I take this mixture of beads that all have ATPs on them, and I pour my column with my buffer, as I did before. And now what's going to happen is proteins that bind to ATP are going to stick to this column, and proteins that don't bind to ATP aren't going to stick. Well, this is a really powerful technique, a very, very powerful technique. Will I only get proteins that bind to ATP? Well, I might get a little bit of other stuff, but for the most part I'm going to get proteins that bind to ATP. Is that only going to be one protein? Well, no. There are many proteins in a cell that will bind to ATP, but I'll have a nice collection of the ones that do, and my protein's going to be one of them. Yes, sir? Student: Can a bead get more than one ATP on it? Kevin Ahern: Yes. Can a bead get more than one ATP on it? It can get thousands, Yes. Yeah. Well, how do I get myójust a second, Shannonóhow do I get my protein off? I would ask you that question. How would I get my protein off of such a column? What would I have to add? Student: Whatever the natural [unintelligible] is. Kevin Ahern: ATP. I could add ATP, right? And so now my protein's going to let go of this and it's going to grab ATP and it's going to come off, right? That's a very cool thing. Because, again, remember, the protein is not covalently bound, so it's going on, going off, going on, going off. And when it comes off, a loose ATP comes in here, it binds to ATP and now it comes off the column and doesn't stay stuck. So I add the natural ligandóin this case, ATPóto the molecule. Shannon, did you have a question? Student: Yeah. Is it practical to functionalize your beads? Or do you usually buy them pre-functionalized? Kevin Ahern: Yeah. Is it practical to functionalize your own beads, or do you buy them pre-functionalized? You can do both. So it depends. If have something that's a very specific molecule, you might do it yourself. Good questions. Alright, so affinity chromatography is really a very nice way of doing purification for specific target proteins. I want to just briefly mention one other because you frequently see it in laboratories. It's called HPLC, and HPLC stands foróand this is commonly misstatedóhigh performance liquid chromatography... high performance liquid chromatography. A lot of people say high pressure liquid chromatography because the columns generate a lot of pressure, but, in fact,the correct name is high performance liquid chromatography. This is a technique for separating, usually, fairly small molecules. But even that's not absolute. That's been adapted somewhat over the years. The way that this technique works is by taking and, instead of using a nice glass tube that's there, these are typically poured into stainless steel tubes that have great strength. And the reason they need great strength is because these are used to, at very high pressure. You don't want them to burst, for example. Well, what's the packing material? The packing material here is also beads, but the beads are microscopic. They're very, very, very tiny. So they're smaller, an individual bead would be smaller than your eye would recognize. They come as powders, essentially. And these powders have on them long hydrophobic sections of molecules, like long fatty acids, for example. A commonly used one is called a C-18. And what that means is that the bead has a whole bunch of 18-carbon units with hydrogens on them, sticking off... very, very hydrophobic. So now what I have, because the beads are so tiny, is I have millions of interfaces, millions of these hydrophobic molecules that the solvent is in contact with. If I pass my material through it, first of all, to get it through, it takes high pressure because these things are packed very, very densely And they're packed densely so I can get as many of these possible things in there as I can. Well, now, instead of having charges, or holes, or specific affinity molecules, now I basically have a bed, alright, that is the column material, I have a bed of hydrophobic side chains. What do you suppose is going to stick to it? Well, the things that are going to interact with those hydrophobic side chains are going to be hydrophobic molecules. And the things that are not going to interact with that support are going to be hydrophilic. So now I can separate on the basis of whether something likes water or doesn't like water. The ones that will come off of a column like this first are the hydrophilics because they don't interact with those C-18 groups. The ones that are going to come off last will be those that are hydrophobic, that do interact with those. The rate with which they come off is actually a function of their hydrophobicity. So, again, we can imagine a range of things, that are very hydrophilic, very hydrophobic, and things somewhere in between. What I've just described to you, and, by the way, there are a couple different strategies for HPLC, but what I've just described to you is the most common form, and it's the only one you're responsible for. It's called "reverse phase chromatography," reverse phase. Now I want to spend a few minutes telling you about a couple of techniques that now get into some really cool stuff with respect to purification of proteins. I'm going to skip down, and I'll come back and talk about polyacrylamide gel electrophoresis later and SDS. What I want to talk about right now is an interesting technique called "isoelectric focusing." Isoelectric focusing is a little difficult to conceptualize, but I'll try to do it here. Imagine, if you will, I now have a bunch of beads. And these beads have, not one property, but they're a mixture of beads, each with their own property. So before, I used all the beads that had the same hole, or they all had the same negative charge, or they all had the same affinity molecule, or they all had the same C-18 group. Now, I have mixtures of beads, each with their own property. What's the property? Well, the property is as follows. Some beads will have on them, let's say, 50 negative charges. And some beads will have on them, let's say, 49 negative charges. And some will have 48, 47, 46, 45. I go all the way down to zero. And then I have some beads that have +1 charge, and some that have +2, and some that have all the way up to +50, just as an example. Everybody envision that? So I've got some beads that have all these different things. So I take this slurry of all these beads and I shake 'em up, and I put them into tube, a glass tube, as I did before. And these beads are relatively mobile. That is, they can move around. They're not like the column I did before. Instead of standing it up like this, I lay it out like this. And now I apply an electrical current to it. What's going to happen? Well, to the positive end, the most negative charged ones are going to race and get over there, right? And at the negative end, the ones that are the most positively charged are going to race and get over there. And right square in the middle, those that are zero are going to stop right there. That make sense? So what I've just made in this tube is a gradient of charge... a gradient of charge, from the most positive at one end, to the most negative at the other end, with zero in the middle. Everybody envision that? So this is called "isoelectric focusing." It turns out that what I have just described to you, in terms of separating charge, also separates on the basis of pI. We talked about pI. pI is the pH at which a molecule has a net charge of zero. And so by setting up a column like this, I actually separate molecules on the basis of their pI, the pH at which they have a net charge of zero. The ones that have the lowest pI's will be at one end, the ones that have the highest pI's will be at the other end, and the ones closest to a pI of 7 will be right in the middle. Everyone with me? Well, to do this kind of experiment, to do this kind of a separation, I take not just the beads, but I take all my proteins and I mix it with the beads. I take all my proteins and I mix it with the beads. My proteins have a variety of charges on them. Some are very negative, some are very positive, and some are somewhere in between. When I apply the current, just as the beads separate themselves, so, too, do the proteins separate themselves... one end very low pI, one end very high pI, in the middle, those that have a pI around 7. So I've separated all of my proteins on the basis of their pI. Yes, sir? Student: Do you really need the beads? Kevin Ahern: Yeah. It's a good question. I do need the beads because the beads provide a support. In theory, I wouldn't need to do that. But if I don't have the beads there, the proteins just come racing off. So, yes, I do need the beads there. Yes? Student: In this slide, does [unintelligible] stand for pI? Kevin Ahern: No. It's a pH gradient. And because it's a pH gradient, that's where the pI's line up. So at a given pHóthat's a good questionóbut at a given pH, if the pI of this molecule is, let's say, 3.2, that means that molecule has a net charge of 0 right here and that's why it migrates to that point and stops. Does that make sense? Student: So, like, everything with a low pI would be towards the positive end and everything with a high pI would be towards the negative end? Kevin Ahern: Actually, it's backwards of that. But, yes. But you don't need to worry about that. All I want you to know, at this point, is that it is simply a separation on the basis of pI. Yes, sir? Student: So are the beads small, like the powder? Are you trying to pack as many in there as you can? Kevin Ahern: Are the beads that small? No, the beads are not very small. The beads are relatively large. Student: On the top picture, I don't understand, like, that there's the plus, there's the plus/minus and the minus unintelligible] the three colors. Kevin Ahern: Well, this is just simply saying that, here these guys are the most positive. They're going this direction. These are the most negative. They're going this direction. And the in-betweens are going to be in here. That's all that's saying. So, keep it simple. Keep it simple. So we've got positive, negative and basically neutral in the middle. I've got a gradient of that, So this is a way of separating proteins on the basis of pI. Now this, in itself, is useful. For example, I say, "Well, my protein has a pI of about 3.2, I could go and cut out the band that corresponding to 3.2, and I would have a mixture of proteins that all have similar pI to my protein, right? That's not the most important or the most powerful application of this technique. But in order for me to understand a more, for you to understand a more powerful application, we have to understand this process first. So I'm separating on the basis of their pI's. I have a whole gradient of pI's. What's the next thing I do? Well, next time I'll tell you a little bit about gel separation, but I'm going to cheat and tell you about gel separation here, Now, keep in mind what I just told you about isoelectric focusing. We're going to use it in a second. But before we get to apply this technology into something else, we need to understand how we separate proteins. How many people here have ever run a gel in a laboratory? Many people have. Gels are ways of separating molecules using electricity on the basis of their size. I'll talk about the theory for that in the next lecture, but today all we need to understand is that gel electrophoresis, as it's called, separates molecules on the basis of their size. The largest ones are the slowest moving and the smallest ones are the fastest moving. It uses electricity to do it. As you might imagine, it involves charge. We'll talk about the specifics next time, but we're going to have gel electrophoresis separating proteins. So if I take my mixture of proteins and they've got a whole bunch of sizes and I apply them to the top of the gel, what will happen is, the electricity will drive them through, with the smallest ones moving the fastest and the slowest ones, or the biggest ones moving the slowest. Now, here's the clincher, and this is the cool thing. The cool thing is, I can combine these two technologies. I do something called two-dimensional gel electrophoresis. It's schematically shown here. The two dimensions are, I do two different techniques. First, I take my mixture of proteins and I mix it with this slurry to do isoelectric focusing. So I take my tube. I lay it out here. I apply the current. I get the separation on the basis of pI. So I have this tube now that has this gradient of proteins separate on the basis of their pI. Alright? I'm very careful and I slice open this tube, and I take that material that's in there and I put it on the top of the gel. And now I run electric current through the column material and driving those proteins into the gel, first I separate it in this dimension on the basis of pI. Now I'm going to separate all those guys on the basis of size. What I will see is something that schematically looks like this. So, if I were to look at this, the molecules that have the most positive charge will be on the left side of this gel. The ones that have the most negative charge will be on the right side of this gel. And those that are the largest will be on the top, and those that are the smallest will be on the bottom. Down here, I would expect proteins would be small, positively charged. Over here, I would expect proteins would be large and negatively charged. Now, in two dimensions, I can separate every protein in this cell. Every protein that was in my mix I can now separate and actually see a spot on this gel. Let me show you what this looks like. This, I think, is a magical technology, This is what one might look like. Now, we see quite a bunch of interesting stuff here. We see dark bands. We see light bands. We see all kinds of mixtures of stuff. But, again, largest and most negative... smallest and most positive. Neutral, small. Neutral, large. Really interesting stuff. You say, "Well, that's cool. That's really totally there for a nerd." Right? Only a nerd could love the beauty in one of these things. And I'm going to make you love 'em, too, Which basically means I'll make a nerd out of you, alright? The beauty of thisólet me finishóthe beauty of this is that what, let's imagine, if you would, that I'm a person who is a medical doctor. And I've got a patient who has a liver tumor, And I want to understand how the liver tumor proteins are different from the proteins in the non-tumorous part of the liver. I could operate. I could remove that tumor. And as I'm removing that tumor, I could scrape off some normal cells from that same person's liver and I could isolate the proteins from each. And then, I could do a 2D gel on the normal liver cell proteins and I could do a 2D gel on the tumor cell proteins, and, guess what? I'm going to see differences. These are reproducible. So I could look and say, "This band right here, look how intense that is in the tumor cell. I don't hardly see this protein, at all, in the normal cell. Here is a protein I see in the normal cell. I don't see it in the tumor cell." I could understand, for every protein that's in these cells, I could understand whether it's more in tumor, more in normal, or no difference. I could understand, at the protein level, one of the mechanisms and one of the differences between a normal cell and a tumor cell. And I could do it in a single gel. That's absolutely phenomenal! Let's imagine that you're a pharmacist. I'm not quite done, yet. I'll be done in just a second. Let's imagine that you're a pharmacist and you want to test a new drug that your company has just created. What's the effect of this drug? Are there any nasty side effects of this drug? Well, I take one group of cells. I treat 'em with my drug. I take the other group of cells. I don't treat them. And I compare. "Oh, my god! This thing's knocking down DNA polymerase tenfold! I'd better be careful with this stuff." Alright? "This thing isn't having any effect, whatsoever." Maybe I'm interested in a compound that somebody says, "Hey! It's carcinogenic." It really affects cells if I have this. One treated, one untreated, and I can look at the entire pattern of proteins... an absolutely phenomenal technology. Alright. That's enough for today. I'll see you guys on Monday. Student: So do they have this stuff archived? Kevin Ahern: Do they have these? There are many places where you can archive this information. Student: So you can, like, do matching that way? Kevin Ahern: You can, but, in general, you'll want to do it yourself, just to make sure that there's not variability from that. Student: It must be really hard. Kevin Ahern: It's a sophisticated technique, yeah. Yes, sir. Sorry. I wanted to get through there. Yeah. Student: That's That very top left corner, marked negative? Kevin Ahern: Uh-huh. Student: Well, was that a natural protein sample, would you think? Or is that an artifact from the actual process? Because it was a large smear. Kevin Ahern: Smears will happen when you've got things that don't fit in well, and they're actually artifacts, in a sense, but they're real things, but they're not [unintelligible]. Student: Do they have a "bible," if you will, of different, like... Kevin Ahern: They do. They do. Isn't it cool? Yeah. Did you have a question? Student: [unintelligible] Kevin Ahern: Yes. Student: Is there a standard that you can look at [inaudible]. Kevin Ahern: Very good question. That's what everybody else has been asking. Kevin Ahern: So, yes, there are, and if you remind me, I'll say that at the beginning of the lecture next time. There are libraries of these where you can actually do that comparison, which is kind of cool. Excuse me. Oh, I'm sorry. Good day. How are you doing? I've gotta squeeze in here. Sorry. Student: Sorry. [END]
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Medical_Lectures
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Stem_Cells_Tissue_Regeneration.txt
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[Music] Stanford University uh welcome to the third Mini Med school session it's great to see you all back um I'm Sher Ren who co-directs the course with uh Phil piso guil chw slides from last week are now up on the website so you're welcome to go and look at them it gives me great pleasure tonight to uh introduce Jill Helms who's going to talk to us about stem cells and tissue reg generation which I think is something you can't turn on the TV or the radio without hearing those words for the past I don't know 8 years and uh Jill has sort of an interesting background for a medical school because she actually is not only a PhD that she got from the University of Connecticut but she's also a dentist and she got that degree from the University of Minnesota and Stanford does not have a dental school so it's uh she has unique talents though and we were really happy to recruit her away from UCSF where she served on the faculty for8 years she's currently an associate professor of surgery uh here and she's very interested in cranial facial development and also regenerative Med medicine she does really exciting research and if you read her CV my favorite thing she does is she works on a set of protein called the Sonic Hedgehog proteins now I think that anybody who can work on something called Sonic Hedgehog I'd love to know how that protein got named cuz somebody was definitely having some with that today what she's going to do is really introduce you to the terminology used in stem cell research and then share with you some of the recent advances that have been made and we'll discuss how stem cells could really be used for Medical Treatments and this whole new field that is now being called regenerative medicine so I'd like to welcome Dr Jill Helms thank you thank you so much what a pleasure to be here I'm I see some familiar faces a lot of new faces I have gotten the best subject to talk about in this whole um miniseries I must say to be honored with the topic of stem cells and regenerative medicine is like being handed the best best topic so I hope to excite you with the same enthusiasm I have for this subject so I'm going to start with a couple of questions first of all by a show of hands how many people know someone who's saved umbilic Cord Blood after pregnancies raise your hands high okay I'd say about a third of you at least a third maybe a half okay how many people know someone or themselves have a condition that they believe could be treated by stem cells yeah that's that's impressive yes Nancy Pelosi said every family knows that there one phone call or one diagnosis away from needing what stem cell research can yield for us I think that's so true as we go along these days last of all how many people in here voted for prop 71 you remember what prop 71 was how many people voted in 2004 well seven over seven million Californians said yes to prop 71 which funded stem cell research in this state and I want to tell you a little bit about what we're doing not only in stem cell biology around the world but also what's happening here in California and most specifically here at Stanford where we lead um the institutions in California in stem cell funding so in New York Times I had a friend of mine look through the last three years and found out how many articles mention stem cells 604 in just the last three years and the amazing thing is the distribution of these articles where in everything from science to business to politics my favorite is 11 mentions of stem cells in the style section which I read religiously so 12 are in the Obits anyway hopefully not so you can see that stem cells are big news there's no doubt about it and the field of regenerative medicine is built around stem cells and I'll tell you a little bit about how those two topics sort of segregate but tonight I want to tell you about what stem cells are what makes them so unique and most importantly do they really have the power to transform the way we practice medicine today and I'm going to hopefully convince you that in fact we absolutely do so first of all I'm going to tell you how the talk is going to be broken out so there if there are those among you who decide one part of the talk is a snoozer you can up and lead and come back for a later part hopefully you'll want to stay first of all I'm going to tell you why we need regenerative medicine then I'm going to introduce you to the Lexicon of stem cell biology make you familiar with some of the key um maybe the key nomenclature that accompanies stem cell research then I'm going to talk about the concept of regeneration what why is regeneration different from repair and what can we learn from animals that do regenerate very well then I'm going to go on to talk about stem cells in particular cloning which is the basis for stem cells and some real life applications that we think are just around the corner and then last I'm going to tell you about Stamford and stem cells one of my favorite topics so first of all why do we need regenerative medicine this was an article from The Washington Post a couple years ago and they heralded all of the advances that have been made in the Machinery to replace missing or damaged parts for the human here you see a clear implant here's a couple of others they mentioned dental implants corneal implants hip implants artificial hearts with all this technological advancement why do we need regenerative medicine and I think that I'm going to start with the concept that as good as these machines are there are limitations to all of them how good is good enough does this young man who came back from Iraq and needed an amputation is this good enough he can walk but there are Li itations that go far beyond what we think about aesthetic reconstructions for example I want to emphasize that Hardware is not equivalent to software this is a very famous athlete does anybody know who this is oops bar Barbaro that's right that's his leg and you can imagine the best veterinary the best veterinarians in the world we're working on his leg and you can see the amazing amount of Hardware that has been put in there to hold those bones together regardless of the strength of that implant that bone plate and those bone screws I can tell you that that material is not as strong as bone and what you see on the right hand side is an illustration of strain Fields within materials and in this case it's strain Fields within bone and the important thing to notice is that bone has the ability to adapt to loads and can bend and form to a load that's excessive these plates have a failure point I think any engineers in the group will understand what I'm talking about and they often times do fail this one failed as well and of course we know what happened to poor Barbaro there's another reason why mad M materials aren't nearly as good as our own tissues and that is adaptability I show here just two examples of the ability of bone to both resorb that is be taken up and and be reformed this process is going on all the time in your bones in your body so every seven years the skeleton you have now will be completely replaced with a new skeleton and the advantage of that constant renewal is that if you have excessive loads the bones will adapt to that load if you have less loading we know that you can lose bone mass but this process means that defects in the bone structure will should be taken out by this remodeling process this of course is not something that a man-made material will do by the way interrupt me if you have any questions there's another important difference between man-made materials and our own tissues and that is the ability of a man-made material to integrate here you see an example of ligaments muscles and tendons that all make bones functional so we can't just replace the structure of a bone with a steel rod no matter how strong it is because we'll never get the functional uh integration of that steel rod with muscles and ligaments and tendons attaching to it so all of these things tell you why man-made materials have limitations and why those won't nearly be as good as our own natural um tissues and organs so then you might ask well what about transplantation because there it is we're not using an artificial heart we're transplanting one heart from a donor into a recipient and you're right those tissues have much more adaptability but we all know the biggest uh disadvantages of these tissue and organ transplants anybody who's on a list will tell you that the limited number of organs and tissues for donation mean that a lot of people die needlessly there's another issue and a lot of people don't really keep this in mind and that is if you receive an organ it's from another person and that means that your cells can recognize the cells that are comprising that other tissue as foreign and that means if you didn't give a patient immunosuppressive drugs they'd eventually reject that that tissue or organ and so these immunosuppressive drugs are something that patients have to be on for the rest of their lives and there's a very high cancer risk associated with drugs like this of course if we're talking about somebody's heart you can't live without a heart so you'll run the risk taking the drugs and having cancer later on but you can understand now why other kinds of tissues that aren't going to be lifethreatening are rarely rarely transplanted things like people maybe you've heard about the transplanted hand or the transplanted phase have you heard about those things so those are things you can live without right but still some people opt to take that risk because they're so aware of the deformity or the limitation of life without that organ or tissue so I hope that I've made it clear that regenerative medicine will address these kinds of limitations but there's one other things besides being able to replace tissues or organs for which we have man-made materials or maybe the ability to transplant materials regenerative medicine also seeks to address diseases for which we know no cure and that is the amazing hope that we hold for the future and I really do believe that it's very close at hand so now I'm going to move on to the second part of the talk and if you guys don't interrupt we're going to be out of here by 8 o' and that is the concept of regeneration and this oh I'm sorry oh yes you said that basically on those girl um what if could a twin donate something and then would the recipient accept that more readily right so the question is can a twin be a donor uh and will the recipient accept those tissues more readily yes yes that's absolutely true have to take the drugs that's a good question would you still have to take the drugs so it it depends on whether you're an identical twin or a fraternal twin if you're an identical twin you're biologically uh the same and actually the first kidney transplant that was done in the United States was done between identical twins then you would not need to take imuno supression if it's a fraternal twin you do have differences that the body can recognize and then yes you would have to take immunosuppression so it's about being genetically identical okay yes I was kind of fascinated when you said that every s years your skeletal system is basically renewed how then can you see in an x-ray a fracture that happened many many years ago right that's a very good question why does a fracture that happened many years ago not get completely remodeled well that efficiency at remodeling your skeleton decreases as you age and so in areas that have fractures that never heal the problem there is that the mechanism for repair and then remodeling is disrupted and so typically those kinds of fractures that don't heal I do so because they lack a blood supply and if you have that kind of problem then that air of your skeleton will not remodel and of course with osteoporosis we know that the bone resorbing activity is greater than the bone forming activity and that's why you start losing bone mass with um the disease osteoporosis okay yeah is the same thing true with scar tissue yeah is the same thing true with scar tissue does it repair or remodel like that unfortunately no and that's a big big problem because scar tissue and I'll talk about this in a little while Scar Tissue has very different properties than the initial tissue that form and a lot of tissues don't remodel quite like bone does but certainly scars do not and so I'll bet anybody can look and say I got that scar when I was seven years old it'll stay there for the rest of your life or the place where you got your vaccination those scars never Remodel and that means that when we have tissues that heal through scarring they'll be there forever okay so now I'm going to oh I'm sorry yep so why did they tell you you should do weightbearing exercise to help prevent osteoporosis how does that affect that mechanism right so why does weightbearing appear to help with osteoporosis so weightbearing is one way your skeleton senses that it needs to build more bone so like a tennis player who always uses his like like Roger feder or not me so when he uses his right hand his his skeleton of his serving arm is actually larger so actually mechanical force like walking or weightbearing exercises or something like tennis will make a skeleton grow it will make that part of your skeleton grow and so that's in an effort to stimulate the bone forming capacity of the body versus nonweightbearing activities question y I see nowadays A lot of people doing knee Replacements and knee surgeries what is your opinion on that right so what's my opinion on knee replacement knee surgeries this is from not a surgeon's point of view is that um wait as long as you possibly can before you do it because unfortunately these implants tend to loosen over time that seems to be the case in almost all kinds of implants like that and so that means when you have the surgery you have about 15 years would you say Sher are is at about 15 years before you need to have another one and the problem is every time you put in that implant you have to take a little bit more bone away and as that goes along of course you're going to eventually have not enough bone to hold an implant in place so these kinds of man-made Replacements I mean sometimes pain is the distinguishing feature which says now it's time but a lot of people try to put it off as long as possible okay so now I'm going to move on to the idea of regeneration and we're very lucky to think about regeneration because nature does it nearly flawlessly in some organisms so first I'm going to talk to you about two features that stem cells have that make them unique so here's an illustration of a stem cell the first unique feature of a stem cell is that the stem cell can become any cell in the body and I've Illustrated that here with a stem cell with a little halo around it and I've showed the kinds of tissue that we typically think about a cell being able to make that is muscle bone fat and cartilage three of those we want one of them not so much but you can imagine that the ability of a stem cell to differentiate into any of those cell types is really quite remarkable now I want to tell you how remarkable that is by telling you that when a bone grows it's a bone cell giving rise to another bone cell that just cause that's happens because the cell divid this is a stem cell that can become any one of those kinds of tissues you see so that makes it very unique now I want to give you an example of that from science and this was published just a couple of years ago and it uses the tissue of the breast now here's a schematic of the breast and this has now been shown in a number of tissues but the breast is a good example and in the breast you know that its primary function is for lactation and you can see the ductal structure in a mouse breast here on the right hand side and that entire very complicated lattice work is comprised of many many different cell types and I'll show you just three of them there are alvear cells and those are Illustrated here as lining a lumen or a tube there are the ductal cells that lie on the outside of that Lumen that serve as sort of the support then there's muscle that surrounds the ductal cells and then there are the milk producing cells in the middle there's one other cell type that's found in the breast besides these and that is the mamory stem cell that is a stem cell that has ability to give rise to all of the cell types I just told you about and here's the proof of that so I'm going to tell you I'm not going to tell you how they isolated the mamor stem cell but I tell you that they did it from a mouth that was genetically U manipulated in a way so all of the cells from that Mouse would turn blue if they're exposed to a particular chemical so all of the cells from this mouse are labeled so they're blue and these researchers isolated the cell that they call the mamory stem cell and then they put it back into a recipient now it turns out the recipient was not genetically marked so it was blue it had normal colored cells and then they looked at the developing mamor gland and remember now the well maybe you don't know this but the mamory gland grows during pregnancy and during lactation and then it recedes and then it grows again so in a mouse they made this Mouse a mouse that was pregnant so the breast was then growing they put the cell back in and when they transplanted a single cell which they call a stem cell into the recipient here's what they found all of the cells in the breast were blue the ductal cells the milk producing cells the alv cells all of the cells were derived from that one cell that was put in and that's proof of the concept that a stem cell can give rise to any cell type we call that differentiating into any cell type and this ability is called Pur potency it's a very unique feature that's reserved for stem cells whether they come from embo embryos or adults y does that mean there was only one M stem cell operating that's a good question does that mean that there's only one mamor stem cell no there are multiple mammory stem cells and they tend to be activated by injury it's just that in this experiment they wanted to prove that a single cell could do it so that's why rather than take the multiple mammory stem cells that they can find they took a single cell only does that make sense but I would think the mouse would have Own St that were not blue very good yesj that's exactly right so that's very good question so the host animal has its own mamor stem cells why didn't they contribute well in fact you do have to have the injury and the removal of part of the breast tissue in order to get the new cell to integrate so if you just inserted the one blue cell in there and didn't disturb the recipients stem cells then you get a patchwork it's called a chimeric mouse but in this case they removed a part of the breast and so in doing so they removed the mount the recipient's own stem cells a very good question if you want a job in a lab I come see me afterwards all right there's another unique feature of stem cells and that is that they can duplicate themselves now that is a very unique feature and I've Illustrated it here um but sometimes people think that what this means is like I was saying when the bone is injured one bone cell gives rise to another bone cell that's not the same this is this is like like if you have children you might think well that's like one cell giving rise to two cells a parent giving rise to two kids that's not what this is this is you giving rise to you again exactly you again and that unique feature is called self-renewal and it seems to be limited to stem cells and now I'm going to give you an example of that ability to self-renew and I think most people have probably heard about bone marrow transplants this is a this is a clearly a way that stem cell research and regenerative medicine have already been established in the clinic so here you see the inside of a marrow cavity and what I hope you appreciate is it's tissue there's not just it's not like your bones are empty and it's a pipe full of blood this is actually cells and tissue and in a bone marrow transplant this tissue is harvested from a donor and here you see that happening right now and then it's injected into a recipient and like in the case of the mouse the recipient is usually very ill and their own bone marrow has to be removed in order to allow the new bone marrow to integrate and here's the reason why that's possible why it works is because in this bone marrow are contain all of the cells the stem cells that give rise to all of these different kinds of blood cells as well as some I don't think you can see it here as well as the precursors of bone so when the doctor ex um takes the bone marrow from a donor and puts it into a host what they're doing is transplanting stem cells now we know that both the donor and the host can survive this so the donor can do just fine because their stem cells can renew these hematopoetic stem cells and their ability self-renew was one of the first kinds of stem cell self-renewal that we really understood and I think it's beautifully Illustrated in this model of bone marrow transplant ation so these are the two unique features again the first is that stem cells have the ability to self-renew and that capacity appears to be limited to stem cells and the second feature that distinguishes them from all other cells is that they're plur poent that is they can give rise to any cell type y that me when they when they go through my mitosis they don't make mistakes so you asked whether when they go through mitosis they don't make mistakes well of course any cell can make a mistake um but supposedly they live forever these still L some stem cells are very long lived other stem cells are very shortlived so we used to think the length of a cell's life was an indicator of whether it was a stem cell or not now we know that that's not true stem cells can make mistakes and we think that cancer is an example of a stem cell making a mistake because type of stem cell well it may be a different kind of stem cell but it appears to be in a tissue so we call them tissue specific stem cells and they're basically stem cells that keep on hearing the signal self renew self renew self-renew and they never hear the signal differentiate and that's what makes it a cancer stem cell we think that they they're a unique kind of stem cell with a defect yes you said the stem cell can self renew now other cells divide that's right each cell is identical so what's the difference here yeah it's it's a difficult concept but it's sort of the the parent giving rise to two kids that's what tissues do when they remodel they give rise to two cells that are already differentiated they're already bone cells whereas a stem cell doesn't give rise to a cell that's different from it gives rise to a cell that's exactly the same does that I'm not sure if I'm answer I can get into more detail any cell the body though individual cell through mitosis there's then two cells are then you have two cells and they're both differentiated already like a heart cell or a muscle cell giving rise dividing into two muscle cells a stem cell a stem cell when it goes through that process it still gives rise to a cell that can give rise to any other cell type whereas a myoblast can only give rise to another myoblast yesul trans replacement do you need IM imun present drugs well often times not because a patient the recipient has had a radiation to remove all of their endogenous or their own bone marrow usually they're very ill as a consequence of this how much bone marrow from the how many cc's Sherry oh she winced I've done a million bone marrow transplants in mice I can tell you that no yeah we can tell you everything in a mouse but what would you six in a mouse okay um it's probably not it's not that much um I don't know I'm sorry it's a fairly small volume yeah I bet it's under 100 Ms for sure and guessing somewhere around 50 yeah sorry mous I could tell you we know everything about mice yeah is the cell man also called mitosis with a stem cell versus differentiated cells so is it is it called mitosis yes I believe it's still called mitosis although people usually differentiate it by saying selfrenewal um but I believe you can still call it mitosis yeah yes with the bone marrow transplants don't they often take it from the person that has the cancer so so you're asking about the and then and then put it back into those so you're asking about the bone marrow transplant where do they get the donation of the new bone marrow no it's never taken from the diseased individual because that bone marrow is essentially diseased and so I'm sorry so there there were autologous bone marrow transplants that's what that's called yeah but and it was the first place you heard about it was it actually in breast cancer there was a huge thing actually court cases because uh insurance companies denied payment for Aus bone marrow transplant to treat breast cancer it then was shown in studies to actually not be of benefit so um to my knowledge right now I'm not there may be some conditions where autologus may be sufficient depending on what you're trying to treat but most of the bone marrow transplants for things like leuk IA or other uh diseases of the uh blood are coming from other donors if that answers your question and again with the issue of compatibility they try to typically use people who are closely related because the cells as they get introduced into the patient still have markers on them that distinguish them from um the donor or from the host yes just speaking about the different kind of transplant I actually work with it we do a significant number of autologous transplant and it depends of uh what is the problem that brought you to transplant um you can have a steam cell transplant not because you have to have a different immune system but because you might have to receive a otherwise uh killing do of chemotherapy and you just need a rescue from that what becomes cure dose that would kill you by un supressing that's yeah that's a very good point sort of a rescue transplant yeah it's called a rescue trans you do transplant neuras you do that a a a significant number of it I would say it's about 40% of at least in the Pediatric population we do aist Transformers come on down here that's a beautiful answer that and that's from real life any anybody else want to yes sorry I'm still confused between confused between mitosis and self-renewal act of replication level last week we heard Jeffrey Chu talk about DNA replication had a good discussion at three prime and PR Prime replication what's different so what's different about this process well I can tell you that the short answer is we're not entirely sure but the longer answer and a little bit longer is that stem cells unlike other cells seem to have to actively repress differentiation when they divide when cells typically divide or undergo mitosis and give rise to other cells they already are expressing genes that have to do with their differentiation and stem cells have to actively repress this now we used to I'm sorry go ahead sorry you see you're saying expressing genes all that's going on is a simple chemical process of replication not expressing the gene it's replacing or duplicating so there's no no protein being formed that's what's confusing me it's just just a simple simple chemical process I guess what I'm trying to emphasize here and I might be missing your point but what I'm trying to emphasize is the difference between self-renewal and differentiation not what is mitosis so what I'm trying to emphasize is that unlike differentiated cells when they divide one thing that I think that the mitosis part or the replication is indistinguishable as far as we know but there's a caveat there but what makes these cells unique is not that they undergo mitosis that's true of all cells but rather that when they do so they're progeny are is a at least one of them one of the two progyny is a stem cell again and there has to be a mechanism in place for making that happen that's very unique we can talk more later I'm sorry well just a if you are a skin cell and you divide you become a skin cell again you don't become a bone cell or a blood cell whereas this cell keeps that ability which is a miraculous ability to become any cell so it is all the mitosis but a skin becomes a new skin and a cell a stem cell becomes a stem cell which I don't see what's different that's they're both doing what they naturally do is replicate what's different about this replic I I hope to have convinced you theend what's so special yeah do we know how we uh initially get our first TS oh that's a very good question yes we do know something about how we get our first stem cells and I'll show you that in um as soon as we finish the mitosis part of the stomach I am going to answer your question sometime yes um so can you describe what's the process when a stem cell decides suddenly to be a muscle cell what's that what initiates the differentiation and after it differentiates does it then after it differentiates into a bone cell or whatever does does it then make stem cells when it under goes mitosis or or other bone cells right okay so two questions how does a stem cell know what to do or how does it get its instructions to okay now you differentiate into bone and then once that cell is differentiated can go can it go oh I'm going to go backwards okay or make baby stem cells right or make more stem cells okay so the first question is very complex we know something about how cells choose ches to become differentiated but it involves a turning on and off of multiple multiple genes Gene regulatory networks that lead those cells to differentiate and I can give you names of genes but what exactly triggers that differentiation is not always clear when we transplant stem cells from one tissue into another we have some ideas of what those signals are because they are those signals that come on very soon after the cell has been transplanted sometimes we can even drive those stem cells to differentiate by adding certain growth factors but it's this complex M of the environment in which a cell finds itself growth signals and differentiation signals that the cells are getting that lead them to differentiate so we know something about how they differentiate now once they differentiate two years ago I would have told you no they're done they're differentiated that's it in the last year and a half we now know that's not it they can go backwards it's phenomenal it's truly phenomenal I'll tell you about an example of that towards uh in the um Le second part of the talk yeah uh you mentioned about the mouse you call it a memory stem cell yes memory stem cell does that refer to the location where it's found or it has this cell already kind of differentiated into becoming any type of breast cell that it very good question so the word that describes the stem cell like mamory stem cell indicates the tissue of its origin so that means that within the breast they found the stem cell there's epidermal stem cells found in the skin there's kidney stem cells Etc so if I change the environment and put that cell into a bone say it become a bone cell right so the question is now if you take a mamory stem cell can you put it anywhere and it'll become a bone cell a muscle cell we used to think absolutely and now just in July was published a paper that suggests that rather these stem cells in adult tissues like this is a mamory stem cell so it's tissue specific that they seem to remember where they came from they remember even if you put them in a dish and you passage them over and over and over again you'd think a cell would not know this but they do seem to remember and it's phenomenal that they have this memory this memory we think is encoded by a set of genes called Hawk genes and that seems to imbue stem cells with some kind of memory of where they came from it doesn't mean that they can't make other kinds of cell but they tend not to do so normally so if a cell came from a certain germ layer called the ectoderm it's most likely to give rise to tissues or cells in tissues also derived from ectoderm like skin or bone whereas cells derived from mesoderm tend to give rise to mesodermally derived tissues this is a restriction that we were unaware of before and I'll show you an example of that in a little bit okay now somebody asked over here I think where do you get stem cells from that's a very good question because of course if you want to study them you must have a source of them and I'll bet a lot of you know that the previous administration had something against using embryos for stem cells and now I think this this happens to be a mouse embryo but it looks very much like a human embryo and you might look at this and say okay so that's why people are upset set because this is where you get embryonic stem cells you get them from an embryo but what I want to emphasize here is this is not the stage at which you get stem shells stem cells this looks like a baby mouse what to me this is where you get stem cells from it's called a morula that's from the Latin meaning Mulberry it does look like a mulberry this is before implantation this is the stores this is the source of embryonic stem cells at this stage and let me show you a little movie now that shows with ex one cell becomes get aula two become four each duplicates the original unique DNA the enchanted progression of cell division continues for 5 days the embryos are monitored finally the division creates masses of cells known as Blas assists okay 5 days 5 days in culture and you get this the blastic it comes from the Greek word meaning bud and as you can see there's a small group of cells right here called the Inner Cell mass and that's schematized here this ball of cells right here called the Inner Cell mass in that little thing called a morula that is where the stem cells reside and that Inner Cell Mass contains all of the embryonic stem cells will give rise to every tissue in the body now there's something very unique about embryonic stem cells they're not found within tissues as you can see they're at the beginning before there is any tissue and the thing about embryonic stem cells which really distinguishes them from adult stem cells are first of all of all adult stem cells have tissue specific properties mamory stem cell bone stem cell but the greatest potential is found in embryonic stem cells that have no TSS tissue specificity and here you can see that some kinds of or adult stem cells can form some kinds of tissues we call them multi potent whereas embryonic stem cells are plur poent or to potent this is what makes embryonic stem cells so valuable because they have the capacity to give rise to all tissues all cell types now something that I want to point out right now is how do you ever get adult or tissue specific stem cells we don't know the answer but what it appears to be is during development of an embryo cells are somehow set aside in every tissue where they stay in a quiescent state until disease or injury seems to activate them we don't know how they get there we don't really know where they reside we have to search for them using very um technologically advanced methods but they seem to reside in every tissue meaning that every tissue has a capacity to regenerate at least a little at least what at least a little there's often times other processes that go much much faster and that's why we get scar tissue and I'll talk about that in a few minutes y that include the heart yes that does include the heart they now have identified cardiac stem cells yep and in the brain where you might have thought that that was it you know you know how you're told don't drink because you're killing your brain sell that's all you got turns out few more few more okay there's another big difference and this is important when you're doing research and that is that adult stem cells when you grow them in culture have a limited lifespan whereas embryonic stem cells can grow continuously this becomes very important if you're trying to study them and the last thing is that in adults in adult tissues the number of stem cells is very limited whereas in the embryos are abundant so so here's here's the news that maybe it's bad news as we age stem cells stop being stem cells they tend to differentiate until the proportion of stem cells is significantly reduced and they make up one very small percentage of our total cells and I say that with resignation as I start to look more and more like that woman with the bathing cap okay now what I just told you is true in humans in some animals they retain the ability to regenerate throughout their life it is a remarkable ability and now I'm going to talk about those animals oh yeah so the Bush Administration yes I think limited it it wasn't that there is no stem cell research there was a certain number of existing lines that's right why why why does the world need more than even one line I mean if they regenerate for and why isn't that good enough I so why yeah so it's a fair question there were a a a few lines uh most of them turned out not to be so good but why do you need more than one if you can make you know the goose that lays the golden egg you can keep on getting more the problem is in the actual culturing of these cells and now I got to explain a little technical detail when you take that Inner Cell mass and you replicate it and keep it growing you have to give it the growth factors that it needs to survive and the way that's done is by putting those Inner Cell mass cells on top of something called a feeder layer those feeder cells are so not supposed to contaminate the Inner Cell mass cells they're supposed to stay separate and the feeder layers are basically um supporting the stem cells by providing growth factors but it turns out and then almost all of the cells lines those feeder layers eventually contaminate the the stem cells so the lines that were produced many many years ago tended to have a lot of flaws in them and so they weren't very robust as embryonic stem cells now there were certainly places where people could generate new cell lines and that has been done and with a lot more care in keeping these feeder layers separate but those were done with institutional money that was not federally supported so for example Howard Hughes Medical Institute supported a lot of that stem the um the development of additional stem cell lines so that's the main reason also some of those stem cell lines were from adult stem cells and I've already told you why those would be limited in their potential some reason they're taking skin cells and then taking them back to an embryonic state to then fast forward into other cells is that happen yes that is happening that is called reprogramming when you take a differentiated cell and make it look like a stem cell and I'll talk about that in more detail but how was that achieved but at the time um I would say this was about 2004 um that was not even on the horizon okay so regeneration some animals have enormous capacity we do not why is that well one way we we can learn about how we heal tissues versus how these animals heal tissues is to study these animals and there's a bunch of them that are really good at regenerating and this is one in particular this is a n or an axel lottle they come by very many different names and I want to show you a video that was taken from the Howard Hughes um Medical Institute and where they videotaped this process of regeneration now I'm going to turn off the sound on this simply because um she the person narrating talks a on for quite a while here you see the Axel has had the limb cut off and now you watch it over the course of 60 days completely reform now this is schematized here you can see now the Axel can move its limb it's showing you the cartilage the muscle the nerves that make it movable and then the experimenter comes along snip okay and now what you see here is the formation of something called a blastema a wound blastema that forms at the tip of the amputated limb and what they're showing you here now this streaming our cells stem cells coming from the cartilage the muscle and the surrounding soft tissues and the nerves that flood into that blastema and then through mechanisms that we're not completely fully aware of they reorganize themselves and regenerate a complete functioning limb as many times as you can stomach cutting off that limb it will regenerate I know that's okay so where does the new limb come from it comes from the blastema here's a picture picture of that little blasa and there's a tissue section through the blastea this comes from the Greek word meaning to sprout the entire ability to regenerate all the tissues of the limb are contained within that little group of cells so here's an experiment that was just published in July 2009 that helps us understand how is this possible so here you SK see I've schematized that limb again there's the cartilage the muscle and the nerves that comprise the limb and there are three populations I've showed you three populations of stem cells one from each of those tissues muscle stem cells cartilage stem cells nerve stem cells and when the limb is amputated what we now know is that each of these tissue compartments contribute stem cells to the Reg generation of the limb and each of them keeps track of where they came from we used to think that one cell could give rise to all of them now we know that each tissue delivers its stem cells to the blastea and then through processes we're not completely aware of yet they reorganize themselves and generate a functioning limb by understanding in a salamander that there are tissue specific stem cells we learned something very important we learned that each tissue contains stem cells in salamanders and in us that's the amazing thing the process is is conserved all the way from salamanders to humans the difference is that they can reorganize these stem cells and most importantly they're appears to be an unlimited number of stem cells like I said you could cut that limb off again and again again and again we seem to have a limited capacity to produce more stem cells what if we could jumpstart the stem cells that we do have within our tissues and maybe make a bigger pool of stem cells that is feasible yeah so are there any long lived animals that have this kind of regeneration capacity because it seems like if you're living longer you accumulate more DNA damage and then having this kind of capacity is a reci for you know horrible kinds of cancer right so the question is if long lived animals maybe don't have this regenerative capacity so the ability to regenerate would be associated with a short-lived animal because then you couldn't accumulate the mutations right so we used to think that that was it that it was just that simple that animals that live for very short periods of time have a lot of regenerative capacity but we've somehow winnowed that out because we live longer and then we could have more mutations it's it's not that it isn't true it is true that when you live longer you accumulate mutations and we heard last week about that but it doesn't seem to be the case that long-lived animals have fewer stem cells than short-lived animals for example squirrels do you have any idea how long a squirrel lives it's it's like something like four years it's it's long maybe even longer than that I mean it's long to me rats which look just like squirrels except for they don't have the furry tail they're much shorter lived and yet they both seem to have the same number of stem cells so we don't know why that is we don't it's maybe just an association but not causal okay so now I remember a question about what's the difference between repair and regeneration so we don't regenerate after a lot of wounding after some kinds of wounding some tissues do have an ability to regenerate but I'm showing you one that does not and that is skin despite the fact that our skin turns over constantly when we suffer a deep injury into the deeper layers not The Superficial layers but the deeper layers we form scar tissue and at first you might think well that's okay you know scar tissue is not so bad well there's a fundamental difference between these two and I'll tell you why and then I'll show you why when we repair something we replace the disease or damaged tissue with more tissue but it doesn't necessarily have the same capacity as the original tissue regeneration is different this means to reform in whole or in part the original tissue so here's what Scar Tissue looks like scar tissue is not like normal tissue as you can see from this soldier who was burned in IRA this Scar Tissue has different functionality it lacks sweat glands it lacks hair it lacks elasticity and as a consequence it is not a functional replacement now besides the Aesthetics here you have to understand that this Scar Tissue the ability to form a scar tissue versus regenerate a tissue may have evolved because we had to develop a way to quickly close off wounds but these are conjectures we have no proof that that's why we generate Scar and salamanders regenerate limbs but I can tell you that there a lot of tissues where scar formation is devastating think about spinal cord injuries there's a scar that forms at the site of injury and neurons can no longer pass that Scar and as a consequence we have paraplegia after a stroke we develop a g scar in the brain and functionality is reduced in that area and we know after heart attacks we form scar as well and you might think well you know that's better than having a hole there but you have to realize that that scar tissue now compromises the ability of the heart to function all of these kinds of healing this mechanism through which scar tissue forms are substandard and obviously not regeneration yes in case of salamander the ability to regenerate is it limited to the limbs or extends to all of your body parts and tissues they have so you were asking a question about is the salamander limb the only part that regenerates no it's not they regenerate their tails if you cut the optic stock they'll regenerate that so they uh regain sight there's multiple parts that you can cut off and replace it's just that limit is used first of all because it's relatively easy to get at versus the optic nerve or something like that and second because so many tissues have to contri distribute and organize themselves properly and then you have a real readout you know it moves of whether you've got full regeneration that's why it's usually the limb that they use right but their internal body parts like uh they must be after their life they must be dying right because certain body parts are not functioning well so do they also regenerate yes internal organs can regenerate as well in zebra fish for instance you can cut I think it's 2third of the heart off and the Heart Will regener generate fully um multiple organs you can imagine that those are kind of hard experiments because you have to do the surgery not kill the animal and then it has to regenerate so I was getting the point that there could be some animals then they could live forever they want to die um so could this make them live forever no uh because there are other reasons why they die not not because they're in the lab but because um um but they develop they also develop diseases so so these are they're relatively short lived animals yes to what extent does a human liver regenerate yeah the human liver is an amazing organ it can regenerate up to 25% of it can be removed and it will regenerate yeah it has a re enormous regenerative capacity and do you know who was that um Prometheus Prometheus who gave the fire to mankind remember and in the Greek Legends he gets his liver pecked out every night by the eagle she already knows this okay and and then it regenerates so they knew something way back when yes how manytimes how many times can yeah like the leg of the C can you do it endlessly or yes you can do it as as many times as you would like yes it will regenerate continuously so you donate once you could possibly donate again yes yes so it regenerates if you remove part of it what if you just damaged it yeah that's a good question so what if you damage it and form a scar you know I don't know too many experiments where they just damage like a crush or something like that but I believe then the regenerative capacity would be reduced because you have to form that blast in order to have regeneration that blastema is key because you can transplant the blastema to another part of their body and it will give rise to the extra limb as well think you like I don't know had too much to drink over your lifetime often develop what curosis right which is liver that's correct so why are you not regenerating rather than so humans don't have this regenerative capacity even in organs like the liver because in the case of excessive drinking you're damaging the entire liver so it's not like you're excising a part of a healthy so diseased tissues don't regenerate yeah also for therosis you cause Scar Tissue within the liver it's called fibrosis and you actually can see scar tissue and within those you'll see some regen you see regenerative nodules but eventually this framework of Scar Tissue scars so much you cannot grow the liver anymore and actually in the end stage of sosis you get an incredibly small shrunken liver and it's just very very scarred it's a fibrotic liver so you replace a lot of the liver with scar tissue from repetitive damage it's not just drinking uh some of the hepatitis virus can do the same thing if you have chronic active hepatitis and other toxins but you will regenerate for quite a while but if you keep the damage up you'll eventually lose that ability but I don't know why that happens but probably because we have liver stem cells yes we do have liver stem cells thank you okay so the basic goal of regenerative medicine because that's what I'm going to shift to now is to somehow figure out how to reinduce the molecular program that made the tissue in the first place and now reactivate that in the disease State and I have to caution there's there's one point of caution here and that is that we don't know much about regenerating tissues that are diseased like Sher had talked about the liver that's a big problem and I'll tell you why in a few minutes why that's such a a difficulty what we're concentrating on for the short term is how do we create new tissues that have been acutely injured say a spinal cord injury and before you have a large gal Scar and the patient has lost muscle mass Etc because of being paralyzed how do we regenerate that that's a much harder question then how do we treat the acute injury okay and so I'm going to emphasize this concept that if we understand something about how tissues develop then basically we're studying tissue regeneration all right and so now I'm going to get to the part of stem cells and cloning and some of the real life applications if anybody wants to stretch or anything do you want to take a break it's 8 o'clock keep on going is there anybody okay go for it okay excellent just like the students haha okay you're so much easier to talk to there they're you're working against Facebook you know because they're all Facebooking okay okay so now the nuts and bolt of regenerative medicine it begins with a story of cloning and so I want to take a step back and talk about cloning and for some of you you might remember cloning in 1973 and can you hear that I have a little beggar right here and what you want basically is a whole entire person connected to that nose right otherwise you get your money back now did you want me to leave room for mustache course there's a there's a nice area in there which it it's um what it's clever she's clever what the doctor is doing here we placing down the garments cuz we're going to clone we're going to make an attempt to clone the patient directly into his suit and that way you know he'll be completely dressed at the end of the operation it's a first in cloning and then we could all just get the hell out of here we're don't to hang around the cloning room while suits up uh I think it's time to check the cell structure yes the checking the cell structure checking the cell structure checking the cell checking cheing cell structure the structure of the cell is to be checked I love that of course we do have to check the cell structure to understand how this goes but do you I I I was not I was not even born then [Laughter] no I'm just brilliant um all right so basically the idea of cloning is what Woody Allen was talking about with that nose is that you make a genetically identical copy of either a cell or an organ now way back when this was first done this was a remarkable feet nowadays people talk about cloning like it's nothing but I want to try to give you that flavor of just how impressive it was when cloning was first achieved and then try to emphasize well what did we learn from all of that way back when so first of all can you believe that the first organ to be um cloned was a carrot 1958 an entire carrot was generated from a single cell that was taken from the root of a carrot now that shows you the ability of a stem cell to give rise to an entire in this case carrot or Oregon you can think about it but in humans I'm sorry not in humans but in mammals this ability had to wait a number of years the first animal to be cloned was done by John giren and he used a frog but I'm going to ex give you an example more recently using a Maman cell now here's the point first you can take a differentiated cell it doesn't matter from what tissue and if you take that fully differentiated cell and you put it you take one of the cells and you remove it from the group of the other cells and then you introduce it into another group of cells that are different in this case we'll say bone cells that differentiated cell will not integrate right that seems almost obvious to us now but back then it was an important control that the cell if you put the cell in there it wouldn't integrate but if you took the nucleus out of that fully differentiated cell then you could get it to integrate and here's an example now of a pipet that's holding a cell and the nucleus is going to be injected into it you see the nucleus is sitting in that pipet this is unbelievably challenging initially now it seems like it's much easier to do you know with experience so here's the experiment you take a fully differentiated cell nucleus so that means you have to take the cell you have to separate the nucleus out of that cell then you have to take another cell and you hold on to its nucleus and then you introduce it into the first cell does that make sense so far so you've just transferred the nucleus of a fully differentiated cell into another cell and that is sufficient to turn that differentiated cell into the kind of cell that the nucleus came from and of course it integrates fully into the tissue so that okay now I went backwards excuse me now I have to go through the animation again maybe it's a good review so that ability means something very very important what it means is the nucleus of a cell contains all the information in the form of DNA to give rise to a complete new tissue organ we know one organis m in particular Dolly the sheep remember when that was achieved so the nucleus of one cell contains all the information that's required to make that cell differentiate and we used to think that it only went one direction remember that you take it from a a a embryonic cell and you do introduce it into another embryonic cell and then you get a fully formed Dolly the sheep now we know that we can do this with nuclei from multiple kinds of cells so what is it that's in the nucleus that makes it so able to do this remarkable feat well I want you to listen to John to talk about mechanisms of nuclear 2006 now the reason we want to know about this is that in the far distant future I'm sure long off to my time it should in principle be possible to take a skin cell and to overexpress in that cell those genes which are needed to reprogram that cell back to an embryonic condition without even having to go through an egg that's the long-term aim but in order to achieve that we have to know what are the mechanisms by which these nuclei are reprogrammed once you take them out and put them into to an ache he said in the in the future what did he say the yeah this 2006 2007 the first induced plur poent cell was created it's phenomenal this set of experiments was done by a number of labs now and what the researchers showed is that they could take and now we know you can take a skin cell a fully differentiated skin cell and you can introduce these genes into the nucleus of that cell and turn that differentiated cell back into an embryonic stem cell like cell four genes we used to think the nucleus is going to be so complex there's no way the gene regulatory networks say four genes now the way they did that experiment first was they introduced those genes by putting them in a virus and then putting the virus into the nucleus and you'd say no no that's not going to work because you can't introduce these viruses and they're right because in their experiments with mice this Gene C mck is called a Proto onco Gene that means an enco Gene meaning it's cancer causing and sure enough 20% of the mice from this experiment develop cancer that was in 2007 in the last two years we've now found that you can use a host of family members from these four Gene families and they can remove semic get it out of there because that's a bad one put in its place a gene called n and they can achieve the same effect we now know just recently published that you don't even need to use a virus to introduce these genes into the cell you can actually use the proteins that means there's no viral particles that you'd have to put into a cell and then put that cell into a human I have to tell you something but you know scientists they have a sense of humor it's not so welldeveloped but anyway um this is a TI nanog that is Irish I think it's Irish myth and it's called the land of Eternal use and that's where the name nanog came from it's a little humor it's like Sonic the Hedgehog yeah okay so this ability to reprogram an adult cell to become like an embryonic cell do you realize what that means that means you don't have to go searching for those one in a million stem cells you could take any cell presumably and reprogram it to be an embryonic cell it could be one of your cells so you don't have to wor worry about immune rejection now you might be saying to yourself yeah but where are we going to get all those cells where would we be able to move these differentiated cells back to a stem cell well I am here to tell you that there are plenty of places where you can get adult stem cells or adult cells yes fat so you might wonder why can't we just take these stem cells there's a few of them why can't we just inject them into the damaged tissues and I think that Sher introduced the concept of why you can't put it into a damaged liver that's fibrotic why can't you put it into a brain after stroke or a heart after heart attack or into the pancreas to make beta cells that produce insulin why can't we just inject them and it's complicated but I'll tell you right now that in a nutshell it's because disease tissues probably aren't producing the right cocktail of growth factors that be will be required to keep these cells alive so that's why we concentrate now on acute injuries because we presume that acutely injured tissues still can produce these cocktail of growth factors that these stem cells will need and I'll tell you about one of those in a few minutes we also know that if cells see the wrong signals uh especially if it's a stem cell that they can go on to do very Terrible Things certainly cancer this is one example this is called a Teratoma this is a tumor it's a benign tumor luckily but it has bone carage hair teeth found within this teratoma so all of those tissues I it's totally disgusting um which is why I like it so much um all of those tissues are able to form so we know that cells can do this it's just you want to make sure that they do it in the right place at the right time did you have a question yeah I I've heard that the veterinary research in stem cells is actually progressing in some areas more rapidly than yes right so so um he asked a question about veterinary medicine seems to be taking advantage of stem cells in treatment nowadays and it's absolutely true NBC had a story not too long ago on the introduction of stem cells from a horse back into the horse to try to treat a ligament or a tendonous injury and you might say well how come why do they get this and we don't because we still believe that you do it first in large animals before you do it in humans and in this case what they're doing is they're isolating cells they usually suck out either blood or fat from the animal then they expand those cells in culture and then they reintroduce them supposedly they're isolating the stem cells so there's a couple questions one how efficient is the isolation of the stem cell from these adult tissues not so easy two when you culture cells outside of the body you can imagine the kind of regulatory hurdles that you encounter if you're going to introduce those cells back into an animal with a horse you know maybe not so many issues as with a human and the third is they're introducing them into an area that's been damaged so is it a degenerative condition is it an acute injury they're also just injecting them what's what's the chance that the cell that you inject stays put these are the kinds of things that we really have to know before we can do it in humans nonetheless they're doing it in large animals yeah if adult St adult uh differentiated stem cells are as good as any other stem cell and we just put an end to the embryonic stem cells sources right so the question is quality point of view right so the question is if these SK cells can be reprogrammed into embryonic stem cells can we just do away with the controversy forget all that to some degree so we know that the difference between embryonic cells and adult stem cells are is different in multi potency versus plur poent things like this there're qualitative and quantitative differences in these cells but the biggest hurdle I think we face in using reprogrammed cells to to treat a person is that we have to reprogram those cells in a way that doesn't make the cell become it becomes a stem cell but it has to be a stem cell that's under control it can't be a it can't um have lost its ability to regulate differentiation in other words those cells if you introduce them back into a human you have to make sure that the cancer risk is very very very low right because a stem cell that no longer listens to the signals that say differentiate that is the cancer stem cell so we have to know how to do that we're still working but you know wait a month there are qualitative differences yeah I would have said in my lifetime but now no okay so the big question is if we knew the cocktail of growth factors that maintains stem cells and presumably we could add that cocktail to cells and we could excuse me sorry we could theoretically expand what would be a limited number of cells stem cells into many more stem cells in this case we're not introducing the cells themselves we're introducing the growth factor and I'm going to tell you about an example of that in the last few minutes now well maybe not last few minutes you're not going to get out early too bad for whoever was thinking that okay so I want to start first with a story that isn't at Stanford and in fact it was from quite a long time ago this was a a research report that was published in 1948 in veterinary medicine and it was done in Ithaca New York Upstate New York and there this is the name of it a preliminary report on the propagation of Aven pneumo Encephalitis virus Newcastle disease in vitro I know you're thinking that's a real page Turner but these scientists were very interested in figuring out what caused Newcastle disease now they were at the vet College in associated with Cornell and they were studying this disease that affected the poultry it turned out it affected all avens so Ducks as well as chickens number of other kinds of birds but most importantly for the poultry industry in Upstate New York was that this disease decimated the poultry and these researchers were trying to figure out what happened in this Newcastle disease so it's 1945 and there were working hard to figure this out and they were very lucky they were very diligent they figured out how to grow it turned out to be a virus how to grow the virus that causes Newcastle disease in these Aven cells they worked years on this isolating the cells from the birds plating them out getting them to grow this very hard this is just a differentiated cells and then infecting those cells with a virus that causes terrible disease and in doing these rigorous experiments they were able to identify this virus called a paramix virus that caused this disease that decimated the chicken U poultry farms in itha New York and you're probably thinking now there they are and they're so happy to have published in veterinary medicine and you've never heard of them around that same time when was a disease that was the scourge of the nation and that was polio most of the victims were children and in 1952 only a couple years after that paper was published polio killed more children than any other communicable disease there you see the numbers the ones who survived often times had to face a life in an iron lung the lucky ones could walk with crutches the public reaction was that this was a plague I only know the stories I've heard about just how terrible and scared people were about this terrible terrible virus and now here is a man whose name you probably do recognize Jonas Suk he was working very hard on what caused polio he joined the University of Pittsburgh in 1947 when he was in medical school and he had to B for a lab so there's always a shortage of lab space and he was working very hard on this problem and you can imagine this was the HIV of the times right only the victims were children it was a horrible disease and his biggest hurdle was he was unable to study the virus because he could not grow it in cells and without the ability to grow the virus in cells that meant he could never generate sufficient amounts of the virus to really analyze it in detail and so he was stuck until he saw this he saw the paper from the two people up in Ithaca studying those chickens that were dying and he recognized that Aven cells might be more susceptible to infection with that virus that polio virus were the cells that he was using he used Aven viruses I I'm sorry Aven cells to infect them with polio virus and in doing so he began to develop the first polio vaccine in 1952 he tried the first vaccine on his wife and child and in 1955 3 years people were getting treated with that polio vaccine 97% of reduction in the incidence of polio what I am emphasizing here is something that I hold very dear to my heart and that is that we stand on the shoulders of giants that's from the Google Website you know that when you look on a scholar his ability to generate a vaccine depended on somebody doing research on chickens and that's in one of the most important lessons that I want to leave you with and I'm going to now introduce this video you need to the money that we spend the public dollars that we already have and are spending this is a matter of reprioritizing we've got a$3 trillion budget in this country and Congress spends some $18 billion on earmarks for their political pet projects and that's right there is more than the shortfall to fully fund idea where does a lot of that earmark money end up anyway you guys have heard some of the examples of where those dollars go you've heard about the bridges and you've heard about um the some of these pet projects that really don't make a whole lot of sense and sometimes these dollars they go to projects having little or nothing to do with the public good things like fruit fly research in Paris France I kid you not so why don't we talk about fruit flies for just a few minutes here I want to introduce you to a faculty member here at Stanford his name is Rule nusa Rule hails from uh Amsterdam and he works on stem cells but that's now and I want to tell you where he came from we're building a lot of bricks and mortar here at Stanford but in those buildings are these individuals that make this possible so there he is a number of years ago I'm sure he'd kill me if he knew I had that picture he doesn't look all that much different um and Rule came from the Netherlands he visited the us back in the 70s he took a Greyhound bus trip around the US and he ended up in San Francisco and he ended up in the lab of Harold varus shown here on the right hand side these two guys were working Harold varis and Mike Bishop at UCSF on how viruses worked and in the uh let's see I guess it was the N 1980s they won a Nobel Prize for their discovery on the origin of retroviral Ana I won't go into what that is exactly but Harold varas wrote a letter to rule after he visited the lab and here's the letter and in this letter he talks to rule about the kinds of experiments that maybe he should do when he comes to the lab and this letter is on um actually you can Google it and find it but there's two points that I want to draw your attention to the first is that he says you can work on anything basically even though your fellowship says to work on a certain thing you can work on anything here and in the second page of this letter he says down in that bottom right hand corner PS I have one project in particular that I think will be interesting for you to work on that PS became Rule's project as a postto rule identified a gene in fruit flies which is called wingless wingless gets its name like most fruit fly genes because of the way the embryos look when that Gene is removed so you see on the bottom that that em or that fly has only one set of wings instead of two and so they gave it the name wingless he worked on this Gene for many many years and he found out that there is a vertebrate Maman homologue of wingless and we call it wi now it turns out here you see a example of how many Wint genes we have in mammals enormous number of them 19 wi genes countless receptors countless parts of the pathway which rule played a big part in identifying and discovering now we have these kind of diagrams that show you how cells interpret a wi signal what they do in response to it it's quite complicated I just want to show you today I did a search I used the criteria here wi and cancer 170 Pages worth of papers on wi plus cancer that's today then I said wi and stem cells 77 Pages worth wi and regeneration let's see how many pages was that 19 that's today he started off working on a fruit fly Gene and today we have hundreds of papers having to do with wi in cancer wi in stem cells wi in regeneration I'm going to tell you one story about wind and that has to do with the development and regeneration of cartilage and Bone I'm going to just focus on bone for the sake of being short what we know now is humans who have mutations that cause too much wi signaling that is too much wit is either being produced or cells sense that there's more than there is they develop a kind of condition called vanam disease and what I hope you can appreciate here is that the skeleton of a person has too much wind signal is there's too much bone we also know that in diseases where you have bone loss this is multiple Myoma that those that bone loss is associated with two little wind signaling I'm being pretty brief here but basically we now have tools in which we can assess how much wit does a cell see how long does it see it and those tools are shown here where embryos or cells are engineered in such a way that when they see a wi signal they turn blue that means in a tissue section you can stain it with this special method and you can see where did the wind signal act on cells and here within bone this is a section through bone and that that little dotted line indicates the periostium that's the tissue that surrounds all bone we can see the blue cells so we know that when those cells that are in the outer layer of the bone if they see too much wind singling then they make a lot of Osteo blasts they make the bone forming cells and you get that vum disease we know that if wi signaling is blocked then bones don't heal we also know you have bone loss so like the principle of Goldilocks you have to have just the right amount and with rule we did some experiments where we found that wind signaling is activated that blue indicates activator turned on when the skeleton is injured so in an intact bone you have a little wind signaling but when you injure a bone the wind goes way up so what's the purpose of that we wondered and we've done a whole series of experiments I can now tell you that if you block that upregulation then you block bone healing so we know that wind signals are absolutely essential for bone repair pair so now this is the Goldilocks principle right you know you have to have a little bit not too much not too little otherwise you get these diseases and in 2003 after nearly 15 years of trying they purified For the First Time The Wit protein itself it took 15 years this person the rule never gave up on this and in doing that he opened the door to be able to ask a simple question what if we take that protein and we put it into an injury a bone injury that doesn't heal and in on my own lab we figured out a mechanism for packaging the protein I won't go into that now and then we introduced that packaged wind protein into skeletal injuries and here you see that in the side on the right which has gotten the wind you see there are more blue cells that's important because that tells us that what we introduce used turned up the wi pathway more than normal and what we found is that bones that were injured healed much much faster than bones that were left alone we know that they they make 350% more bone than injuries that are just left go now you might say why do you care about healing a mouse bone I don't what I care about is healing a human's skeletal injuries that are slow to repair most of us you know you can say we're lucky when we s when we have an injury we regenerate bone fairly well if we're healthy but as we get older as we have a disease if we're imuno compromised then that ability is really repressed this may be a mechanism whereby we can stimulate bone formation and we now know why it works because it turns out that the stem cells within an injury site respond to the Wind by proliferating that's the key feature and then those cells differentiate now the important thing here is this balance we know everything's a balance but we know that cells tend to make more bone faster when they see a wi signal and so this cell in the center is the stem cell that seems to be pushed towards a bone forming fate now this is the state of art of cartilage regeneration that's it right there so there's a lot of room to improve here and see if we can regenerate cartilage tissues through a mechanism but I want to leave you with a very important Point here I told you about one tissue bone and went in one tissue but here's how far that fruit fly research that has no value to humans right how far it's gone here's in nature went dependent hair folc regeneration for those of you are a little all on uh it also seems to accelerate skin wound healing here's another one it mediates the proliferation of neural stem cells after stroke another paper the role for wind signaling in the Regeneration and self-renewal of homat poetic stem cells so these are in the top rated journals in the world and you can see that the regenerative potential of this fruit fly Gene is tremendous now I want to leave you with this recent paper that says this wi pathway is required for patterning in plenaria does anybody know what a plar is a flat worm now you're probably saying to yourself why should I care about a plaria and these animals you can cut them in half do you see the eyes here the little dots up there in the corner those those are the eyes if you cut this worm in half it'll regenerate a tail but if you tweak the wind pathway it'll regenerate two heads pretty cool and if you cut off the head and get rid of that and keep the tail it'll regenerate two tails wi signaling regulates that all the way down to Flat worms and you might say why do I care about a flat worm oh but I hope you don't because nobody knows where this is going to lead so I think in conclusion we are living in the most exciting time in stem cell research I I can't imagine a better time to be involved in science I think the hope of regenerative medicine is firmly established in basic science research it needs that foundation so people who think about funding research for all kinds of animals whether it be fruit flies flat worms or chickens it does make a difference for human human biology and I want to emphasize that it's not necessarily a straight path from the bench to the bedside and being willing to go along for the ride is one of the most important attributes of a scientist who's in this field because you have to believe in the long term that it is an enormous potential we hold and this is the last quote from Governor Schwarzenegger so you can see on both sides of the political uh debate that he thinks said it's a he's a great believer it will improve not only the human condition but our economy as well how good is that thank you very much thank you so I think you can see why we asked Dr Helms to speak tonight and thank you for telling us why pinaria are so important and thank you to our bone marrow expert in the audience there we have uh some time for questions so please uh ask away yes you spoke most of the stem cell reing organs happening within the body I've seen work on on organs being grown outside of the body also either a ear being grown on the back of mouth so is that the same phenomenon how is that different is that any further along than you know I lose my suppos to right right right so the idea can we regenerate tissues outside of the body and then bring them back in and he uh reminded me of the mouse with the ear growing on its back so first I have to tell you what that Mouse growing the ear on its back so that is actually not in ear so what that is is a piece of cartilage that was tucked under the skin of a mouse and the skin then formed around around the cartilage now that mouse was you remember it didn't have any fur so you could really see the ear that's because the cartilage came from a human okay and so the mouse had to have a suppressed immune system so that wasn't an organ it didn't have any hearing capabilities it was merely the architecture the outer architecture of the ear and now you have to realize that while the cartilage was human the skin was Mouse so there's no way to introduce that back into a human so so that was just a demonstration I think that is it possible to grow tissues that have sort of shapes to them that look appropriate but not functionality so there's no hair follicle regeneration inner ear hair follicles um there's no functional restoration of hearing nothing like that all right so that's it now can we grow an organ in a dish the answer is no right now and that may be because of what we need in order to grow an organ nothing will grow without a blood supply and we simply cannot create that in a dish so try as we might most of our experiments have to be done in an animal and that's another reason why I think anybody who does research has to think about you know what's the balance between I'm using a mouse this mouse will die what's the advantage of it because that animal will die for the hope of doing something that will benefit mankind so without a blood supply we can't grow an organ in a dish there are a lot of other limitations like we need a lot of different kinds of cells um and there also the contribution of things like um they get carried along with the blood not only the blood itself but also all the growth factors we can't recapitulate that in a dish yeah yeah a couple years ago I heard of someone doing research into big hearts they were taking the big hearts and using detergent to wash out the cells and leaving the structure and then somehow repopulating that with human cells do you know anything about that yes so he was reminding me of the pig heart where they isolated a pig heart then they remove as many of the cells as possible leaving sort of the structure of the heart and then they want to repopulate that that scaffold with human cells and then introduce it back these are called xenographs and so they've attempted them in a number of different kinds of tissues from valves to an entire heart and the basic limitation of that there's a big Advantage the structure is there right and that's hard to rebuild right I mean mechanically that man-made heart is you know a far cry from the functional the functionality of a pig heart the real limitation is though that again one has to think about the immune rejection part right so if you're replacing an entire heart yeah that might be the strategy to use but what we hope is that the diseases that lead to needing a complete heart transplant may be circumvented by treating illnesses very early on like you remember the picture of the young man with the amputated leg and the artificial leg some of say well you can't regenerate a whole leg but what you have to realize is the amputation just like this heart transplant is not the result of a single uh you know damage to the leg it's an infection uh at the site of injury the infection can't be controlled so they have to cut out tissue which means that the wound doesn't heal leading to further infection another surgery etc etc Etc and then you get amputation same way with some of the heart diseases that require transplantation can we intervene early enough to prevent the need for that but yes how's the wi delivered is that by injection right so how is the wi delivered so we've now treated bone cartilage skin muscle a heart muscle um brain retina and cornea with this approach and in some cases it's topically applied if it's the skin in other cases is injected through a catheter if it's the heart in bone we can put it on things like collagen sponges that are put into injury sites that's what's currently used um it not because the collagen sponges to anything special but just because it's a sort of a delivery vehicle we have to figure out ways to optimally package it and for each application and we're still learning that so in some cases it's injected like following stroke uh into the injured area of the mouse brain in other cases you know topically or in some other solid support yes I thought I'd give you the opportunity to do like in that video that we showed um when do you think of that we might have humans regrowing feet or legs like like you saw that what were the words he used not in his lifetime not in his lifetime maybe not in my lifetime but in the lifetime of children that are alive today yeah I think yeah I I can't believe I could actually say that but I I really do I see the progress is at a speed that's almost unbelievable I had slides from from a lecture that I gave 6 months ago on the the Regeneration you know with a different cell type that's already changed that was 6 months I mean these are huge huge advances so I never believed a differentiated cell could be turned back in time so I don't know I think it's the limitation is not is not like it used to be we have questions from this house I'm sorry it's it's hard the the lights hit you here that's why nobody wants to look this yes I'm just wondering why somebody hasn't taken sarup aside and explain that sounds but that really is a serious question I mean we we sort of you know it doesn't take that long what's what's the obstacle that keeps the door opens and I can walk through um only have well I think there's two things to be to be um kind there's two things first you have to be willing to listen and I think the fact is is a lot of people do not want to listen so and then you have to well besides being willing to listen you have to be smart enough to understand yes is there any possible rejection of it's added or similar to already body and then in also in an instance you remove the the nucleus of the cell and position what happen to the cell the residual C after the nucleus is no longer present in it right do it does that regrow or so in all right so two questions one is um if you take the nucleus out of a cell does it regrow a nucleus the answer is no you've remove you've removed all of the information for survival and that nucle Nu so that cell is gone all right and then you asked a question about the wind about what was it now I'm sorrya if you add it is there any re oh is there a rejection no there isn't a rejection for the simple fact that you're not introducing a cell you're introducing a protein it' be like injecting insulin we don't develop immune response to proteins so that's the big difference yeah no cells y you mentioned the spinal cord is regeneration a couple of times and I know there there is going to be the first human trial um in related to a cute spinal cord injury that should be starting soon is a spinal cord a particularly good model or why has why is the spinal cord the first FDA approv right so why is the spinal cord going to be the first and is it some somehow a particularly good model well first of all it's a devastating injury second there's no treatment these are the kinds of conditions that make it easier to get FDA approval versus doing something that might uh person could live without so there's no treatment third acute injuries are the very are they going to be the first things get tried versus chronic degenerative diseases because of this uh the acute injury site shouldn't have developed too much of a scar there still will be a scar but it will be a better site than say one where chronic disease has led to uh the tissue's atrophying ETC so it's not only good because it's so devastating but it has no treatment and acute injuries are a good model how about our last two questions the lady with glasses and then there's a gentleman sitting right underneath the very bright light wouldn't be able to see him yes so my question uh to start at the beginning of the lecture what's the dealio with people saving umbilical blood what's Theo um so why do people save umbilical cord blood because in the cord blood are stem cells now to what extent are those stem cells viable through the process of harvesting that usually this tissue is discarded haven't they been differentiated I thought it was only five days so some of the cells you could call them tissue specific stem cells right so they aren't embryonic stem cells they're more tissue the at that point they're called adult stem cells meaning anything post embrionic day five so they appear to have some regenerative capacity and so what people are doing is banking this umbilical Core Blood for their child on the chance that technology will have progressed to a point that should they have a disease or an injury that could could be treated by stem cells they have a bank of their own stem cells stem cells from them in a drawer at home or something not in a drawer at home no there are you have to rent space for them one gnarly drawer um yeah it's exactly right you have to rent space for them so there are companies yeah yeah yeah and the gentleman in the blue shirt I was wondering why we had tissue specific stem cells versus universal stem cells everywhere right what's the advantage of universal stem cell well why do we have tissue specific stem cells why not Universal we have Universal stem cells when we're 5 days old in that Inner Cell mass that could be considered the time when we have stem cells that are Universal why do we have tissue specific stem cells it's a good question I don't know why but it does appear that every tissue Harbors stem cells so tissues that we once said there's no way that they have any regenerative capacity the brain the heart it appears that in fact they do and in some cases those stem cells in the brain and heart and other tissues like Photo receptors of the eye they can be reinduced to somehow give rise to more photo receptors say that have been lost because of a acute injury so why we have them I don't know thank God we do and they seem to be the key to maybe how particular tissues might be able to heal so what I'd like to do is you're going to see Dr Helms next week because neither myself or Dr piso can be here so she's going to be introducing our next speaker who's going to be speaking on the human nervous system I would highly encourage you to read the uh chapters assigned uh for next week so you can really follow along and again I'd like to thank Dr Helms for such a fabulous thank you for more please visit us at stanford.edu
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Medical_Lectures
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Hemostasis_Lesson_3_Coagulation_Cascade_and_Fibrinolysis.txt
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[Music] this is the third video in this series on hemostasis and is part two of the discussion of normal physiology specifically covering the topics of secondary hemostasis and fibrinolysis the learning objectives of this video are first to be able to describe the general features of the coagulation Cascade second to be able to list the anti-thrombotic control mechanisms which terminate clot propagation next to describe the process of fibrinolysis then to describe the role of vitamin K in coagulation and last to diagram the inter relationship between platelet plug formation and the coagulation Cascade so here is the overview of hemostasis we saw in the first two videos of the series in this video I'll essentially be discussing the right half of the screen that is everything from the coagulation Cascade Downstream as with platelet activation it's easy to get lost in the many specific details of the coagulation Cascade but first let's begin with a few general principles the coagulation Cascade is a series of enzymatic conversions of inactive pro-enzymes also known as zymogens to activate enzymes for example and these are not the actual enzyme names but just for illustration imagine activated enzyme a comes along and induces a confirmational change in pro-enzyme B thus activating it then activated enzyme B might cleave something off of a pro-enzyme c activating that and maybe activated protein C then requires a co-actor of some kind in order to activate pro-enzyme D and then comes along an inactivating enzyme which binds to enzyme D preventing its action each step along the way gets multiplied in that each copy of enzyme a might activate 100 copies of enzyme B and each copy of activated enzyme B might activate 100 copies of enzyme C and each enzyme C might activate 100 copies of enzyme D so from one molecule of activated enzy enme a we end up with 1 million molecules of activated enzyme D this process is what is known as the Cascade its initial trigger is normally vessel wall injury it ends with the formation of fibrin strands and the termination of the Cascade by various anti-thrombotic mechanisms remembering from before it requires a careful balance between procoagulants and anti-coagulants too much procoagulants and and spontaneous clotting will occur resulting in deep Venus thrombosis and Pulmonary emilii too much anti-coagulants and excessive bleeding and hemorrhaging will occur finally the process of fibrinolysis eventually dissolves the clot we saw this table briefly in the first video of the series it demonstrates that most clotting factors in the Cascade are designated with a Roman numeral in addition to having one or more alternative names the clotting factors are almost always known by the ran numeral with the exception of factors 1 through four which are always referred to by their names fibrinogen or fibrin prothombin or thrombin tissue factor and of course calcium a lowercase a after the Roman numeral designates the active form of the factor most activated factors are enzymes with the major exceptions being 1 a commonly known as fibrin 5A and 8A the traditional model of coagulation was first described in the 1960s as it means to explain lab findings when studying the process of coagulation in vitro that is in test tubes outside of the body it was not necessarily intended originally to be a description of the processes actually occurring in the body however since it provided a relatively straightforward way to visualize a highly complex system and allowed one to understand abnor noral results of coagulation tests clinicians rapidly adopted it it consists of parallel extrinsic and intrinsic Pathways so-called based on whether or not the trigger was a compound that was extrinsic or intrinsic to the endothelium the extrinsic pathway involves an integral membrane glycoprotein called tissue Factor along with clotting factors 7 and 10 the intrinsic pathway involves clotting factors 8 9 10 11 and 12 the end result of both of these is the common pathway which involves Factor 10 along with thrombin and fibrin unfortunately this traditional description is no longer believed to be the most accurate model of the physiology it nonetheless is still wiely taught in textbooks and University lectures it's not that the traditional model is literally wrong but rather that it is an oversimplification whose division into these distinct extrinsic and intrinsic Pathways is artificial in addition it ignores the distinct phases of coagulation and has not typically included a thorough discussion of essential multicomponent complexes in the Contemporary model of the coagulation Cascade there are four key multi component complexes Each of which consists of an activated clotting Factor enzyme a co-actor and the enzyme substrate the first is called the extrinsic Factor 10 a and is composed of activated Factor 7 tissue factor and inactivated Factor 10 the intrinsic factor 10as is composed of activated Factor 9 activated factor 8 and inactivated Factor 10 the prothrombinase complex is composed of activated Factor 10 activated Factor 5 and prothrombin and last the protein C complex is composed of thrombin thrombomodulin and in activated protein C these complexes are assembled on an anionic phospholipid surface for which calcium is required the Contemporary model's most important distinction from the traditional one is the central importance of tissue Factor as a primary trigger for the whole Cascade tissue factor is a membrane glycoprotein expressed in vascular adventia during the initiation phase when it's exposed by vessel injury it binds to Factor 7 which is then activated the tissue Factor activated 7 complex which combines with Factor 10 on the plasma membrane forms the first of the four key multicomponent complexes which is referred to as the extrinsic tenas the extrinsic tenas activates Factor 10 which then binds to activate at Factor 5 to form the prothrombinase complex which in turn converts a small amount prothombin into thrombin although there is a small magnification effect as each extrinsic 10as can activate many copies of factor 10 and each activated Factor 10 can activate many copies of thrombin the amount of thrombin produced from the initiation phase is still insufficient to lead to significant generation of fiin threads and thus a blood clot for this there's the amplification phase the limited amount of thrombin produced so far activates more Factor 5 as well as factor 8 and 11 while the 7A tissue Factor complex as well as activated Factor 11 is able to activate Factor 9 activated Factor 9 binds to activated factor 8 which then forms the intrinsic factor 10as thus activating 10 the amplification effect at each step is such that the intrinsic 10as ends up activating as much as 100 times the factor 10 that is activated by the extrinsic 10 each activated Factor 10 converts more prothombin to thrombin which in turn further accelerates the amplification process until there is a virtual explosion of throm generation Vonda brand factor which we encountered in the last video on platet activation plays a key role in the amplification phase as it binds to inactive factor 8 that is circulating in the bloodstream greatly increasing its halflife in the final steps thrombin converts fibrinogen both that which is circulating and that which has been recently released by activated platelets to fibrin soluble fibrin monomers spontaneously polymerize into relatively weak threads Factor 13A which is also activated by thrombin then cross links and strengthens the overlapping fibrin strands the cross-link fibron which forms a three-dimensional mesh in which red blood cells and platelets become trapped represents the final step in the conventional coagulation Cascade referring back to the traditional model of coagulation for a moment although it has some limitations it will be helpful to know that the traditionally described extrinsic pathway also known as the tissue Factor pathway incorporates these steps the intrinsic pathway also known as the contact activation pathway incorporates these steps and the so-called common pathway includes everything that's left now there are several factors which have been historically included in the coagulation Cascade in addition to what I've already mentioned as specifically the intrinsic pathway Upstream of factor 11 which I've decided to not include in the diagram those are factor 12 cacan and something called High molecular weight kogen while they have been identified in the lab as having a role in coagulation I haven't listed them here because defects of these proteins do not result in a clinically apparent clotting disorder for me this races doubt that these factors play any significant role in actual physiology to prevent either spontaneous or runaway intravascular coagulation there are three main anti-thrombotic control mechanisms the first is antithrombin formerly known as antithrombin 3 which is a serine protease inhibitor that inactivates thrombin as well as factors 7 9 10 and 11 binding to heprin either endogenous or exogenous greatly increases antithrombin proteas activity though the physiologic role of heprin as an endogenous antithrombotic mechanism is unclear the second antithrombotic control mechanism is the protein C pathway in this an integral membrane protein of the endothelium called thrombomodulin induces a confirmational change in thrombin the alter thrombin is incapable of activating platelets or converting fibrinogen to fibrin but can Now activate a pro-enzyme called protein C activated protein C in association with its co-actor protein S inactivates factors 5 and 8 inhibiting the function of the prothrombinase and intrinsic factor 10as complexes respectively the last anti-thrombotic control mechanism is tissue Factor pathway inhibitor this is a single chain polypeptide which can reversibly inhibit Factor 10 as well as the 7A tissue Factor complex a small amount of tissue Factor pathway inhibitor circulates in the bloodstream While most is found attached to the microvascular endothelium so what is the final end result of this incredibly complex series of reactions starting with platelet adhesion and including platelet activation and secretion platelet ation the coagulation Cascade polymerization of fibrin and last the anti-thrombotic control mechanisms it's this this is a colorized scanning electron micrograph of a thrombus or blood clot red blood cells are obvious The Irregular gray blobs are platelet Aggregates the green cell right in the middle is a white blood cell and the hundreds of brown strands into which the cells are entangled are fibrin for me seeing a picture like this really emphasizes what an amazing process hemostasis is as amazing as clot formation is we don't necessarily want blood clots to hang around forever eventually we need our blood vessels to become patent again clots are eventually removed by the body in the process of fibrinolysis there are two major steps first plasminogen which is a circulating pro-enzyme is activated by conversion to plasmin by one of two similar enzymes tissue type plasminogen activator also known as TPA which is secreted from vascular cells and itself activated by thrombin or urinary type plasminogen activator also known as urokinase or UPA which is secreted from a variety of cell types plasman then Cleaves cross-link fibrin essentially severing the fiin thread holding the blood clot together this results in a variety of fragments known formally as fibrin degradation products one of which is the D dier which is composed of two D domains from adjacent fibrin monomers which are linked together the D dier will be important when discussing certain pathologic conditions such as disseminated intravascular coagulation as it can only be produced by plasmin cleaving fibrin and thus its presence indicates intravascular clotting in addition to fibrin plasmine also Cleaves fibrinogen and several other clotting factors as with the main coagulation Cascade fibrinolysis has additional levels of Regulation specifically three Inhibitors the first is plasminogen activator inhibitor 1 which as its name implies inactivates TPA and UPA then there is Alpha 2 antiplasmin which in in activates plasmin and finally thrombin activatable fibrinolysis inhibitor which actually acts on fibrin by cleaving the end C terminal residues which are important for the normal action of plasman in other words it makes fibrin relatively resistant to degradation the final topic to discuss is the important role that vitamin K plays in the coagulation Cascade in fact the K of vamin K is derived from coagulations vitamin which is the German word for clotting vitamin there are multiple types of naturally occurring vitamin K vitamin K1 also known as Pho quinone is found in green vegetables with the highest concentration in spinach kale collared greens and brussel sprouts in addition there is vitamin K2 known as menone which is synthesized from vitamin K1 by normal gut bacteria there are many similar forms of vitamin K2 which are named after the number of double bonds present on the side chain which can become quite long Vitamin K is fat soluble and requires intact biliary and pancreatic function for Effective absorption the role of vitamin K is to act as a co-enzyme in the post transational modification of several clotting factors specifically prothombin factors 7 9 and 10 as well as protein C and protein s it aids in the carboxy of glutamic acid residues to form gamma carboxy glutamil residues which allows the proteins to bind to calcium and thus allows them to be activated deficiency of vitamin K can lead to clinically significant bleeding this is seen in a variety of conditions most notably malabsorption syndromes vitamin K deficiency is also common in newborns leading to a higher risk of bleeding in the first week of life this is due to a combination of an immature liver that cannot effectively utilize vitamin K relative lack of vitamin K in breast milk and a sterile gut devoid of vitamin K synthesizing bacteria thus parental Vitamin K is typically administered at Birth so that concludes our grand overview of normal hemostasis let me return to this diagram once more the very first reaction to vascular injury is vasil constriction followed by platelet activation which is largely mediated by exposure of collagen platelet activation results in a change in platelet shape platelet aggregation mediated largely by Von willbrand factor and fibrinogen and results in the platelet plug this phase of hemostasis is called primary hemostasis the second phase of hemostasis is largely triggered by exposure of tissue Factor during vascular injury which triggers the coagulation Cascade the end result of which is thrombin conversion of fibrinogen to fibrin fibrin polymerizes generating fibrin strands which are superimposed on the platelet plug and trap red blood cells to form a blood clot this is secondary hemostasis there are many critical points at which the platelets and coagulation Cascade rely on one another in addition there are an anti-thrombotic control mechanisms which prevent both spontaneous intravascular coagulation as well as runaway coagulation in response to actual injury and finally the enzyme plasmine is responsible for Cleavage of the fibrin strands and eventual clot degradation that concludes part two of the normal physiology of hemostasis if you found this video to be interesting and helpful please remember to like it and share it with your colleagues and classmates the next video in this series will discuss lab tests of hemostasis
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Medical_Lectures
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The_Medical_H_and_P_Part_2_of_2.txt
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[Music] this is part two of two in this video series on the medical history and physical in this part I'll be presenting an hmp using the format and principles that were discussed in part one remember that the purpose of the oral presentation is to convey information to colleagues rapidly in order to Aid real-time decision-making therefore it will be necessary to be briefer than the written h&p would be for the same patient but all of the same structural components should still be present as I mentioned among the final tips from the last video unless explicitly instructed otherwise you should keep your oral presentation to 5 to 7 Minutes in length this can be very challenging particularly for patients with complicated medical histories or those with long differential diagnosis however as anyone who is rounded in the hospital has likely observed there never seems to be adequate time to accomplish what needs to be accomplished things are almost always falling behind schedule and even when they aren't all it takes is one unanticipated emergency sick colleague or upset patient and your team will end up behind schedule very quickly so it really is critically important to keep your presentations Within These time bounds if the listener wants more information information he or she can ask you for it or can refer to your more uh detailed written Note One seemingly simple but guaranteed helpful recommendation in order to keep your presentations under 7 minutes is to speak quickly you should be speaking slightly faster than you would in normal conversation or if you were giving a lecture to a room full of students but don't overdo it if you find yourself running out of breath while presenting that that's probably too fast given the speed of the presentation some of the annotations in this video on the side may seem like they're quickly flying by I encourage you to make liberal use of the pause button the source of information is the patient Mrs Jones with additional info provided by her husband both appear reliable the chief complaint Mrs Jones is an 80-year-old woman presenting with two episodes of Syncopy over the past week Mrs Jones reports being in her usual state of health until four weeks ago at which time she noted the onset of occasional lightheadedness these episodes usually occurred while walking lasted for a few minutes at a time and spontaneously resolved upon sitting down they initially occurred about once every 2 to 3 days there were no Associated symptoms including chest pain shortness of breath or palpitations over the next 3 weeks they became more frequent eventually occurring several times a day one week ago she stood up from the dinner table to walk across the kitchen suddenly felt lightheaded for a few seconds and passed out she woke on the ground several seconds later where she stayed for another minute while her lightheadedness passed she then got up and rested in a chair for another 5 minutes before feeling completely back to normal she denies hitting her head at that time the event was witnessed by her husband who reported no jerking motions of the arms or legs no incontinence and no significant confusion after she woke her husband wanted to bring her to the ER but she declined because she was afraid of being admitted to the hospital since then she has continued to experience intermittent lightheadedness continuing to become more frequent until recurrent episode of passing out on the day of admission that was identical to the first her husband called 911 and paramedics then brought her to the hospital Mrs Jones currently reports feeling fine and is asking to go home when asked what she thinks might be causing her symptoms she states that she should be staying better hydrated for her past medical history Mrs Jones had an MI in 2010 but has had no history of heart failure she also has had diabetes for 20 years with a recent hemoglobin A1c of 88.5% diabetic Bron neuropathy and osteoarthritis her surgical history includes only an appendectomy 40 years ago she has no significant gynecologic or psychiatric history medications include aspirin metool linil sytin metformin and am amitryptiline the last of which she was recently started on for her neuropathy she takes no herbals or supplements and she reports 100% adherence to all medication she's had no adverse drug reactions for her social history she is a non-smoker and drinks one to two glasses of wine per night she denies any history of elicit drug use she currently lives in downtown paloalto in a single family home with her husband her family history is non-contributory review of systems was negative aside from what was covered in the HPI on exam she is a well-nourished elderly woman who appears her state of age and is in no apparent discomfort temperature is 98.4 heart rate 58 supine blood pressure 134 over 70 which decreases to 110 over 65 upon standing respiratory rate 14 and O2 set of 96% on room a she has no corate brewes her cardiac exam reveals a normal sinus rhythm normal S1 and S2 2 out of six early systolic murmur at both upper sternal borders without radiation no S3 or S4 her jvp is about 6 cm pulmonary abdominal extremity and skin exams are all normal a thorough neuro exam was unremarkable with the exception of diminished sensation to light touch throughout both feet along with absent ankle reflexes bilaterally her gate is slow but without other abnormalities Labs demonstrate an unremarkable CBC and complete metabolic panel BMP is 220 and a troponin is less than 0.07 a UA shows only one plus protein chest xray demonstrates mild cardiomegaly and probable osteopenia nkg reveals non respiratory sinus rhythmia with a rate of 56 and first degree AV block with a PR interval of about 250 M seconds she has Q waves in 23 and avf and has evidence of LVH by voltage criteria so in summary Mrs Jones is an 80-year-old woman with a past medical history of Mi and diabetes who presents with Subacute Progressive positional lightheadedness culminating in two recent episodes of Syncopy her exam is notable for mild orthostatic hypotension an early systolic murmur unremarkable labs and an EKG with evidence of mild conduction system disease problem number one is her lightheadedness and Syncopy given the combination of orthostasis by history and exam and recent medication change orthostatic hypotension secondary Amat tripoline is the most likely diagnosis particularly as this is one of the most frequently observed meds to cause this problem closely related to this possibility is a chance that she may have autonomic dysfunction from diabetes as the presence of neuropathy suggests her diabetes has been long-standing and not optimally controlled less likely but still an important consideration is a Brady arhythmia such as severe sinos pericardia or intermittent High deegree AV block her EKG suggest the presence of conduction system disease and Brady rhythmia are a relatively common cause of Syncopy in the elderly however this is not typically positional as she describes her symptoms a don't misdiagnosis for Mrs Jones is ventricular tardia which she is at risk for given her prior Mi but otherwise nothing else is suggestive of this diagnosis her heart murmur is consistent with the atic stenosis though the murmur character is not consistent with the severity of as that would be necessary to cause Syncopy the diagnostic plan for her Syncopy includes Telemetry monitor for 24 hours followed by a twoe ambulatory monitor if the diagnosis remains unclear at discharge she will receive an echo to roll out aortic stenosis and as something which spans both the diagnostic and therapeutic domains we will DC her amitryptiline and monitor for resolution of the orthostasis over the next several weeks as an outpatient for Education we will instruct Mrs Jones to always move from a lying to standing position over the course of several minutes problem number two is her CAD for which we will continue all of her previous cardiac meds in the event that her Telemetry picks up more significant brto cardia we need to discuss the risk benefit ratio of discontinuing the mopol problem number three is her diabetes as she will likely be eating normally and we do not anticipate any upcoming contrast studies we will continue her outpatient metformin for her neuropathy as stated above we are discontinuing the amitryptiline to avoid confounding a presentation we will hold off on adding any new meds for now but would consider gapa Penton at some point in the future for diet she'll be on a standard carb control diet for prophylaxis we are encouraging ambulation and will start subq heprin and finally her primary state of goal of care is to get home as quickly as possible preferably with her lightheadedness resolved she clearly states that in the event of a cardiac Ares uh she would not want to receive attempts at resuscitation and would be strongly opposed to an ICU admission as such we have placed a DNR dni order in her chart so that's my example of a typical hmp oral presentation you should not expect to be able to present a case this fluidly and efficiently when starting out or even after months on Awards as a clinical student this is essentially what you should be striving to achieve by the end of your intern year which is the first year after completion of medical school for anyone viewing this video from a country with a different structure to Medical Training I hope you found this annotated demonstration helpful concur with practicing your presentation skills you should also be working on your clinical reasoning skills that is how to generate the differential diagnosis that you will discuss in the oral presentation you may find my three-part video series on that skill helpful as well [Music]
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Medical_Lectures
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30_Cancer_2.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. TYLER JACKS: OK, good morning everybody. Morning. Good morning, good morning. All right, we're going to continue our discussion about cancer today as well as on Friday. In between you have an exam on Wednesday. Partly in preparation for that I'm having office hours tomorrow between 3:00 and 4:00 in 76453. And as the note says, come with questions that you have about immunology, which I taught you, and the introductory cancer class that I taught you. If you have questions about other subjects that will be covered in this exam that Professor [? Sid ?] covered for you, it's best to go to her office hours, or to your TAs, or the other sections that will be held. All right so towards the end of last lecture we talked about smoking and cancer, and I warned you against the evils of smoking. And I hope you were paying attention. This slide shows you some pretty startling statistics that relate the increase in smoking, shown here, among men in this country, which began around 1900. And you can see it rapidly increased over the first part of the last century. And you can also see that about 20 years later lung cancer rates rose equally quickly. And this was some of the evidence that smoking caused lung cancer. And we now know, as I mentioned to you last time, that there are lots of carcinogens in cigarette smoke that cause lung cancer. Lots of mutagens. Either mutagens in their native form or promutagens that can be converted into mutagens, that we expose ourselves to through the process of tobacco smoking. That was men. But lung cancer has also increased in women. You can see this also precipitous increase in lung cancer among women in this country. And lung cancer has now passed breast cancer as the leading cause of cancer deaths among women in the United States. You can see that in the case of men, smoking levels actually started to drop quite some time ago. And coincident with that, lung cancer deaths among men in this country also have been dropping. It's a direct result of smoking cessation and people not starting to smoke. That had not happened in the case of women until very recently. In fact, just last month a study was published that showed that also for women lung cancer rates are now starting to go down. And that's a direct consequence of the fact that fewer women are smoking. It takes a few years to see that effect play out because people who were smoking 20 years ago are getting cancer based on that now. So examples of why smoking is bad for you. Cigarette smoke can be considered a environmental carcinogen, although it's one that we expose ourselves to. I talked to you about a variety of other environmental carcinogens that we get exposed to. I also described this phenomenon by which chemicals that are not in their native state, mutagenic, can become mutagenic through metabolism in our bodies. Promutagens can be converted to mutagens in our body. And I showed you the example of benzo [a] pyrene which is in cigarette smoke and gets converted into a mutagenic form in our bodies. And there are many other such. There are also examples, and I didn't say this explicitly, of nonmutagenic carcinogens. This goes against the rule that carcinogens are mutagens. These are actually nonmutagenic but they're cancer causing so we call them carcinogens. Examples of this would be alcohol and asbestos. These affect cancer rates, we think, because they cause tissue damage. Alcohol in the liver, for example, causes tissue damage in the liver. Asbestos, if you breathe it in, can cause tissue damage in the linings of the lungs. This tissue damage then results in increased proliferation. Cells are recruited to grow in order to repair the damage that was caused, and this can then indirectly result in mutations. Because as I mentioned to you, every time your cell divides there's a chance that a problem will happen, a chance that a replication error will occur. There's a chance that an important gene relevant to cancer will incur a mutation. So more replication, more proliferation, the greater the chance that a problematic mutation will take place. I didn't say this explicitly, but likewise problems in chromosome segregation can occur every time a cell divides. We have intricate systems to ensure that chromosomes separate properly so that each daughter cell gets the right complement of chromosomes. But sometimes that breaks down. And sometimes cells get the wrong number of chromosomes. We talked about this in the context of meiosis, but non-disjunction events can occur in mitosis as well, leading to chromosome imbalances, which are also hallmarks of cancer. As I showed you last time, if you look at the chromosome content of a cancer cell it is often very, very abnormal. And these abnormal chromosome numbers occur due to defects in chromosome segregation, including non-disjunction events. OK, so various things that we expose ourselves to or that just go along with the normal process of cell division can lead to mutations. And as I emphasized in the last lecture, cancer is the consequence of the accumulation of mutations in critical genes. So to summarize what I told you about last time, we now think of the process of cancer development as going from a normal cell through multiple steps to the development of malignant cells. This process, for most types of cancer, will take years to accomplish. Cells acquire these alterations over multiple years. And these arrows represent mutations to cellular genes. Mutations in processes that are important in determining the normal behavior of these cells and allowing these cells to behave abnormally. This process unfolds over time, as I mentioned. And gives rise to this phenomenon, or this hypothesis really for which we have great evidence now, the so-called clonal evolution of cancer, which I drew on the board for you last time. This is a nice version of that same concept from a figure from a book from Bob Weinberg who teaches this same class in the fall. Here you have a row of normal cells. Within these cells a mutation arises. This mutation gives that cell the ability to expand, perhaps better than its neighbor cells. So a number of daughter cells carrying this mutation are born. Within this now clone of singly mutant cells, a second mutation can arise, say in this cell here. This mutation, likewise, confers upon these cells a greater ability to grow, divide, survive, out compete their neighbors. So you get a lot of these cells too. And then within that expanded clone of cells, a third mutation arises, and so on. Until one has enough patients to have a fully malignant cell. So what are the genes that are relevant here? Well, let's think about the processes that we know are important in cancer development, and those are listed in the bottom right. Proliferation is the most obvious one. Cells proliferate abnormally in cancer. So mutations in genes that regulate proliferation are likely to be important here. Cancer is a disease of cell number, too many cells. You can get that because you have too much proliferation, but you can also get it if you have too little cell death. So mutations in genes that regulate cell death processes are also found in cancer. I told you last time about angiogenesis, the process by which blood vessels are recruited into the tumor. These two are recruited by virtue of signals that the tumor sends, some of which other product of mutations. I told you that cancer cells move in the process of metastasis. They break their interactions with their neighbors and they begin to move throughout the tissue, and ultimately throughout the body. So they have increased cell motility. And they also can invade. They can invade through the basement membrane. They can invade into the local tissue. So they have increased invasiveness. And there are other changes that take place within cancer cells. And so these mutations then collectively increase proliferation, decrease cell death, increase angiogenesis, increase motility, and increase invasiveness. It's important to know about these because today we are able to target some of these mutations. We are able to target the gene products formed by these mutant genes and thereby create better therapies. So our goal is to understand these processes such that we can ultimately control them more effectively in the context of treatment. OK well, that then leads us to what are the genes? What amongst the 22,000 genes in your genome get mutated in the development of cancer? How can we find them? Nowadays, we sequence the genomes of cancer cells, that's how we find them. But that's not always been true. It's actually been true for only the last few years. And so I'll give you some examples of how they were found previously. And why do we care? Well, I've already indicated some of this a few minutes ago, but the reason we care to understand the disease at the molecular level. Ultimately we'll be able to provide, in very accurate detail, improved diagnostic information, improved prognostic information. We'll be able to tell that an individual has cancer by detecting abnormal genes in their blood. Circulating DNA in their blood carrying specific mutations associated with particular cancers. We'll be able to diagnose the disease at an earlier stage that way. We'll be able to figure out exactly what mutations are present in the cancer cell to know whether that's a tumor that's going to go on to kill the patient, or sit there and do nothing. Should we treat the patient aggressively? Or should we leave them alone? Only with this molecular information will be will be able to figure that out in detail. We will, and we already, use this information to make better cancer drugs, molecularly targeted therapies that will replace conventional chemotherapy, which I'll teach you about on Friday. Chemotherapy can work, but it's highly, highly toxic and we'd like to be able to get rid of it and replace it with drugs that are much more specific, much more selective, and much less harmful except for the cancer cells. And this will usher in a new era, which I would argue is already here in some small way, called personalized medicine. The individual's disease will be diagnosed at the molecular level and a specific therapy will be designed for them based on those alterations. We'll dial up the right therapy based on that information. And again, hopefully, hopefully, yield better results and less toxic side effects. Cancer is leading the way, actually, in personalized medicine. But it will be true for lots of diseases in the not too distant future. OK, so what are the genes? Well, I'm going to introduce you to two broad classes of genes today. The first of which are called oncogenes. Onco for mass, genes. Oncogenes, cancer genes, cancer causing genes of one sort or another. The first of these oncogenes was identified in the context of oncogenic viruses. For example, rous sarcoma virus, which is a retrovirus. We'll learn more about those next week. Retroviruses are viruses with an RNA genome. HIV falls into the same class. Rous sarcoma virus, or RSV, has been studied since about 1910 or so. It was discovered by a virologist at Rockefeller University by the name of Peyton Rous. Peyton Rous was a virologist and a Long Island chicken farmer came to Peyton Rous with a prize hen from his collection. A prize hen that had a tumor mass growing on its breast muscle. And the farmer brought this hen to Peyton Rous and asked him to cure the hen because it was very valuable to him Peyton Rous took the hen. The farmer said, thank you. The farmer went away. Peyton Rous then promptly killed the hen and isolated the tumor. And was able to isolate from the tumor a virus. A virus that could infect another bird and cause cancer in that bird. So this was the first example of a virus associated with tumor development. There have since been many examples of viruses associated with tumor development in animal species. A few examples in people, but relatively few. Most cancers in humans are not virus associated but some are, like human papilloma virus associated cervical cancer as an example. But these viruses studied in laboratory animals were extremely important in teaching us about how cancers arise normally. It was known using rous sarcoma virus that if you took a normal chicken cell, fibroblasts from a chicken and infected it with rous sarcoma virus, it would cause those cells to round up and begin to proliferate abnormally. And these cells were given a term called being transformed. Transformed cells that had the appearance of cancer cells. After that work, a great deal of effort went into figuring out what were the genes of rous sarcoma virus that allowed the virus to cause those cells to become transformed. And it was determined that the virus carries a single gene called SRC, S-R-C, for sarcoma, which is responsible for this transformation process. You could basically just add the SRC gene and the same process would occur. Trying to understand the origins of the SRC, two investigators at UCSF, Mike Bishop and Harold Varmus in around 1975, determined that the SRC gene had a homologue, a related copy, in chicken cells. And they went on to hypothesize correctly that rous sarcoma virus stole this gene from the chicken cells that it was infecting and incorporated into its genome. So the SRC gene, this cancer gene, had a cellular origin. Moreover, they were able to show that SRC exists in human cells. And this was quite shocking. This cancer associated gene was present in our DNA. This discovery led Bishop and Varmus to win the Nobel Prize in 1989. And a funny story happened that day. This is a true story. I actually worked for Varmus as a PhD student, so I heard it from the horse's mouth. Varmus's sister-in-law was standing in a cafeteria line at Berkeley waiting to get her lunch and two guys were standing in front of her. And one guy said to the other what are you going to have today? And the guy said, I don't really know. And the first guy said, well don't get the chicken because two guys just won the Nobel Prize for showing that chicken causes cancer. This is sometimes how the public perceives what we do. But anyway, they went on to win the Nobel Prize for this important work. And they proposed that there were genes in our DNA, which they referred to as proto-oncogenes, which could undergo mutation and become oncogenes. In the case of RSV, the mutation was to take the gene out of the genome and stick it in the genome of a virus. But as we'll see today, there are other ways to do that too. Now, what the heck are we doing with proto-oncogenes in our genomes? What are these genes doing there? Why do we have the SRC gene? Why do we have other such genes? Who can tell me? Why is it beneficial to have a cancer causing gene in our DNA? What might these genes be doing? AUDIENCE: They might be just for normal metabolic processes. TYLER JACKS: Yeah, normal metabolic processes. Or perhaps normal proliferation. Our cells divide too. You go from a single cell when you're at fertilization to 10 to the 13 cells. It's a lot of cell division. That's a controlled process. There's lots of genes that are devoted to teaching your cells what to do when they're supposed to do it. And these genes are presumably involved in those normal cell division processes but they get corrupted in the context of cancer. They get altered so they don't work properly. OK. So RSV was the first example, SRC was the first example. But it still led to some skepticism, some concern that in fact what was seen in the context of these viruses might not be relevant to real human cancer. Which as I told you a few minutes ago, rarely involves viral infection. And so along came Bob Weinberg who was, and is still, at MIT. His lab is in the Whitehead Institute. And Weinberg did a critical experiment. He started with an individual. [LAUGHTER] Sorry about that. This is a person. Sort of. And this individual had bladder cancer. Weinberg isolated the DNA from the bladder cancer to ask the question, were there cancer genes in there? Were there oncogenes in there? Were there mutationally altered genes which the reason that this tumor arose? And so he did an experiment. He took this isolated DNA and he introduced it into some cells in the laboratory. These were immortalized mouse cells. Immortalize meaning that they would grow in the lab forever. Which by the way, is not normal. Normally you take cells out of your body or out of a mouse, put them in the lab, they'll grow for a while but they'll stop growing eventually. These cells were immortal. They could continue to grow. But they were otherwise pretty well-behaved. They laid flat on the dish. They had normal boundaries between cells. They were non-transformed. They didn't look like cancer cells. And they were non-tumorigenic. If he introduced these cells into an experimental animal they wouldn't cause a tumor, OK. They were immortalized but they were otherwise pretty well-behaved. He introduced the isolated DNA from the tumor cell through a process call transfection, where basically the DNA gets sheared up and introduced into the cells such that, roughly speaking each cell is getting individual genes spread out amongst this population of mouse cells. And then he waited. And what he found was that at low frequency, whereas most of the cells state in their normal morphology, occasionally he got this. [PHONE RINGING] Can somebody get that? A transformed colony of cells. And he assumed, correctly, that this colony arose because one of these genes was a cancer gene that allowed these cells to divide abnormally. He further could show that if he took those cells and introduced them into an animal they would now cause a tumor. So they were not just transformed, they were also tumorigenic. He went on to isolate the human gene. Which was not difficult to do because the cells themselves were mouse, so he's looking for the human DNA sequence. And he found, eventually, that it was a mutant version of a gene called RAS. A gene that you've actually learned about already in class. And I'll tell you more about in a second. This discovery made by Weinberg's lab and a few other labs at the same time in the early 1980's is the reason I'm standing before you. When I was your age Weinberg came to my college and gave a lecture on this work. And I was so excited about the potential of learning about cancer at the molecular level that I decided right then and there to start working on cancer, and have been doing ever since. So sometimes the things you learn in class actually change your life. Not to say that's going to happen today, but sometimes it does. OK, so Weinberg isolates the RAS oncogene from these bladder cancer Cells. They went on to sequence the gene to determine how it was different from the normal copy of RAS. And that's illustrated here. Here's the normal RAS sequence. This is now not working. Here's the normal RAS sequence. It's a version of RAS called H-RAS. That's insignificant. You can see the codons, the encoded amino acids. Here's the change that has taken place within the cancer cells. They didn't have a G in this position, instead they had a T in this position. This gene wouldn't encode glycine like it's supposed to but instead valene at codon position number 12. This change, this single nucleotide change, this single amino acid substitution caused this signaling protein to go from a regulatable state as shown here. And hopefully this is familiar to you because you learned about it already in the signaling part of cell biology. RAS is a GTP binding protein, which normally cycles between an active GTP bound state and an inactive GDP bound state. It goes from on to off through this hydrolysis of GTP. In the context of this mutation the GDP hydrolysis is inhibited. So the protein stays in its GTP bound active state. It gets stuck on. And rather than signaling in a regulated fashion, signals constitutively. Rather than telling the cells to divide when they should, it tells the cells to divide always. And that's presumably why this mutation is selected for in this type of cancer and many, many others. OK, I remind you that RAS is a signal transduction pathway. Here's RAS itself. RAS interacts with upstream receptor molecules and growth factors. There are intermediary adapter proteins that help that interaction. There are downstream signaling proteins, like kinases. And ultimately there are nuclear transcription factors and target genes that they regulate. This is a signaling cascade. And actually what we learn in cancer is that many of these genes, not just RAS, but many of these genes can be mutated in the development of one or more cancers. For example, some genes like these are amplified. And I'll tell you more about that in a second. Others are the product of translocation so that they're expressed abnormally. Others have structural mutations like deletions. And others have very subtle mutations. The RAS mutation is a subtle mutation. It's a single nucleotide change that allows this gene to function abnormally. OK, so importantly oncogenes dominantly transform cells. Weinberg added a single mutant oncogene and it transformed those mouse cells. The mutations are a gain of function mutations. They're dominant mutations. Gain of function mutations. There are now about 300 or so, and the number is growing, known oncogenes. Within the 22,000 genes in your genomes, about 300 of them can be converted to an oncogenic form through mutation. What types of mutations do we find? Well I've listed them on that slide but I'll just write it down as well. Subtle mutations, RAS is the classic example. A single amino acid change will convert the protein to an oncogenic form. We can also find gene amplifications. Your genes are present at two per cell, one from mom, one from dad. You're supposed to have two. Sometimes in cancer amplification of regions of the DNA occur. So you go from having not two, but four, eight, 50, 100 copies of the gene. And you can imagine having too many copies, leading to too much protein product, would actually lead to inappropriate signaling. So here the structural gene may not be mutated, the amino acid sequence may be the same. You've just got too much of it. Also, genes can be rearranged. Gene rearrangements. For example, translocations. And I showed you some pictures of translocations. Chromosomes that get joined inappropriately together. This can break two genes and form a new gene in the context of the translocation. Perhaps a gene is expressed from a weakly acting promoter, normally. But because of a translocation event a very strongly acting promoter gets stuck in front of that gene by mistake. And now the gene is expressed at very high levels, inappropriately. It's not unlike the consequences of gene amplification. So translocations, likewise, occur frequently. And we'll hear about consequences of amplification and re-arrangement next time when we talk about therapies. OK, so oncogenes act dominantly. They can transform cells all by themselves. But this should be in your minds creating confusion. Because I've also told you that cancer is a multi-step process. And if cancer were a multi-step process, how could it be that single mutations can transform cells? But in fact, we now know that single mutations are typically and maybe never sufficient to produce true cancer. So given that, can somebody explain how the Weinberg experiment worked? How was it that Weinberg was able to transfer a single gene into these cells and cause them to become transformed and tumorigenic if single mutations aren't enough? Why did this work? Anybody? The key is that they weren't normal cells. The cells he started with are already abnormal. They were already immortalized. They'd already been growing in the laboratory dish for a long time. So they were sensitive to a single mutation but normal cells in your body are not. And that's a good thing. Because as you sit there today you probably have about a million RAS mutant cells in your body, scattered around. And that's just based on the normal mutation frequency. The likelihood is that we all have about a million or more RAS mutant cells in our body. But because RAS mutations are not sufficient to drive cancer formation, that may be in a tumor initiated cell but it's not yet a full cancer, other mutations have to take place thereafter. We now believe that there's probably something like three to 20 mutations are required to make a full blown cancer. OK, so I've told you about oncogenes and I've related their function to normal cell division, and that's appropriate to do. Normal cells do get signals to divide, make more of themselves. And moreover, they get signals to stop dividing when it's time to stop. In embryogenesis once you form the liver you want to stop cell division within the liver cells. When you wound yourself you recruit cells to divide, but once the wound is healed you want it to stop dividing. And so there's complimentary signals, stop signals, that come into play to cause the cells to stop. Cancer cells have defects in both of these classes of signals. We've been talking now about the oncogene signals. They are the go Signals. The signals to tell the cell to divide. And because of these alterations like in RAS the cells are more capable of dividing. They make more of themselves. Moreover, the brakes on this process, the stop signals, are also typically lost in the development of cancer. Such that now the cells lacking the breaking signals will continue to divide still more. And these two classes of genes are called oncogenes, that we've been discussing already, and tumor suppressor genes that we'll discuss from now to the end of the lecture. Tumor suppressor genes. There are a number of tumor suppressor genes now as well, more than 200 known. So that's 500 or so known cancer genes already. The first one, the very first one described occurred in this type of tumor. This is a child with a tumor of the eye. The tumor is called retinoblastoma. And based on the examination of the genes within those tumor cells, it was hypothesized that the retinoblastoma gene that was responsible for this disease was one of these tumor suppressor genes before any of them were actually known. That led ultimately to the cloning of the RB gene, the retinoblastoma susceptibility gene called RB. Also in Bob Weinberg's lab. Which encodes a protein called pRB. And further work in Weinberg's lab and many others showed why cells get rid of the RB gene in the development of that cancer as well as other cancers. The RB gene is now-- or the protein that it encodes-- is now known to be an important regulator of the cell cycle. We learned about the cell cycle. Cells go from my mitosis G1, into S phase, and then G2, and then into mitosis again. This is a regulated process. And the RB gene or protein is important in controlling it. The RB protein acts here by blocking the transition from G1 into S, and I'll tell you a bit more how it does so in a moment. This is when the protein is in its active state. It can be inactivated through phosphorylation. A phosphate group, actually many phosphate groups, can be transferred onto the RB gene through kinases, which are stimulated by growth promoting signals. Growth factor binding to a cell will ultimately result in the stimulation of these kinases to phosphorylate the RB protein, taking it from this active state to its phosphorylated state, which is inactive. So the break is released because the kinases inactivate the break, OK. Just a little bit more detail about RB. A lot is known about this process now. This transition from G1 into S requires some transcription factors called E2F transcription factors. And RB binds and inactivates the RB transcription factors, keeping them silent. When it's an active state RB has a little pocket to which the E2F transcription factors bind. Here's pRB, and here are the E2F transcription factors and they are sequestered by the RB protein and can't otherwise do their job. But when RB gets phosphorylated by those kinases these phosphate groups interfere with the binding of E2F, and E2F is then liberated to carry out its normal function driving this transition. OK. So this is how the break functions and it's a very, very important break. RB is mutated in a very high percentage of cancer cells, not just retinoblastomas but many others as well. And the pathway that it is controlled by is likewise mutated in a high frequency. Tumor suppressor genes, like RB, are the breaks. They negatively regulate proliferation. As such, are these genes normally hyper activated or inactivated in cancer? Hyper activated or inactivated? Inactivated. That's in contrast to the oncogenes which get activated, hyper activated. These get inactivated or lost. These inactivating mutations are recessive. They are loss of function, again, in contrast to the situation with oncogenes which are dominant mutations, gain of function. And this actually creates an interesting challenge for us. If these are recessive mutations and a cell acquires a mutation in one copy of the RB gene-- and I'll just show the chromosomes on which RB exists. Happens to be chromosome 13. We have a normal cell, say it's a normal cell in the developing retina. And that cell incurs a mutation. Random chance, cigarette smoking, probably not in this child, but random chance. Leads to the inactivation of one copy of the RB gene. What's the phenotype of this cell? It's normal. These are recessive mutations. Having one normal copy is sufficient to provide RB protein, to provide control of the cell cycle. So the cell is normal but the cell is predisposed because now it only has one functional copy left. We could call this the first mutation. And now within this clone of cells if a second mutation occurs then we're in trouble. And the second mutation can occur in one of a couple of different ways. For example, it could be that by random chance, bad luck the normal copy on the other chromosome gets mutated. Or it could be, and it's actually more frequent, that a chromosomal event takes place. For example, a cell arises which only has one chromosome 13 through chromosomal non-disjunction. And it's the one with the mutant copy. These are functionally equivalent. Both of these cells are RB deficient. And they are on their way to becoming cancers. Tumor suppressor genes therefore require two hits. Two hits, two mutational events to inactivate the two normal copies of the gene. They can be two mutations within the gene or they can be one mutation within the gene plus some chromosomal event. And I've shown you one example here, there are others. OK. So that looks good. Everybody gets it right, no problems? Let me show you this picture, which I actually showed you on the first day of class. This is a child who has bilateral, multi-focal retinoblastoma. This child has 12 different retinoblastoma tumors affecting both eyes. Not only that, this child comes from a family in which retinoblastoma is passing through the generations. The grandfather had it, he passed on the predisposition to multiple of his children, who passed on the predisposition to multiple of their children. This is an example of a familial cancer syndrome. Familial predisposition to cancer. Most cancers don't have this kind of pedigree. Most cancers are sporadic, but about 10% of cancers look like this with a clear family history. Familial retinoblastoma, familial breast cancer, familial colon cancer, and there are others. These are caused by mutations in genes like this. This individual that I showed you, the individuals in this family have inherited a defective copy of the gene from one of their parents. And as such they have this genotype. They are heterozygous for an RB mutation. Heterozygous for an RB mutation. And because they're heterozygous for an RB mutation they are highly predisposed to developing retinoblastoma. And the reason is that within their cells, within the developing retinal cells in their body two mutational events are not required. Instead in them, all of their cells look like this. All of their cells are in that heterozygous predisposed state. And therefore in them, a single hit is required to give rise to a cell that is lacking the function of RB altogether. And that's why they're predisposed. And that's why BRCA1 patients get breast cancer, and why APC patients get familial colon cancer. This slide also raises for you some interesting questions and we'll talk about them next time. Here's a person who has the right genotype but he didn't get the disease. Why not? And another question, what would happen if two individuals who were heterozygous for the mutation were to marry, have a child who was homozygous for an RB gene mutation? What would happen then? We'll talk about that next time.
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Medical_Lectures
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15_Biochemistry_Enzyme_Regulation_II_Lecture_for_Kevin_Aherns_BB_450550.txt
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Kevin Ahern: Everybody had a good weekend? Student: You ruined it. Kevin Ahern: I ruined it? Student: Yes. Kevin Ahern: How did I ruin your weekend? Student: You gave us our tests back on Friday! Kevin Ahern: That's supposed to be good news! Well, okay, I'm sorry. You didn't have to pick it up, okay? Nobody said, "Go pick up your exam." Student: Is the curve shown now or at the end of the course? Kevin Ahern: The curve is right here on the web page. If you look right there, there he be. I showed everybody that. So it's on the web page. You can see it. I have a fair amount to get through today. I probably won't make it all the way through, but that's fine, too. I want to continue our discussion that I started last time about mechanisms of controlling enzymes. If you recall, I finished just about the place where I was talking about a model of control. When I talked about hemoglobin, I talked about how binding the first oxygen affected the binding of the second, affected the binding of the third, affected binding of the fourth. I said that this was a sequential model, that is, that each one influenced the one after it. That's one way of explaining how it is that cooperativity that we see in hemoglobin exists. When we talk about enzymes, of course, we don't talk about cooperativity so much as we talk about control, and there are other models that are used to explain that, and I want to explain a couple of those models to you. The first of these models is called the "concerted model." The concerted model is a little odd, at least to understand, well, let me back up. It's a little odd to understand, to begin with, because it has a different setup than the sequential model. The sequential model looks like what you see on the screen. The sequential model said that we started out with a protein, maybe it was hemoglobin or maybe it was an enzyme, and that protein was in one state. We can think of this as being in the T state. And binding of the molecule to one subunit caused it to affect the other subunit. So you see how this guy has bound a substrate and you can see how these other subunits next to it have changed from being squared off to being rounded off. That is, their structure has been affected by the binding of the first one to the very first subunit. We can think of this as a cause and effect. This causes these to change. This causes these to change. This causes this to change, et cetera, et cetera. So we see a cause and effect. Binding of one causes the next subunit to change, causes the next subunit to change, et cetera, et cetera. So that sequential model is sort of an intuitive one that we think about, but there are other ways of explaining cooperativity in hemoglobin, and also control in enzymes. The second one that I want to talk about is called the "concerted model." The concerted model, I don't have a good one to show you here, visually, unfortunately, but I'll have to describe it to you. The concerted model says... let's go back and look here. The concerted model says there's not a cause and effect. There's not a cause and effect. The enzyme can exist in two states. It can exist in a T state and it can exist in an R state, and the flipping from T to R has nothing to do with the binding of the substrate. You think, "Well, how does it do it, then?" It has nothing to do with the binding of the substrate. It has nothing to do with the binding of the allosteric effector. What the concerted model says is the flipping is an inherent activity of the enzyme. The enzyme is capable of flipping from R to T, completely independent of another molecule. The sequential model says it was a cause and effect. The concerted model says it's flipping independent of that. Well then how do we see the phenomenon exhibit itself? What the concerted model says is that after the flipping occurs, then something combined, and that something that binds locks it into one state versus the other state. So if have an enzyme that flips into the R state, and if this enzyme is ATCase, and this enzyme binds to ATP, it locks it in the R state. Therefore, that enzyme will remain in the R state and be very active. If the enzyme lets go of ATP and it flips back to the T state, all on its own, and it binds to CTP, it locks it in the T state, and it stays locked in that state until it lets go of CTP. So these are fundamentally different ways of explaining R state versus T state. In the sequential model, the R-to-T flip occurs as a result of binding. In the concerted model, the flipping is independent of binding, but binding locks it into one or the other, okay? Now, I know that's always a little conceptually difficult to get your heads around, so I'll stop and take a brief question for that Student: Is one or the other more accurate? Or is it a hybrid of the two? Kevin Ahern: Is one or the other more accurate, or is it a hybrid of the two? It varies from one protein to another, in terms of which one it tends to be more like. So there are some that will be very much more on the concerted side, others will be very much more on the sequential side, and all I'm expecting you to do is not memorize which one is which, but to just understand what Student: Is there something that causes it to unlock then, later? Or is it time driven, or...? Kevin Ahern: When you say "unlocked," do you mean to let go of the molecule? So is there something that causes it to let go of CTP or something that causes it to let go of ATP or aspartate? The answer is, like every other binding we've talked about that's not covalent, these are associative things. They come on, they go off. They come on, they go off. So we always see reversible binding of these intermediates. If they're covalently bound then, of course, we don't have any way of getting them off again. Yes, sir? Student: Would the state change be spontaneous of itself, or would it result from a reaction of something other than the intended substrate, like the [unintelligible]? Kevin Ahern: Good question. So his question is, is the change R to T completely independent of everything else? And the answer is, yes, it is. Yes, it is. And we will see preferences for one versus the other. I told you that when we started studying ATCase, we didn't even really know the R state existed until we locked it with PALA. That's because the preference was something like a couple hundred to one of T over R when there's nothing there, a couple hundred to one, when there's nothing else there. But if we lock one of them in the R state, then that equilibrium starts moving in the direction of the R state. More and more start flipping into the R. Yes, sir? Student: Does this rate of T-to-R transition switch, is there any trend to where something that switches states spontaneously, very quickly, favors the concerted model? Kevin Ahern: I'm not sure I understand the question. Student: If an enzyme spontaneously changes states very rapidly... relative to other enzymes, would that be better described by the concerted model since there would be more opportunity for [unintelligible]? Kevin Ahern: So his question has to do with how rapidly the T-to-R state occurs within an enzyme, and does this tend to favor one model versus the other. The answer is, if it's independently changing, it already is in the concerted model. So if the changing is independent of a substrate binding or an allosteric effector binding, then it automatically is a concerted model, okay? Okay, so I think that you should, as I say, understand the generalities of those models and be able to explain them to me in words. That would be useful. I want to turn our attention now from talking about models to talking about some interesting enzymes. This first enzyme I'm going to talk about is one you're going to hear a lot about. Yeah? Question? Student: So that graph was not so important? That you kind of, for that concerted model graph? Kevin Ahern: You're talking about here? Student: Yeah. Kevin Ahern: Yeah, so all this graph is showing is, okay, so here is the ratio of T to R of 70. This is when ATP's present. Here's the ratio of T to R that's 200 when there's nothing present. And here's the ratio of 1250 to 1 when there's CTP present. This is consistent with what I told you before. That is, ATP favors the R state more, so we see less of a ratio. "L" is the ratio of T to R, okay? Nothing is 200, and when we have the inhibitor, in this case, CTPóit's 1250 to 1, okay? Now, as I said, I'm going to talk a little bit about an enzyme you're going to hear a lot more about this term. We're going to see that this enzyme plays a very, very important role in a phenomenon we refer to as "signaling." Signaling is a way for cells to talk to each other. For one cell to talk to another, it must release something like a hormone. That hormone goes to that other cell and binds to that other cell, and causes changes to happen inside the target cell. One of the enzymes that plays a role in those changes in the target cell is protein kinase A. You're going to hear a lot about protein kinase A, so it's important that we understand, first of all, how it works. Protein kinase A is allosterically affected by a molecule called cyclic AMP, or, as we will refer to it in this class, cAMP. You see it right here. The structure of cAMP is shown here. No, you don't need to know the structure. It's related to AMP, AMP being adenosine monophosphate. The cyclic part is this little ring out here. So AMP doesn't have a ring. It just has a phosphate hanging out there, but when we make it cyclic, we actually rejoin it to this and make a circular structure. Cyclic AMP turns out to be very, very important for this communication process that cells go through, very important. The reason it's very important is because cyclic AMP affects protein kinase A. As I said, it is an allosteric effector of the enzyme, and the way it allosterically affects the enzyme is different than we've talked about other allosteric effectors. So far, with allosteric effectors, we've talked about the T to R or the R to T conversion. We've seen shape changes that occurred as a result of binding of allosteric effectors. Cyclic AMP is an allosteric effector, but it's not doing T to R changes. It's doing something like shown here. It turns out protein kinase A has multiple subunits, like you've seen before in other enzymes. However, this guy has catalytic and regulatory subunits, like you saw with ATCase, but, in this case, the binding of cyclic AMP causes something different to happen. The something different that happens is binding of cyclic AMP causes the catalytic subunit to let go of the regulatory subunits. This is, literally, an on/off switch. Remember I said that with T and R we don't really have on/off switches? I said we turn the volume down or we turn the volume up. This is an on/off switch. When the regulatory subunits are bound to the catalytic subunits, as we see on the left over here, this guy is dead in the water. It's not catalyzing anything. I'll tell you what it catalyzes in a second. When cyclic AMP is present, it binds to the regulatory subunits, releases them from the catalytic, and these catalytic subunits are active as all get out. So cyclic AMP is literally turning the enzyme from the off position to the on position. How do we turn it back off? Well, if we start degrading cyclic AMP in the cell, when this guy comes off, it'll get degraded. Remember, this is a reversible binding again. These are coming on, coming off, coming on, coming off. As we start degrading cyclic AMP in the cell, these guys will move back to this state and they will grab a hold of the catalytic subunits and turn them off. Now, these guys work by literally blocking the active site of protein kinase A. They block the active site. They physically provide a barrier. The active site can't get at what it normally catalyzes and, as a result, is dead in the water. It doesn't do anything. What does protein kinase A catalyze? Well, first of all, I need to tell you a term that you should absolutely know, and that's the term "kinase." First of all, whenever you hear the name of a molecule or the name of a protein in this class ending in "-ase" it is always an enzyme, always. So the fact that "kinase" ends in "-ase" tells us that it's an enzyme. Kinase is an enzyme that puts phosphates onto molecules. It puts phosphates onto molecules. So when you hear the term "phosphate," it's putting a phosphate onto something. The name "protein kinase" tells us that this enzyme puts phosphates onto other proteins. So protein kinase A, that just happens to be the very first one, protein kinase A catalyzes the addition of phosphates to target proteins. Now, on these target proteins, the effects can be enormous, as we shall see. But for the moment, all you need to know is that protein kinase A, when it's active, is putting phosphates onto target proteins, okay? Everybody got that? So kinase puts a phosphate onto target proteins. There have to be enzymes that take phosphates off, and I'll talk about those in just a few minutes. But a kinase puts it on. A protein kinase puts it onto other proteins, and, as we will see in this signaling process that I have described to you, addition of a phosphate to a protein can make a protein be much more active or much less active. So now we start to see another level of control of enzymes. Allosteric control was one, and we've seen a couple of examples in the form of ATCase and also protein kinase A. Protein kinase A covalently modifies targets, those targets being other proteins, and that covalent modification is a regulatory mechanism, as well. Covalent modification in the cell occurs in many ways. Putting a phosphate on or taking a phosphate off is one covalent modification. There are other covalent modifications, and you've already seen one example of those when I talked about the proteases. Chymotrypsin is covalently modifying its target because it's cutting it in half. It's breaking a covalent bond. These are different modifications that can occur inside of a cell... phosphorylation, acetylation, meaning putting on an acetyl group, myristoylation, putting on myristic acid, blah, blah, blah. I'm not giving you this table to give you something to memorize. But I do want you to be aware that there's many different kinds of modification to proteins that can occur. Those modifications affect the proteins in significant ways, because frequently they will change the charge of the protein. If we put a phosphate onto one section of a protein where there wasn't a charge before, now, all of a sudden, it might be attracted to another part of the protein that's positively charged, it might be repelling a part that's negatively charged. So we can see then, it's a mechanism of changing protein shape, and changing protein shape changes protein function. Now, let's talk about protein kinase. As I said, protein kinase or protein kinases, specifically, protein kinase A, takes a phosphate and it puts it onto a target protein. Where does it put it on? Primarily it puts it onto the amino acids that contain hydroxyl groups. These include serine, threonine and tyrosine. When we look at the enzymes that catalyze these, we see that they can be grouped into different categories. Some protein kinases strongly prefer or absolutely prefer to put them onto serine or threonine. Protein kinase A is one of those enzymes. It strongly prefers to put a phosphate onto either serine or threonine side chains. Other enzymes, other protein kinases, prefer to put it onto tyrosine. We'll see when we talk about signaling that the tyrosine kinases are really interesting. They're intimately involved in controlling cells' decision to divide or not divide. This plays a role in cancer and in many other processes. Kevin Ahern: I'm sorry? Student: Protein kinase A prefers serine and threonine? Kevin Ahern: Protein kinase A works on serineand threonine. So that's the reaction being catalyzed. We see that ATP is the source of the phosphate. Here's our target hydroxyl group. Here's the phosphate that's put onto it. That leaves us with ADP as a byproduct. Putting phosphates on is one thing. Taking phosphates off is another. Student: Is ATP always the source of that phosphate? Kevin Ahern: Is ATP always the source of the phosphate? Never say "always," but mostly I would say "yes." Removing phosphates is a phenomenon known as dephosphorylation. It's not the reversal of a phosphorylation. Instead, separate enzymes are used, and the enzymes that remove phosphates off of molecules or proteins are known as phosphatases. We'll talk a lot about this enzyme called protein phosphatase because it reverses the action of protein kinases. It catalyzes the hydrolysis, that is, the use of water, to split off a phosphate off of a target. So here is the phosphorylated protein that we had from the protein kinase. If we treat that protein with protein phosphatase, we see that we get phosphate and the protein back in its original state. This is important, because, as I've said before, cells are control freaks. If they have a way of turning something on, they darned sure better have a way of turning it off. Just like you better have a way to turn off your lights, or your electric bill is going to go crazy, so, too, cells want to turn off a process or they may burn up all their resources. So if they have the ability to put the phosphate on, they darned sure better have the ability to take the phosphate off, and that's what we see in the form of protein phosphatase. Understanding signals that cells are getting requires us to understand which of these enzymes is active at any given time, and that's a fascinating process. Okay, I'll stop, slow down and take questions. Clear as mud? Am I going slow enough for you? Student: [unintelligible] Kevin Ahern: What's that? Student: [unintelligible] Kevin Ahern: Clear as mud. Oh, you like that saying. Well, that's good, alright. So, phosphorylation is important. Dephosphorylation is important. Let's now think of some other covalent modifications, some other covalent modifications. I've already alluded to the fact that breaking peptide bonds is a covalent modification, and this covalent modification turns out to have some really cool implications for human health, very, very cool implications. They're involved in digestive processes and controlling the enzymes of digestive processes. They're involved in blood clotting. I'll be talking about both of these. The first one I'll talk about is digestive processes. If we think about it for a moment, we sort of take digestion for granted until we get an upset stomach. We get an upset stomach and then we become very aware of digestion. But in the absence of that, we eat our turkey, and we go to bed, and we fall asleep and that's that. We wake up in the morning and we haven't given a thought to digesting something, right? But what's been happening after we eat is that we have digestive enzymes that are breaking down those things that we've been stuffing down our gullet the whole day. With Thanksgiving coming up, I have turkey on the mind, you can see. Now, it's important that the body control those digestive enzymes. I don't want those digestive enzymes digesting me. And those digestive enzymes are really, really good at digesting things. When we look at the digestive enzymes, they're made in a variety of places, but one of the more common places they're made is by our pancreas. Our pancreas makes digestive enzymes. These digestive enzymes come in a variety of categories, but we're going to focus for the moment on ones that break down proteins. These are proteases. We eat meat. We break the protein in meat down to amino acids so we can use those amino acids to make our own proteins. I don't want my digestive enzymes to digest my pancreas. I'm fond of my pancreas. So my pancreas is not stupid. It doesn't make active enzymes. It makes inactive enzymes called "zymogens." In other words, it starts out making these enzymes that will ultimately become active, but it starts out making them in an inactive form so it can get it away from itself without digesting itself in the process of being made. So these inactive enzymes are known as zymogens. We can see them being made by the pancreas, and they're actually secreted outside of the cell so they can go into our digestive tract, where they get activated and start breaking things down. You might say, "Well, don't they sort of digest "the digestive tract itself?" And the answer is, to a limited extent, they do. Some of the cells that have the most rapid turnover in our body are those found in our digestive tract. When you hear of ulcers and people getting ulcers, what's happening is they're not replacing those cells as rapidly as they're being degraded. You have damage that's happening as a result of that. Student: Aren't those primarily attributed now to Helicobacter pylori? Kevin Ahern: His question has to do with ulcers relating to Helicobacter pylori, and the answer is, yes, but that plays a role, also, again, in this turnover of cells. The turnover of cells is a very important consideration in the digestive tract. We used to say it was stress that caused that phenomenon. Now we now there are other factors that play a role in there. What happens if we release our digestive enzymes from our pancreas and that activation process, which I'll show you about in a second, starts backing up the tubes and getting up to the pancreas? Will that cause a problem? Oh, yeah, oh, yeah. You create a condition that's very painful, in some cases fatal, called pancreatitis. What's happening is those zymogens are getting activated in the wrong place, at the wrong time, and the pancreas is getting chewed up. It's treatable, but it's a very painful condition. Has anyone in here ever had pancreatitis? We usually have one or two that have. Nobody that will raise their hand. It can be life threatening, in some cases. It's a very painful condition. It is treatable, but it has to be treated fairly early. Well, how do these things get activated? Here's one example. It shows how our body activates the enzyme you learned about last week known as chymotrypsin. Chymotrypsin is made in an inactive form, and the inactive form is known as chymotrypsinogen. Whenever we see the suffix "-ogen" on the end of it, we know we have a zymogen, and a zymogen is inactive. Unfortunately, that "-ogen" is not consistent. We see that that was used after they named some other enzymes, and I'll show you those in a second. But for the moment, we have chymotrypsinogen. Here's how it's made. It's made as a 245-amino acid piece and it's inactive. It doesn't do anything. It goes off into the digestive system where it encounters proteases that start to chew it up, and that starting to chew it up actually activates it. The first thing that happens is an enzyme known as trypsin makes a single cut between amino acids 15 and 16, and no, you don't need to memorize these numbers or anything like that. This creates something called "pi-chymotrypsin" that is partly active, meaning that pi-chymotrypsin will cut other pi-chymotrypsins. They're sort of a self-digesting set of enzymes. Pi-chymotrypsin cutting pi-chymotrypsins causes other bonds to be broken. We see that the bond between position 13 and 14 got broken. So now we lose a couple of amino acids there. We see that the bond between 146 and 147, and also between 148 and 149 got broken, and now chymotrypsin's in three pieces. Why don't these three pieces go flying away? Any thoughts? Student: Are there disulfide bonds present? Kevin Ahern: There are disulfide bonds. Absolutely, so this enzyme has to fold properly. The disulfide bonds have to form, and then the cleavage occurs, because if that doesn't happen, then these pieces all go flying away and we have no active enzyme, at all. It's prime evidence for the process of folding. It's prime evidence for the importance of disulfide bonds. Now, after we've got this pi-chymotrypsin making these various cleavages, this guy down here is what we call chymotrypsin. It's completely active, and now we can go out and start breaking down those things that you've been eating over Thanksgiving. What's happening in this process is we're looking at the inactive form in red. We're looking at the active form in blue. For the most part, the structure doesn't change much, but we look at what happens as a result of this cleavage. Look where this amino acid number 16 is, up here, and look how far it gets moved after that cleavage has occurred. Why is that important? Well, what this is doing is it's opening up access to the active site. The movement of this amino acid this distance allows substrate to now come into the active site and be cleaved. Shape change is essential for protein function. This shape change happens as a result of proteolytic cleavage. Student: So those three pieces do stay linked because of their disulfide bonds? Kevin Ahern: The three pieces stay linked because of their disulfide bonds. That's correct, so that's pretty cool. How am I doing on time? We're doing very well. Now, I've got to tell you a cool story. Actually, before I tell you the story, let me show you also the bigger process. I was showing you the activation of one of these guys. It turns out that a bunch of the proteolytic enzymes in our body, as well as other enzymes in our body in the digestive system, are all synthesized in the inactive form. Trypsin itself is synthesized as trypsinogen. Here's chymotrypsinogen that you saw, here. Here's procarboxypeptidase. That's another enzyme that breaks down proteins. Here's proelastase. Generally, when you see a "pro -" on the front, that's the equivalent of an "-ogen" on the end. It means it got named before they started using "-ogen" to name them. Proelastase gets converted to elastase, and I'm going to talk more about that one in just a minute. There's a master switch on this scheme, and that master switch comes from an enzyme known as enteropeptidase, and, no, I'm not going to tell you how enteropeptidase gets activated. We could imagine we could follow this scheme back quite a ways, and we can. The control of this system is important. If we lose control of this system, we end up making things like pancreatitis. We have enzymes active where we don't want them to be active and that's not a good thing. Notice that we have, over here, prolipase going to lipase. A lipase is something that breaks down fat. We're also controlling enzymes that break down fat. So it's important, in this big picture that we've been talking about in terms of regulation, that cells be able to control enzyme activity. If they don't control enzyme activity, then they've got real problems, whether it's an enzyme that breaks down proteins, or it's an enzyme that breaks down fat, or, as we'll see later in the term, enzymes that break down glycogen and glucose. Being able to control these is a very important thing, and the different ways in which cells do that are important for us to understand. Yes, sir? Student: [unintelligible] Kevin Ahern: Oh, yeah. That crazy, damn thing. There we go. Okay, questions about the scheme? Yes, back here. Student: [unintelligible] Kevin Ahern: So, as you can see, trypsin is a master controller of all of these down here. That's correct. Over here, you. Student: So, in this, would... is the trypsin more important [unintelligible] that we want to worry about, or the enteropeptidase? Kevin Ahern: Well, I don't want to say "more." I mean, this obviously controls four of these guys, and we'll see trypsin plays roles with other enzymes, as well, and enteropeptidase controls that. So there's just a hierarchy of these is really all that there is. Did you have a question? Student: Yeah, do these other molecules follow a similar scheme, where they start off as a single long [unintelligible] chain which is disulfide cross-linked, and then selectively cleave to be activated? Kevin Ahern: His question is, do the other enzymes have this phenomenon of getting chopped into pieces, hanging onto pieces, like with disulfide bonds, and then becoming activated? The answer is, there's a wide variety of schemes that are used. In some cases, loss of the piece is important. In some cases, simply structural change is important. So there's no one answer to that question. You had a question back here? Student: With the arrow going back from trypsin back to itself... Student: ...right there. How does that work? Does trypsin also regulate itself? Kevin Ahern: Good eyes. You have good eyes. As the figure implies, trypsin actually does play a role in helping to activate itself, and it's a complicated process. So trypsin can, once you've made some active trypsin, trypsin can then act on trypsinogen, kind of like we saw with chymotrypsinogen becoming partly activated. So, yes, that can happen. So that's cool. Now I said I was going to tell you something else about elastase, and this is a really cool story. So let me tell you this story. Student: Someone had a question. Kevin Ahern: Question? Student: So the trypsin [unintelligible] it seems like if trypsin can activate itself [unintelligible]. Kevin Ahern: Yeah, so, as I said, it's a complicated process. Trypsin can activate itself, to some extent. A lot of this is going to be controlled by when and where that activation occurs. In the digestive system, you want to be able to activate as much as you can very quickly. So if you don't activate any of it until it gets to the digestive systemóthat is, you don't have any of the master one, the, I can't remember the name of it, but the master enzyme, if you don't have any of that present where trypsin is being made, you're okay. But, as you could imagine, you could see things kind of go haywire. Maybe things do back up a little bit. Maybe you're activating trypsin where you don't want it to be activated, and that's what's going to be a mechanism for making pancreatitis. Connie? Student: On the [unintelligible] of the protein that you just showed, do they all just digest proteins except for lipase which digests [unintelligible]? Kevin Ahern: Let me make sure I understand and answer your question properly. So, if I talk about this, all the enzymes here are proteins. All the enzymes from here over to here digest protein down into other proteins, and this one simply breaks down lipids, in this case, fats. But all this, including this, is a protein. Yeah? Student: Could these be used to regulate themselves? Say you have excess lipase or something, but you don't want to shut down all [unintelligible] inhibition further up on their own chain that would maybe, say, get rid of the prolipase or something so that you didn't [unintelligible] the other ones? Kevin Ahern: I'm not sure I understand your question. Are you saying if you'd want to be able to activate only this one and not these guys? Is that what you? Student: Or something like that. Or maybe you just want only [unintelligible]. Kevin Ahern: Yeah. It's, as you could imagine, a fairly complex system, and in the context of our digestive system, we basically want all these guys to be active. We want them there breaking stuff down. So it would be unusual that we would have a situation where we'd want to make this one be active and not these, and as you can see, it would be hard to do that because we've got one master enzyme that's catalyzing it. Does that make sense? Student: Yeah. Kevin Ahern: Okay. Glad to see you guys are thinking about this stuff. I want to tell you my story. This story is a very cool one. This story has to do with what you see on the screen here, and it's something called "trypsin inhibitor." More specifically, it's called "alpha 1-antitrypsin." What is alpha 1-antitrypsin? Alpha 1-antitrypsin is a protein that acts kind of like that regulatory protein of protein kinase A. It will sit on trypsin, in the active site, and prevent trypsin from breaking other proteins down. Alpha 1-antitrypsin is another way of controlling trypsin. So your question back here about if I have trypsin active and I don't want it to be active, do I have a problem? Yes, you could have a problem. But if you have alpha 1-antitrypsin present, the body has another way of controlling trypsin once it's been activated. Once it's been activated, you have alpha 1-antitrypsin that can bind to it and you're set. Well, it turns out that alpha 1-antitrypsin is misnamed. It was named because the first enzyme that people discovered that it worked on was trypsin. But it turns out it works on some of the other proteases, as well, and the one that it works on really well is elastase. Some people say, in fact, that it's better named as anti-elastase instead of alpha 1-antitrypsin, because it's really good at knocking down the activity of elastase, and this is where the story gets interesting. Elastase is an enzyme that breaks down proteins, and it's present not only in our digestive systems, but it's also present in our lungs. Why do we have a proteolytic enzyme present in our lungs? Well, think about what our lungs are exposed to all the time. We're continually breathing in crap from that air. That crap from the air includes bacteria and viruses and all kinds of things. Our lungs are an interface between the rest of the world and our bodies. Having some sort of a protection against those things coming in is very good. Our lungs have elastase in them to break those guys down. They also have in the lungs alpha 1-antitrypsin. Now why do they have both? Well, it turns out that, just like we worried about having too much trypsin be active, so, too, do we want to control how much elastase we have active. Because if we have too much elastase active, the elastase starts doing something we don't want it to do, which is attack our lung tissue, and that creates emphysema. So having the proper balance of alpha 1-antitrypsin in our lungs that's functional is very, very important. If we don't have enough alpha 1-antitrypsin in our lungs, we will develop emphysema. If we have too much, we may be more susceptible to infections. Balance is good. Balance is important. What disturbs that balance? Well, it turns out smoking disturbs that balance. Smoking oxidizes a very critical methionine side chain in alpha 1-antitrypsin. When it oxidizes that side chain, we create the side chain that looks like this. This is what we started with, over here. Remember how I said alpha 1-antitrypsin fits into the active site and blocks it? This guy doesn't fit in the active site anymore. The more you smoke, the more you damage your alpha 1-antitrypsin, the more elastase starts doing the thing that it's supposed to, which is attack proteins, but those proteins are unfortunately in your lung tissue, and you're much more likely to develop emphysema as a result of smoking... very nasty thing. Many of you who know me know how anti-smoking I am, and with good reason. This stuff is designed to kill you. Yes, sir? Student: So is protein that elastase attacks, I'm assuming elastin or something along those lines, is that present in large capacity in, like, the alveoli? And that's what leads to the alveoli collapse [unintelligible]? Kevin Ahern: Yeah. His question is, does elastase preferentially attack a protein in the alveoli, and I don't know the answer to that question. I don't think it's preferential. No, I think it's a byproduct of that. But it is a concern and consideration. It is the alveoli that are affected by emphysema, but I don't know if it's really targeted that one. Are you guys ready for a song? Student: Oh, yeah. Kevin Ahern: I just happen to have one. It's actually relevant to this, and it's a song... I'm not going to sing it. I've got somebody who's recorded it for me. It's on YouTube and it's called "I Lost a Lung." I've got to set it up here... and here we go. If you want to sing along, you can sing. [David Simmons performing "I Lost a Lung" (to the tune of "I Lost My Heart in San Francisco")] Lyrics: You've gone and left me breathless, so especially today an absence from my body makes it hard to say, I'm so horribly upset and reminded evermore. Kevin Ahern: I'm not done yet, in case you guys are interested. Lyrics: I'll not forget, how you took my breath away. I lost a lung from smoking Camels. Emphysema kills, it seems to me. You see those nicotines and tars, leave alveolar scars. My raspy throat will often choke from the smoke. I've no respect for RJ Reynolds and its cor-por-a-toc-rac-y. Now when I hear the name RJ Reynolds, I only think malignancy. Kevin Ahern: So that's my little sermon of the day, I suppose. I've got about five minutes and so I want to finish up talking or at least introducing one last topic that I'll finish up next time, and that's another important process that's controlled with zymogens. Now this process turns out to be absolutely critical for us to function, because it controls our blood clotting. Again, our blood clotting is kind of like our digestion. We don't think about it, but by golly, where are we without it? Let's think of the challenge that our body has. Our body wants to deliver blood to our tissues, and it wants to repair itself if there's damage. So if I poke a hole in one of my arteries or one of my veins, I want to be able to stop that flow from happening. Not only do I want that flow to stop from happening, I want the repair to be done in minutes, because if I don't I will lose all of my blood. So I need to have a system that is extraordinarily rapid. The way that I make it be rapid is I load my bloodstream with zymogens that are capable of forming that clot. We can think of our bloodstream as kind of like a loaded gun. A loaded gun, because it's ready to clot as soon as the sense of damage occurs. Can we see problems happening with that? You betcha. If we clot when we don't want it to clot, we may be blocking blood flow to a critical tissue, like our heart or our brain, and we have a stroke. If we don't make that clot, let's say that we are a hemophiliac, we will bleed ourselves to death. So, again, balance is important. We have to be able to stop that blood flow and stop it in minutes, and, this is the most amazing thing to me, it's got to be water tight! We have to assemble this block from molecular pieces. They have to be self-assembling and it has to be tight enough to hold water. I'll stop with that and that's where we'll pick it up next time, but I want you to think about that. Yes? Student: [unintelligible] Kevin Ahern: Can [unintelligible] activate zymogens? Generally, no. Captioning provided by Disability Access Services at Oregon State University. [END]
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Medical_Lectures
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Immunology_Lecture_MiniCourse_6_of_14_Humoral_Immune_Response.txt
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okay so so this is part two of antibody response and again this is based on a lecture that given by dr. Mattie shark at Einstein I don't give all the lectures at Einstein so I the kind permission of some of the faculty and Einstein they let me use their slides that just change them a little bit and okay so this this slide is a review slide from the previous lecture and I just wanted to make some points and use this to kind of kick off this particular lecture one point I wanted to kind of review and I'll be reviewing this over and over again is that in order for B cells to make antibodies ficient lee it has to interact with a t-cell however it is possible to make antibody in the absence of t-cells and that's something called a T independent antigen something that many of you may have heard about what that methi independent means you don't need a t-cell however in general those antigens tend to be ones that link multiple immunoglobulin molecules on the surface of B cells because they have repeating motifs same antigen repeated over and over again by cross linking multiple immunoglobulin molecules it enables the generation of a very powerful signal that in the absence of T cells can still trigger the B cell to make antibody however in the absence of T cell power the only isotype of antibody that will be made will be IgM and I'm the only way that you make otherwise and types of antibody IgG IgA IgE is solely with T cell health and we know that's decay because as I'll discuss when we talk about the mean of the efficiency if an individual does not have T cells they don't make anything other than IgM however with the presence of T cell he'll now this B cell could be more efficiently and the other point too is that it will undergo the process of somatic hypermutation to increase the affinity of the immunoglobulin molecule you also need T cell health so in the absence of T cells you'll only make IgM and you won't have so Matic mutation so you won't be able to make those high affinity antibodies that are that much more effective in the immune response so clearly that's why T cells play such a critical role the second point the other point I wanted to make is that in this central blast that's where the isotype switching is going to take place and as I'll discuss a little bit later ad not only plays with whether and developmental so that somatic hypermutation but NID exactly the same enzyme also plays a critical role in isotype switching and again that a little confusing at first how the same ends like with two different things but it turns out as I'll show you to be triggering the same repair mechanism and the other point I wanted to make is actually kind of to reinforce this idea of the antigen selectivity several people asked me a question after the lecture about antigen selectivity and the point I wanted to make is that think of the B cells basically require binding of antigen to the antigen receptors the same way we need food so if we go through a reasonable period of time without food or water we basically can't survive these cells also they continuously need to by the antigen to trigger the antigen receptors to trigger the signal transduction pathways to keep the B cell alive in the absence of that antigen those B salt will die now why should that be why would you want these cells to die if there's no antigen any suggestions right what's the point of keeping them if there's no antigen why is it no man vision because you've cleared the inspection and if we don't need to be sell anymore so it's a very efficient way you know it's like a third step once you reached a certain temperature if you turn off the thermostat this so there's no antigen the beetles have done a great job that makes a lot of sense right so now if we think in terms with what's going on in this competitive process for antigen you have some immunoglobulin molecules that are mutated if they're very very high affinity you have other molecule including a lot of molecules that I don't have not mutated or maintain that they recognize antigen not as strongly think about a zoo where you have like twenty Lions in a cage and you throw out one piece of meat into that cage what's going to happen the biggest strongest most powerful lion that's the big one that gets the the meet the other weaker lions or maybe they're not lines now let's say a mixture of lions and and monkeys and I know what some of the animal you know huh yeah you okay great but that could be deserves but so so you throw that one piece of needed who's going to win the strongest that what won't the highest level of affinity and the same thing is happening in the lymph node you throw in a limited amount of antigens especially because only a limited amount of answer is actually getting into the lip no because that's not where the infection is and the antibiotic the highest affinity is going to be like the lion that grabs that antigen and that one is going to be the one that survives as the mutations occur and new higher affinity antibody gets generated think about throwing and even bigger and stronger animal into that same cage and that's now going to be the one that wins or if you take the Lions and you put them on growth hormone and you put them on an anabolic steroids those going to be the ones that now take over and get the food and grow and survive and the other ones unfortunately will undergo apoptosis ok does that kind of clarify that concept a little bit better ok so now if we think you know these are going to be the same questions again well how do we begin advise every possible pathogen so this we've covered in the first lecture highly avoiding Auto antibodies this actually will be discussing in the lecture this afternoon on b-cell and t-cell maturation and selection how do we rapidly increase the amount of antibody mobilization again we'll be discussing that t-cell interaction this lecture is really going to be focusing on how do we switch from making IgM to IgG isotype switching and the functional activity of the different isotypes of immunoglobulin as well as the role of FC receptors in the immune response and again we discussed previously how we increase the affinity of antibody affinity maturation and how do we generate remember again we're going to be discussing that in greater detail in a subsequent lecture so just keep that on the side so again just to kind of review this initially you may i GM as the immune response continues multiple exposure to antigen now you make IgM undergoing class which which we'll focus on and as we discuss in the previous lecture you increase the antibody affinity so this right now I'm going to start focusing on Gizem of class switch from IgM to ITT IgG and IgA well why do we want to class switch well it reduces why we want to class which is maybe those isotypes work better and this is again I show this slide previously this is just to kind of review exactly what immunoglobulin molecule does neutralization optimization complement activation but for virtualization and optimization how tightly that antibody binds to the antigen is critical in determining how well it rose now let's look at that in mind this is a slide that I took a little bit out of work but wanted to kind of just feel River to put a demonstration of what IgM molecule looks like well what strikes you about this you know Guatemala idea molecule how many kind of a new model does it have on it as five it has ten binding sites well sometimes big is good and sometimes big is bad so for example if you need to get into like tight small crevices or areas a big molecule is not going to be very good at doing them you can I have a small agile molecule to do that I mean think in terms of like naval warfare you know giant worships are very powerful instruments but if you start having to navigate it to rocky areas or shallow areas it's completely useless you might have a light quick small boat to be able to navigate so IgM is huge and therefore this big clunky antibody so in a sense one of the limitations is is because it's so big as I'll discuss a little later it can't get into peripheral fluids as efficiently as itg which you can see is only 150 can get in so that's a sizes can be an issue the second point to make is that I GM as I just said does not undergo somatic mutation so what does that mean that means that what is the infinity of each one of these binding site is going to be is it going to be high or is it going to be low it's going to be low right so there is not going to be able to bind well so but how can you make up for that low affinity so the example I always think about is anybody here play basketball okay can you palm a basketball can you hold it with one hand just about okay that's cooling up I thought having basketball here I would like you know give it to you having demonstrates everybody but what that's showing by palming a basketball with your hand is your affinity of your hand how strongly you can hold on to that basketball I have these short stubby fingers it's genetically challenged again you know what one what a great thing is you know years ago when everywhere anyone hear of parents right I assume that's yes so unless you know we have Dali but it and your parents are always like saying why can't you be this you know why can't you be that you know why can't you know always kind of trying to of course make you a better person but there in a sense just criticizing all the time and my father was going on his whole brand why can I do this why can I do that etc etc and looked at my father and I said daddy I said I'm only as good as my genes and when they get my genes from from you so in essence you know he didn't appreciate that you know response but it did cut down on these you know Rico criticisms but again you only you're only as good as you DNA and your genes so similarly I'm short stubby fingers and I have very very low affinity so if we had a contest for example where we had a basketball and I tried not with my hand and you try to pull it away from me who do you think would win you would obviously win because you have higher affinity with your stronger hand or make it even better if we had a contest between me and Shaquille O'Neal I mean Shaquille O'Neal has huge hands I mean he could probably palm a basketball with two fingers right so he be holding on he has high affinity receptors there's no way that anybody in this room could possibly grab a basketball from Shaquille O'Neal does that make sense right but let's say five of us got together and he was holding on to the basketball with his one hand that has high infinity binding and five of us came with our smaller hands and tried to pull away from Shaquille O'Neal who would win we would win right why even though our hands individually have a much lower affinity than Shaquille O'Neal the additive affinity of all of our smaller hands is greater and that's what avidity means avidity means the summation of the ability to hold on to something and that's dependent upon how many molecules you have binding so when I Jam is basically said is I'm resigned to the fact that have short stubby fingers I can you know I can't mutate how my fingers grow longer and longer to get higher affinity but in order to make up for that I'm going to have ten binding sites and that's how my GM still has reasonable functional activity because of the fact that IgG undergoes somatic mutation has stronger ability to grab on to a munich lock toe to the antigen it can now function equally well as a smaller molecule with only two binding sites and then high affinity makes up for so again that's the kind of explanation of why I GM has to be this with ten binding sites in order to make up for its lower affinity out of the individual binding sites but since IgG is evolved into the Shaquille O'Neal of immunoglobulin with a higher affinity it doesn't need as many hands in order to grab on to the antigen okay that makes sense that clarify things so therefore now let's talk about the process of how we get from IgM to IgG so we've talked about somatic mutation now we'll explain also why do any high affinity and the reason we need high affinity is because we need to do things like neutralizing virus we need to grab onto buyers very very tightly now what this slide is kind of reiterating is the mechanism by which immunoglobulin is generated and also the process of class switching which we'll discuss again in a little bit more detail but just to take you through this the b-cell has expressed was on its surface in this case it's expressing an IgM molecule on the surface an important point to realize is while the soluble immune ajm a question right so the question that you're asking is a very common question because affinity and affinities always very confusing so if you have one antibody with high affinity right that just talks about the unique binding site but it ultimately determines how well that antibody will bind to something is really the ability because the ability is the summation of all the binding sites so IgM even though it's individual immunoglobulin binding sites may be lower affinity by virtue of having so many that could bind to antigen that a divinity allows it to bind reasonably well okay but again it but the limiting factor is that the antigen has to have a lot of binding sites available for the antigen for the for the antibody so for example if my pen for example is the antigen and the binding site is the tip of the pen well I can have 10 binding sites but since there's only one site to bind to the other 9 binding sites are dangling in the breeze and I'm only going to get the binding of that one binding site so for I Jim's not going to be very effective that's why you need itg for antigens that have only single epitopes however what antigens do we know that have the same antigen motif repeated over and over again what type of compounds okay bacterial cell models and what about your cell walls made out of well from politics out I mean LPS is in it but it's mana polysaccharides what is poly saccharide poly saccharide is the same saccharide or glucose or motif repeated over and over again so when you have an antibody that recognizes that particular saccharide they'll be lined up with one next to each other and in fact that's what IG m is actually quite effective against polysaccharides because there's a lot of those antigens wall next to each other so all the end of the binding sites to grab on to it whereas protein antigens was tend to have not that many antigens repeated more likely to just unique that's why you need I to G says another kind of conception or point for polysaccharides IgM works actually quite well because you could take advantage of the ability of left hand binding sites but for something like a protein which may only have a single antigen binding site the fact that you have the extra arms won't help you and that's why you need a G have a high affinity of that one binding site does that answer your question okay so that's how we're very thanks for that question because it allowed me I think to really explain it in a way that people may have understood it better and again that's just to encourage people to ask questions because not only by asking questions do you make it more understandable to you but you also help the other people who are trying to learn okay so so now the the antibody binds this is a let's say this is the IgM on the surface so even though the soluble IgM is a pentamer the surface IgM is just a single molecule the same way an IgG is a single molecule the same way IG a remember the service Interlagos or a single molecule all in time you have the pentamer or IgM is a soluble circulating molecule which makes sense because that's what has to grab on to the antigen it internalizes miss case the virus and what's going to happen to the virus it gets digested in a lysosome like yesterday and peptides are identified and then again what MHC model is going to be presenting it MHC class 2 and now what kind of T cells are going to be presenting it towards a cd4 T cell that is a harbor T cell this T cell sees this peptide and now is trigger in addition there was another service model called cd40 ligand expressed on the surface of the t cell and CD fruity which is expressed on the surface of b-cell these two interact with each other and the cd4 T on all these cells is exactly the same the CD Porter line in on all T cells is exactly the same and this interaction also causes signals to occur in the B cell when combined with cytokines that's what's going to be triggering isotype swishing so the similar Teletubby cell to stop making IgM and to start making a different isotype is the T sub having cd4 T cd40 ligand signaling how do we know this is true because they're individual that last sitting for T line yet and they only make IgM and what these cytokines are doing they are providing the fine tuning to say made another MYSA type but different cytokines will drive you to make different iso types of immunoglobulin so the cd4 easy for likeit is the global signal class switch and the side of kind of a bond of fine tuning to telling you what specific isotype you made this piece of undergoes proliferation class swishing now cd4 t cd4 tonight will also trigger somatic mutation and now this B so makes high infinity antibodies that only need to bind to one site to bind very tightly and therefore they can now neutralize the virus are prevented from being infected okay that in a nutshell is the process and why you need T cells in addition antibodies play a critical role in neutralizing toxins so to anyone come up with a toxin that all have been vaccinated against that protects us from stepping on rusty nails tennis Thompson and in fact you know when I was a young and you get a shot I thought that when you get immunized against tetanus you get a tetanus shot I thought that we were being immunized against the bacteria that causes the infection did anyone think that makes sense because you guys again for vodka polio from easels mumps rubella what are you being immunized against the pathogen you're being immunized against pneumococcus home office influenza you're being immunized against a pathogen so I thought the tetanus shot was against cluster the attendee that you're being infected with and why you're getting immunized because that way we step on the rusty nail then your antibodies are going to find that bacteria and eliminated before to make Thompson right then I was at the NIH and I was generating tea salt plums against tetanus and I pull out the vial to immunize and look at it and it says tetanus toxoid and I said this can't be the right stop it says tetanus toxoid I want Clostridium 10 name the real stuff so so in you know the personnel room was looked at me like I was a and it's never a good feeling when someone will set you like that next beaker slowly and carefully and and enunciate their words and favorite is it the idiot ah you what's going to kill you when you step on the rusty nail is not this this detecting plus the the cluster t10 a because it's anaerobic it stays in that dead tissue it doesn't get into your virtual circulation because the oxygen kills it but it would have done what is there is make the Thompson that toxin gets into the circulation and that's what ultimately is going to kill you so therefore the antibody that you want to make is not against the bacteria but against the toxin because what you want to do is hear the Thompson it binds to a receptor on the cell gets internalized and then poisons to cell but what the antibody does is it bodies it somebody important before it was bond to its receptor and protects it and therefore sequestered it so it doesn't cause damage so in fact that's what the immunization is it's actually a toxin they take the toxin they heat it so it's no longer active so you're not going to inject someone with the deadly constant that wouldn't work very well and and then make antibodies where you want those antibodies to be high affinity because one of them to grab is Thompson very quickly before it could bind to the receptor and therefore what bison type or you to want those antibodies to be IgG and you want them to have a very high affinity have undergone somatic mutation again that's why you want a class switch okay is that clear so now we going to be focusing now again here's the b-cell cd4 t cd4 T by n and the cytokines are being made and other would be some proliferation and also class switching now we're winner for this process to occur what does the B cell have to present to the T cell a peptide weird of the peptide come from the antigen but what what does it is to cost you what's the molecular arjen of the antigen what kind of compound is it what is a protein because peptides come from proteins right now what compound doesn't have peptides that we just discussed an IgM is really good against huh polysaccharides polysaccharide there's no peptide it so now let's what the scenario would be if a antigen is a polysaccharide the immunoglobulin molecule will bring it in digest the polysaccharide but what does it have to present polysaccharide T cells don't recognize polysaccharides polysaccharides can't get into MHC molecules so therefore if you immunized with the VDA polysaccharide would you be able to recruit t-cell help no and therefore what if I devised the type would you be stuck making on IgM because you can't class switch and you can make low affinity IgE antibody which may not be as effective and protect you from infection with bacteria like new with pneumococcus so this is actually one of one of the most brilliant inventions in terms of vaccines because what we're now doing and it's really not it's a little tricky and we've kind of apologized an immune system for doing it but we're fooling the t-cell and what we're doing is we're taking the polysaccharide antigen we're cold family linking it to in this case tetanus toxoid so because it's currently linked when the antibody bonds will recognize the polysaccharide it binds to it it internalizes it but it brings in a cold family linked tetanus now it digests up the polysaccharides but whereas when it only ingested the polysaccharide that's all it had to present now is sucked in the tetanus toxoid it chops up the tent is talks about into peptides and now can present tetanus derived peptides to attend a specific T cell the T cell doesn't know what to be so recognized as it would probably foolishly things that this b-cells making antibodies against tetanus never I want to help it so provided all the appropriate signals cd40 ligand cytokines generate class switching so mad and mutation so now their b-cells able to make very high-energy antibodies against polysaccharide that are IgG and are far more effective in terms of protecting the individual from being infected with the polysaccharide so that's the kind of vaccine that we now use for pneumococcus form offals influenza is actually basically completely resolved all home offals influenza infection the new or the pediatrician you remember 20 years ago a moth was influenza meningitis was devastating disease septic arthritis was devastating disease pneumonia was devastating but invention of this conjugated vaccine allowed us now to immunize children and protect them from infection so again this is ruined showing how the power of understanding the physiology of the immune system can be harnessed to figure out how to get around it and now bounds it appropriately very high immune response when physiologically one would not have been able to be generated is that clear does that make sense that again you'll hear over and over again in terms of vaccines conjugate vaccines the idea is is that even though the antigen itself may not be very strong if we can link it to something that's a more powerful antigen we can recruit better help to make better antibodies and again for HIV that's one approach that people have been using okay so now again it's just to review the back we had class switching to IgG from IgM over time and how does this occur and this is basically showing that the different isotypes have different constant regions so even though the variable region stays relatively the same by virtue of having different constant regions so again think of like different screwdrivers that have different handles that are good for different sized hands that has converted to the indian vlog with different functional activities so the IgM may have one functional activity because if it's FC is FC portion this constant region whereas an IgG molecules we have different function and an IgE map molecule may have a different function despite the fact that all of them can recognize exactly the same the antigen and again later on an electoral discuss this different effector functions of the different isotype right now I'm going to focus on this how the molecular rearrangement occurs how do you switch from making IgM to IgG IgE to IgA and this is again reviewing the structure of the even applied one molecule and what you see is that here's where the edje junction is going to be these variable region but right downstream from it you have all these sequence regions that in color this case to the IEM constant region this case the gamma 3 gamma 1 this is from my so a gamma 2 B into a for humans is gamma 3 gamma 1 IgE alpha 1 of a gamma 2 gamma 4 they're like stacked up waiting to be bound so the first antibody that you made is going to be IgM why is that because you make just transcript off your vtj sequence and the first conative region downstream is IgM so you basically exhibit it on make a transcript is IgM and then that's the antibody that you make it turns out that IG delta which is IgG can also be made of that transcript that is at an RNA level is a splicing event which will allow you to make either IgM or IgG and discuss that in the lecture this afternoon the implications of that but that's why I GM is made further its first in line and the genetic sequence is that clear that's that's that's the only the magic you know why should i GM be first its first on the line of all the genes of the cons and regions and that's why it's made it first in order for these I sometimes to be made what do they have to do this is like on when you try to get on to a bus and it's the last bus out and is 10 seats left and is a hundred people waiting on line how well of course none of you would do that because we're all polite but how would an impolite person try to get onto the bus he pushed and cut ahead in line and the same way to these constant region is basically saying I'm going to push the IgM out of the way and it will allow me to be adjacent to the vdj sequence to now make my isotype antibody and that's the molecular mechanism that really needs to occur is get everything else out of the way so that any one of these canal may align adjacent to the variable region sequence in coding sequence okay is that clear and this is now just to kind of illustrate all the different functions of the isotypes and this is a kind of slide that you can spend hours going over but it's going to make a couple of points these are showing the different functional activities and these are showing the different ISO types well one thing jumps out very very rapidly is that IgM actually turns out for them it's not as good a neutralizer as any of the IgG why do you think that is as low of an affinity and therefore again it can neutralize as well and whereas IgG has a higher affinity that's why the neutralizing antibodies but what jumps out about IgM is it activates complement the best and again we're not we don't have a lecture on complement but think about it what pathogen do you think IgM is going to bind to is very very well you mentioned it before bacteria that have polysaccharide walls that allow all ten binding sites to bind to it very very well and it turns out that is complement good at killing viruses now not at all it's greatly killing bacteria so their portion is IgM vine so well to bacteria it turns out that it also activates complement the best to facilitate its activity however other things IgM is not as good at so opposite ization is not as good because it turns out that in order for optimization to occur efficiently with antibodies you need FC receptor binding and IgM does not bind very well to FC receptors and the important is not as effective the other thing to point out as well is is transported cross the placenta IgM does not get transported across the placenta so however IgG one two and three get transported across the placenta very well so maternal IgG crosses the placenta so now the baby when the baby is born has the same level of IgG as the mother has which means that the baby has all protected by GG's that the mother has so the baby comes out is protected from infection it doesn't have IgM though why not well IgM is this wimpy in a low affinity maybe I want if it's also huge and there are the FC receptors used to transport IgG across will set that don't bind IgM well how do we use that clinically to help us make diagnoses and children so a baby's born and mom for example was exposed to measles rubella or toxoplasmosis and you want to ask was the baby infected what tests do you routinely do in baby's pediatrician what tests do you do right so how what antibody do you look for in the baby can you look for IgM antibody because the baby is going to have the same identity the mother has but only if the baby has IgM is telling us the baby's been acutely infected in utero and is mounting its own IgM response because it can't be the mothers because IDM doesn't cross the placenta so that's how we use that clinically the flip side of it is if a baby is born to an HIV infected mother mom has lower than ygg specific antibodies against HIV because she's infected now the baby's born that may be baby may be infected or the beta may not be infected well looking at IG g and the baby be of help to you into turning where the baby is infected or not know Western law and why's that anything with IgM won't help but in the old days before PCR Barlow's what we would look for the baby potentially would be IgM against HIV to look potentially whether the baby had been infected so the fact that IgM does not cross the placenta we use that clinically to make diagnosis in intrauterine infections okay and the other point is is that as I mentioned before diffusion into extravascular sites a lot of infection occurs in extravascular sites IgM because it's so big doesn't get there very well so it's not effective it's great for blood-borne infections but it's not as good as infections in the tissues where ye TT gets into that extravascular site very efficiently because it's small its mobile agile and that's why it's critical in fighting infections that occur there whereas IgM is not as effective okay so now this is just again to review the fact that IgA and IgM of polymeric and that now by giving them 10 binding activities increases their ability and increased his functional activity so now how does class switching a curb well what you're seeing here is is that cytokines are secreted by the T helper cell and different cytokines now will determine what isotype the B cell will class switch into and this is one of those you know tables that give you a lot of information but the reality is is that for you is we still don't know a lot about what cytokines cost cause class switching the one cytokine that we really can take to the bank in humans in terms of class switching is aisle four I'll force absolutely required for class switching to IgE if you don't have file for you will not make IgE and in the developed role that's really irrelevant in terms of allergies because IgE tends to mediate allergies but in the less developed world IgE immunity is critical for parasitic infections so therefore having the ability to make out 400 discuss this and later lectures is critical because IgE is what binds to mast cells that's what gets triggered in the face of a parasitic infection to cause this global response that allows you to basically detach and spit out the parasite these other ones are less as less understood but modified for example this is actually route by a lawyer because in older literature you had seen the auto-5 actually caused iga production it turned out that it really doesn't cause class switching per se but it just somehow increases the production of IgA so you will frequently see this linkage of iga to aisle five but it's not that on five itself causes class switching like Florida's variety it's somehow all commands production Gavin interferon it's also in mice has been shown to induce igg3 and is g2 and again and tgf-beta has been shown to induce to be and iga advice again it's still a lot that we don't know about the many different cytokines that caught it but the concept is is that cytokines are the fine tuners that allow determine which isotype d b cell switches to now how does this cytokine do that and the way the cytokine does that is by triggering something called a sterile transcript and a sterile transcript is is not you know a transcript of a lecture that's been put in autoclave and sterilized but what it basically means it's an are natives been generated that doesn't in color protein RNA basically designed to encode proteins if it doesn't poke proteins it's just being done to open up the DNA that's called a sterile transcript so again to orient you to vbj sequence the initial transcript made by an IEP cell is going to be off the IgM and the IG D and through splicing it will even make IBM or IG d and that's why naive B cells Express under surface both IgM and IgD the exact function of IgG we don't know of but it does play a role and we'll discuss in the next lecture this afternoon on the illumination of autoimmune b-cells so now you're making this and everything is fine but if you now expose the B cell LPS again you'll make a lot a larger amount of antibody but it always will be IgM you can't class switch because what do you need to class switch what cells t-cells and so therefore now you have 98 B cell plus of the aisle for being made by the by the T cell and cd40 ligand interaction now what happens is that this side up on aisle 4 transcription to occur specifically on the gamma1 and the epsilon cost constant regions so it's making this short transcript of all of these of genes the contribution and the presence of tgf-beta is better making a stereo transcript off of the g1 and the epsilon it's going to make it a sterile transcript all of the off of the tubing or of the Alpha okay so now your sterile transcript what is the implication of making sterile transcripts the implication is is what this does is it now opens up the DNA double strands into a transcriptionally active site and now it opens up into what's called a transcription bubble and what the bubble is is basically is basically this region here that's opened up but what this now does is it makes this available for a ID and now a ID this mischievous enzyme comes and it deaminate s' the cytosine to uracil and now this can get cleaved out this can get form and now what do you need to remove into this area in order to fix things what kind of enzymes repair enzymes right so you like you call the repairman and now this is what now causes class switching to occur because in the absence in the absence of any kind of active transcripts just notice there's no way to get the process started however once you start making the mistakes now what you do is you're bringing these repair proteins however once these repair proteins get brought in and also the chromatin now is wide open now you have the ability for the constant regions that are downstream that have been waiting for this opening to push ahead of the IgM in line and now become the ones that get transcribed they have pushed their way forward loop out the IgM IgD or whatever and now the repair enzymes come cleave this process cleave this piece out and now what's adjacent to the bdj is whatever in class which in this case it would be the IG this would be the IgE and you'd be making IgE but again you can understand now why the cytokine is a clip play the preys plays a critical role in this process because what sterile what sterile transcript gets made determines now what can be the reparent i'm going to come to very close together so the sterile transcript specificity is what's it gives you the cytokine specificity for class switching so now you make you have these repair enzymes come and they loop it out and now you've class switch so now to look at it in the big picture so you have edj IgM and delta the initial transcript however stimulation of the sterile transcripts opens up this region depending upon the cytokine the AIT comes in makes the mutation now the repair enzymes come but what that happens over the whole region and allows the adjacent switch region similar to the ones in the video sequence to come push out the loop out in the area and now you can class switch in this case you know may become igg3 but now the transcript is be DJI create or the class which is giving you this case bdj and n iga but again the antigen specificity is exactly the same here as it was here just the isotype is swish so recognize exactly the same antigen but it recognizes it in a context of a different constant region that clear any questions ok so now who's discuss a little bit in terms of what what what what these isotypes can do so in addition to IgG as you will know another critical antibody is IgA and IgA is the important antibody in mucosal immunity as you know IgA is secreted in very large amounts in vehicles and secretions so what secretion that's very common in childbearing women after they've had a baby has high levels of IgA milk if very high levels of IgA why is that because now babies born the baby is at high risk for developing intestinal infections because the baby is eating everything you know getting swallowing everything and the baby's immune system is not yet well developed so maternal milk is given the maternal milk has IgA against all the pathogens that the mother has been exposed to and that now gets into the gut and functions as kind of like a immunological protection agent so any pathogens that come into the gut the baby has pre-existing IgA that's able to protect it that's why breastfeeding pres fed babies have much less infections than non breastfeed babies but and in terms of if you're in a normal gut what how does the IgA get from the lamina propria on the other side of the epithelium in the intestine to the area of the lumen where you want to have AI GA function because the the the epithelial cells have very very tight junctions to protect you from infection all the infectious agents are here you basically are sealed over here so again the same way if you want to protect your house what do you do you have high walls around it in order to protect it well that's all well and good but if you want to have AI GA go from the V cells that are making it and if lambda appropriate of peyer's patches how can they get across these walls they be stuck so it turns out that is secretory to JJ and secretory component is linked to the IgA and this body's to apologize ET receptor and this now allows it actually to get transported inside of the little self and now it gets extruded out into the lumen and that's where IgA can be present in order to bind to any kind of activity that a present that it's specific for to protect you from infection so that's why I GA is unique in terms of antibodies in terms of its binding capacity to the secretory component to allow it to be transported across the epithelium IgG doesn't have to do that so it doesn't have that capacity IgE IgM doesn't have to do that and that's what IG a is the critical mucosal isotype because it has this capacity to be secreted across the epithelial barrier okay is that clear now FC receptors are also both incredibly interesting and incredibly complicated so the first thing when you see a slide like this is you basically say this is too much information I'm never going to know this I'm not going to bother looking at it and thinking about it right and and the answer is you really don't need to know all the information on the slide I just want to kind of convey the concepts behind this to let you know how the immune system has been generated to allow to do multiple different things with antibodies so if you have antibodies for example and you want so basically just to show you there are many different FC receptors and they won't have names now again in enologist are your friends okay so when you see FC what letter is that gamma so what isotype do you think this is going to bond IgG exactly if now you go here and you see this one what letter is that so what give you a lot of you think this is going to bind e IgE so again the two I sometimes that are bound and if you're going to be here what I have to type is that IgA and that's going to bind IgA also here with you it also binds IgM so again just just looking at the name of the FC receptor a lot of your it is because see tells you a finds and see this tells you what the isotype is and are tells you as a receptor and then you have like the numbers which are you know not helpful at all but this is telling you that they're different okay so someone asked you is FC gamma R to a the same as FC gamma bar - B - what would you answer no they're not and the way that they're different is really in terms of what cells express them and therefore what they particularly valid but the one point that you want to make though is that there are some population of FC receptors that Express on their surface a sequence pull item and these are actually inhibitory molecules in terms of signal transduction and I'll discuss this in terms of T so transduction but these actually divert off cells so I'm going to give you a blob would bind to these receptors it turns on the ITM and this actually allows in the vision of this particular cell whatever is expressing this so this is actually a novel regulatory system that allows antibodies to actually down regulate cell function so these FC receptors that lack the idea they have a positive that they induced killing they actually enhance the immunity but this allows for some reasons again you wouldn't have to have a braking system - stomping them in inhibiting and that's what these particular things do so whenever you see an i10 motif that immediately tells you that it has an inhibitory effect and different subpopulations expressed different FC receptors so as I'll show you in a minute what cells are going to be having a lot of IgE bound to them that mediate allergies mass it cells so how did that cell specifically find eyes you eat what FC receptor would you want them to express a FC Epsilon receptor and indent those cell then express the MassHealth use of those baseballs are also associated with allergic responses behind GE on the other hand if you want to have a cell that can utilize a immunoglobulin props increase phagocytosis you want them to express a FC receptor that clients IgG with very high affinity as I'll show you in a minute so again this is best facility of this allows you therefore to dramatically increase the ability of antibodies to modulate cellular function so in addition for in addition to antibody binding to antigen and doing stuff with the antigen killing habit is the antibodies also have the ability to interface with cellular activities and modulate them so it dramatically increases the power of the antibody response okay is that clear so now to show you a specific example so here we have here we have a bacterium and this bacteria if you want to eliminate it you want to get its phagocytosed well yesterday I had mentioned that macropods is expressed like mannose receptors on their surface that are present in on bacteria that's a very low affinity interaction would you much rather have the high affinity of an immunoglobulin molecule binding to bacteria absolutely but how is that going to help in terms of it getting into the macrophage and the answer is these macro files are expressed FC receptors that have very high affinity for antibody and particularly it turns out that when the antibody is not bound to anything the FC receptor has a very low affinity for the immunoglobulin molecule but when the antibody binds to an antigen and changes the structure of the FC receptor to one that the of the FC to advise the FC receptor with very high affinity so now what happens is is that these great antibodies that you've generated for example against Hamas influenza using your conjugated vaccine now bind telemachus influenza very tightly but now what does facilitate is these Hamas was influenzas being taken up by macrophages phagocytosis and lemonade so in essence the by using having the FC receptor now the analyze can recruit phagocytic cells to help eliminate pathogens okay is that clear IgM doesn't bind well this FC receptor so if you didn't have that conjugated vaccine and we're only making IgM you couldn't absolutely lack this critical limb of the immune system so again that explains why that conjugate vaccine making IgG dramatically protects you from infection against bacteria like a mafioso influenza work so well NK cells play a critical role yesterday I discussed the role that they play in terms of cell to down regulate class 1 expression it turns out that that's critical for cancer cells but that may also be critical for HIV infection and they actually use antibodies to facilitate their function so in addition to killer cells that lack MHC expression class 1 they also can use antibodies and FC receptors so here you have a target cell and let's say this part of our cells infected with HIV so now there are gp120 on the surface for example getting ready to bud and release HIV but you have antibodies that bind to it the NK cells have FC receptors and see a lot of the antibody bound to the cell and they know normally this antibody going to be bound to the cell when antibody normally be bound to a surface your own cells know because your anybody shouldn't recognize your cellular antigens so the NK cell says whoa if I see antibody bound to a cell that's bad that means it's something wrong with this cell they're going to investigate is FC receptors bind to the FC portion because the FC portion is stuck out that's because the antigen binding region is bound to the cell and now it releases its granules and kills the cells so again FC receptor antibody harder to use now since it sells for killing and this is a way again very effectively killing either two or cells that have to or images or infect itself that have antigens from the pathogen expressed on the surface and in addition this plays a critical role in allergies because as you mentioned before we have the FC epsilon receptor it binds circulating IgE and now in this case this is like a antigen from a room for example it cross-links the IgE on the surface of the mast cell this is specifically whatever antigen the IG recognizes so let's say it's to a helmet so now it releases all the mediators and now for example in the intestine you got swelling epithelial cells gets lobbed off and whatever that parasitic worm is grabbing on to to keep it in the intestine gets lopped through and the amount of fluid is generated the diarrhea and disgusting stuff it roses out the gut and therefore the worms get extruded into the periphery and that helps protect you from infection but again this is because the SC Absalon receptor expressed in the mast cells bind to the IgE which gives it the specificity which tells the mast cell only to granulate when whatever antigen it is that this IgE recognizes gets crossed length and therefore is environment okay is that clear okay so therefore in this lecture basically I think we pretty much now finished everything almost we still have to sew these two lectures I think people are hope clear and now how do we generate diversity with the antibody how to generate specificity again like I said this guy knew all talk about watery antibodies how immobilize immune system now in this particular lecture I focused on having an item type switching now if you didn't have a ID so let's say you had a mutation and you lack a ID what do you think your immune system would look like what isotypes would you lack everything except on GM exactly because you can class twitch and again patients that have mutations in genes are incredibly instructive because they teach us what our proteins are important in this process we can do a lot of experiments in a test-tube we can do experiments in mice but at the end of the day if it's not the case in humans is not instructive and it turns out there are individuals that lack a ID and those individuals only have I GM it's called hyper IgM syndrome I'll discuss that in the lecture on immune deficiencies but again it underlines that this is the critical protein in this process and how do we increase the affinity of the antibody again through affinity maturation and again we started talking about memory because you make these these cells that are classic ways undergoing somatic mutation and then they are long-lived and again the specific details of discussed in a later lecture thank you very much for your attention well get nutrition from lunch to get more energy to come back in the afternoon thanks a lot
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Medical_Lectures
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Hemostasis_Lesson_5_Antiplatelet_Meds_Part_1_of_2.txt
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[Music] this is the fifth video in this series on hemostasis and today I'll be discussing antiplatelet medications because there's a lot to cover in this topic I'll be dividing it into two parts this one will be an introduction and we'll discuss aspirin which is a historically one of the most important medications of all time part two will cover Every Other antiplatelet Drug the learning objectives are very straightforward to describe the structure mechanism pharmacology indications and side effects of the antiplatelet drugs including aspirin the P2 y12 Inhibitors such as cigil 2b3a Inhibitors Di peritol and costasol here's a diagram we first saw in the second video on the normal physiology of platelets it shows how vascular injury leads to exposed collagen triggering platelet adhesion and activation which then leads to a process of amplification of the activation through platelet release of fibrinogen Von willbrand Factor ADP and thromboxane A2 along with other factors such as thrombin helping to mediate the effects of each of these compounds are specific receptors such as the P2 y12 receptor for ADP and the 2b3a receptor which binds to fibrinogen and Von willbrand factor to support platelet aggregation and the eventual formation of a platelet plug different medications interfere with this complex process at different points for example the most commonly prescribed antiplatelet drug aspirin inhibits the production of thromboxane A2 the p2y 12 Inhibitors clil pril Codine and and tagore all predictably block the P2 y12 receptor the GP 2b3a Inhibitors over here include three drugs eptifibatide Tyran and apomab the drugs dimol and costasol are both phosphodiesterase Inhibitors which have somewhat more complex and obscure mechanisms which partially involve modulating levels of cm and cgmp finally although this new class of medication is not yet widely used and thus I won't be covering it in this video Vora paxar is an antagonist of the P1 receptor the first drug to discuss is aspirin aspirin also known as acetyl salicylic acid which is occasionally abbreviated ASA is one of the world's first effective medications and remains one of its most common because of its widespread use and diverse indications I'll spend a bit more time on this drug compared to the others the history of aspirin begins about 3,000 years ago in ancient Egypt when preparations of willow bark were used to relieve pain and fever the first remotely formal description of the use of aspirin was 400 years before the Common Era when hypocrates described the use of various parts of the willow tree for treating pain in the early 19th century within the span of a few years numerous organic chemists isolated compounds called salicin and salicylic acid from the Willow and the use of these to treat pain and fever grew modestly however their use was limited by significant side effects and an awful taste then at the turn of the 20th century several chemists synthesized a similar compound called aceto salicylic acid which had fewer side effects popularity rose significantly during the Spanish Flu pandemic of 1918 which Co ided with the expiration of Bear's original patent at the time aspirin was essentially the only medication known to reliably reduce fever now the mechanism of aspirin wasn't actually discovered until the 1950s before which it was believed that aspirin's pain relieving effects occurred directly within the central nervous system it was worked on a group of biological compounds called eoso that led to aspirin's Discovery in the second video in this series we looked looked at this diagram which outlines the synthesis and regulation of prostaglandins and thromboxane A2 to remind you thromboxane A2 which helps to activate platelets is produced from arachadonic Acid by three steps the first two are catalyzed by different sites on the same enzyme which can be either cyc oxygenase 1 or cyc oxygenase 2 both of which are occasionally referred to instead by the name pgh2 synthes 1 and two or the name prostaglandin endoperoxide synthes 1 and two although less common use of these names highlight the fact that the cyc oxygenase activity is only responsible for the first step the conversion of arachidonic acid into pgg2 while the enzyme's peroxidase activity is responsible for conversion of pgg2 into pgh2 the general mechanism of aspirin is that it irreversibly acetates a Serene residue in the cyc oxygenase active site thus ultimately inhibiting production of pgh2 because the activity of aspirin on the Searing residue in question varies between the Cox one and the Cox 2 ISO enzymes the consequences of this inhibition depend upon the aspirin dose at low doses of 75 to 100 mg per day aspirin acts primarily on cyc oxygenase 1 which inhibits plate the generation of thromboxane A2 the consequence is a modest generalized anti-thrombotic effect at higher doses such as above 325 MGR aspirin acts on both cyc oxygenase 1 and two leading to an anti-inflammatory effect this is the reason for the significant dose dependence of aspirin's clinical effects so what are aspirin's indications there are many I'll talk primarily about six the clearest indication for aspirin is acute coronary syndrome numerous trials show that patients given aspirin immediately after presenting with either an st elevation Mi non-st elevation Mi or unstable angena have Improvement in various markers of cardiovascular morbidity and mortality the recommended dose in the situation is 162 to 325 milligrams daily starting immediately the first dose should be crushed or chewed in order to maximize speed of absorption what is commonly done in the US is to give four chewable 81 MGR tablets since they are believed to be faster absorbed than a single 325 MGR tablet the next indication is acute ischemic stroke a meta analysis of 12 randomized trials found that aspirin of dosages 160 to 300 milligrams daily started within 48 Hours of symptom onset resulted in an absolute risk reduction of death and dependency a near negligible increased risk of intracranial hemorrhage as that perhaps implied in practice initiation of Aspen following stroke is generally not felt to be as time critical as it is in acute coronary syndrome though I'm honestly not sure where that's the case because they would seem to be somewhat analogous situations next is the secondary prevention of cardiovascular disease which is more or less an extension of the first two indications a meta analysis looking at a variety of end points in patients with known cardiovascular disease already who are randomized to either aspirin or Placebo found that Aspen reduced rates of Mi rates of stroke and all cause mortality comparison of appropriate doses indicate that low doses of 75 or 81 milligrams daily is likely as effective as higher ones and associated with fewer side effects an obvious question that might be raised by this is if acute Mis and acute Strokes are treated with 162 to 325 milligram daily but for secondary prevention patients can be treated with just 75 or 81 milligrams daily at what point after a cardiovascular event should the dose be lowered there seems to be some variability in practice I personally lowered it down to 81 Mig when the patient is discharged from the hospital but there are others who continue a higher dose as an outpatient for some arbitrary length of time aspirin is also used for the primary prevention of cardiovascular disease particularly in older individuals with cardiovascular risk factors a huge Med analysis looked at this question in 2009 and found that aspirin reduced the composite end point of any serious vascular event but the effect was largely due to preventing nonfatal Mis there was no statistically significant difference in mortality and even data on primary prevention of cardiovascular disease in diabetics is conflicting most guidelines currently suggest calculating the 10-year risk of cardiovascular disease based on statistical models which can be easily looked up on the internet and balancing it against the risk of side effects on the whole I'd say that the use of aspirin for primary prevention of cardiovascular disease is not nearly as supported as it was a decade ago for example when I was in residency it was a given that a 60-year-old man with hypertension or diabetes but no known cardiovascular disease should be on a daily aspirin but now in 2015 that's actually quite debatable how about using aspirin to prevent stroke in patients with atrial fibrillation it's occasionally used in place of anti-coagulation in patients with nonvalvular apib when either embolic risk is believed to be small that is the patient has a Chad's vasque score of either zero or one the patient has a cont indication to anti-coagulation or the patient declines anti-coagulant therapy if you're not familiar with the Chad's vas score it's a simple system for predicting the probability of developing an arterial clot in patients with a fib I'll be discussing it more in the next video on anti-coagulation a meta analysis has found that compared to Placebo aspirin conve a 0.8% absolute risk reduction in the primary prevention of Strokes in patients with non non valvular aib and a 2.5% absolute risk reduction in the secondary prevention of stroke although those numbers may sound pretty small considering that aspirin is very cheap well tolerated at low Doses and the consequences of a stroke potentially so profound that both the risk benefit and cost benefit ratios are still quite favorable however because most patients with aib have Chad vas scores above one and have no strong contra indications to anti-coagulation anti-coagulation is still recommended over aspirin for most patients the final aspirin indication I'll discuss is the possible prevention of cancer there is some emerging data that suggests long-term use May reduce the risk of developing numerous types of cancer and may lower the risk of cancer related mortality the conditions which have been most studied are colonic adenomas and coloral cancer although most studies including prospective randomized control trials have found some sort of benefit significant heterogeneity in study design has resulted in no clear consensus as to which specific patients are likely to benefit and what the optimal aspirin dose should be due to these uncertainties and the small but non- negligable risk of GI Hemorrhage from aspirin both the American Cancer Society and the United States preventative Services Task Force currently recommend against the routine use of aspirin for the purposes of preventing colal cancer so that was a lot of information regarding the indications for aspirin let me summarize it in a quick table strong indications include acute coronary syndrome acute stroke the secondary prevention of cardiovascular disease and nonvalvular atrial fibrillation but only in those patients whose Chad vas score is z or 1 or those in whom anti-coagulation is contraindicated modest indications for aspirin use include primary prevention of cardiovascular disease but only in those patients with higher than typical 10year risk of developing such and aspirin is not generally indicated for cancer prevention now some of you might wonder why I have not mentioned the two indications for which aspirin originally became popular 100 years ago that is what about taking aspirin for pain relief or to reduce Fe fever well in the 21st century aspirin is not best used for either of these indications while it can be effective the non-steroidal anti-inflammatory agents such as ibuprofen and Naproxin are as effective and have fewer side effects in Aspirin when aspirin is used at the high doses necessary for its anti-inflammatory and antipyretic effects to be evident so what are the side effects of aspirin at normal therapeutic doses including the relatively high doses for pain relief the primary adverse effect is upper GI bleeding there are two mechanisms of this the major mechanism is systemic Cox inhibition which is because the healthy gastric and duino mucosa constitutively use Cox one to produce its mucosal protective prostaglandins the minor mechanism of aspirin induced GI bleeding is local direct toxicity on the GI mucosa because the major mechanism is a systemic one it should come as little surprise that so-called enta coded or buffered aspirin has no effect on clinically relevant GI bleeding though there is anecdotal evidence these formulations may be associated with a decrease in subjective GI upset risk factors for aspirin induced GI bleeding include a history of peptic ulcer disease or gastritis Advanced age concurrent use of steroids such as prazone Neds other antiplatelet drugs or anti-coagulants and concurrent use of alcohol aspirin is also associated with increased risk of intracranial hemorrhage but the absolute risk is still small enough that this is typically a relatively minor concern in addition to the bleeding side effects acute aspirin overdose is associated with a relatively distinct pattern of toxicity symptoms and signs of a mild overdose include nausea vomiting diarrhea idus and vertigo symptoms and signs of a severe overdose include fever lactic acidosis a respiratory alkalosis non-cardiogenic pulmonary edema confusion and coma the danger from aspirin overdose is primarily related to the amount ingested relative to body weight assuming it's all ingested at approximately the same time the treatment for aspirin overdose is largely supportive care but also includes activated charcoal if the ingestion was within the last 2 hours alkalinization of the urine with sodium bicarbonate infusion to promote excretion of salicylic acid and supplemental glucose to prevent cerebral hypoglycemia which can be present even in the setting of a normal serum glucose so that was a lot about aspirin and it brings us to the end of part one
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Medical_Lectures
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Immunology_Lecture_MiniCourse_7_of_14_Signaling_Through_Lymphocyte_Receptors.txt
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but it's also one of the most complicated aspects of Immunology which is signal transduction and a lot of people see signal transduction they see all these different molecules involved and they kind of like get all this Panic of seeing too many molecules and what they do so what I really try to do is take you through this process at a very step-by-step conceptual approach and then at the end fill in the details of how it works so the questions to consider is basically after recognition of its cognate MHC peptide how does the t-c cell receptor activate immune response genes the t- cell has to turn on genes within the te- cell in order to have its effective functions how does that process occur what's involved in that process what are the structural motifs that are used by signal transduction molecules that allow specific interaction and activation of of eector proteins and this is the most important question the nucleus where the chromosomes and genetic information are is inside this nuclear envelope it's inside this bubble outside all sorts of stuff is going on infections terrible things and yet inside this nucleus bubble the genes may have no idea what's going on outside how does that occur because at the end of the day the genes have to be be turned on to make whatever needs to be made in response to the outside environment yet how does the N the genes know what's going on in the outside environment that's really the critical question that uh is involved in terms of signal transduction process so in know before we kind of get into t- cell receptors I first want to provide some general signal transduction Concepts and again a lot of you probably are familiar with this but again as I've learned there's no no one ever uh is against reviewing some of these Concepts the first concept to appreciate these are the the t- cell specific response and this we're going to expand upon more in lectures tomorrow is that it's a multiple signal process so the paradigmatic class 2 in this case MHC peptide t- cell receptor CD4 interaction the same thing would be the case for cd8 interaction that provides the initial antigen specific signal however that signal is not enough to turn on the t- cell the t- cell needs in addition a second signal and I'll be discussing a little bit maybe some detail today and a lot more tomorrow exactly what's involved in that second signal process but the question that I want to address that I'll answer today is how do you integrate two signals and what happens is if you get one signal nothing happens and in fact the t- cell gets turned off if you get this second signal in the absence of the antigen specific signal nothing happens only when the t- cell gets two signals is a T Cell activated to make the appropriate response well why is that how could two signals integrate at the genetic level to allow an immune response to be generated and conceptually you know what is the reason for having a two signal system so the example that I give in the states is which may be applicable here is a safety deposit box so did they did anyone here ever use a safety deposit box anybody you don't have safety deposit boxes here well in America we're very afraid of crime you know so so so people take all their valuable stuff and bring them to the bank in this giant bank vault and they rent a box and they're given a key to use for their specific box so nobody else can get into the box however for some reason the bank doesn't trust you they think in the middle of the night you'll break in and try to get to your box without them knowing it so they also have a key so in order to get into a safety deposit box you have to insert your key into your specific box and the bank manager puts their specific key into their Keyhole and when both of you turn it at the same time then you could take your box out um so that's how it it works the reason is clearly the bank can't get into the box without you you can't get into it without the bank and that's the the protection mechanism the bank utilizes Sals do the same thing by having this kind of two signal process you have this specific signal that's your key to your specific box that's unique from person to person and this signal is unique for MHC peptide to TCR but then there's this generic second signal which represents the key the bank officer uses that's the same for every box but unless that's turned no one could get their safety deposit box so this combination of a specific sequence and this generic sequence allows you now to really give you an extra level of control in terms of turning the t- cell on and again to tomorrow I'll explain why actually turning off a t- cell when it only sees one signal actually can be a good thing in terms of eliminating self-reactive te- cells okay so just keep in mind two signals how can we explain this at the molecular level now there are some cellular receptors so now this is extra membrane this is inside the cell this is a receptor these uh is in this particular case would be a homodimer because these proteins are exactly the same when they're separate their kinas domains cannot interact with each other and phosphorate each other and again the example I always think about is um anybody have kids here more than one kid okay you how many kids you have two how old are they four and seven four and seven do they have a fight with each other yes they do okay when they fight with each other how do you resolve the fights generally shut both of them but when your wife's around you have to do the right thing what do you do you know act like a parent not sure what the right answer is um uh well you find out what the problem is I guess oh but I mean like let's say you have to you you have to work and you want them to be quiet very very quickly how do you stop the fights you separate them right you know you go in that room you go in that room and then we'll deal with this later when when your when your mother comes home right what she does yeah oh yeah till you come home and Daddy will take care of you um so so the same way with receptor molecules if you don't want any transduction to occur you separate those molecules they're not doing anything however if now let's say you want them to interact what do you do you bring them together and what's going to bring them together is whatever the Lian that they bind to now the Lian comes it brings the two receptor chains together because it's a now forms a a a dier but now these kyate domains when they're separate which couldn't interact with each other now can phosphorate the substrate which is present in the opposite chain and therefore start inducing a signal transduction pathway but in this case the kyes domain is an integral part of the receptor itself okay is that so that's a very simple straightforward transduction pathway however other cellular receptors are transduction molecule challenged the politically correct way of saying that they lack kyes domains and even though the kise domain is not part of the receptor protein itself self they have associated with them physically A cise protein that is now the same concept when they're separate no interaction no phosphorilation no signal transduction when now the Lian comes brings these two molecules together the kindas Protein that's associated with The receptors now interact with each other phosphorate each other and now cause signal transduction so those that concept of basically bringing two chains together is one that's utilized by many signal transduction Pathways and either the receptor has its own integral kinas or it has to have a kinas molecule associated with it uh through some other interaction okay now why do we have uh phosphorilation events and the other major Concept in signal transduction is you need to amplify the signal so if you have one receptor for example or two receptors or three receptors and you want to turn on hundreds of genes that's going be very very difficult to do unless you can mobilize a Amplified signal and what the kinases can do is kinas could amplify it dramatically because if you turn on one kinas molecule then that now could phosphorate say 100 substrates if those substrates themselves are KY molecules that by phosphor you're turning on now you have 100 new molecules that can fate 100 new molecules if their kindness has turn them on and that way a single molecule for example can turn on thousands and thousands of different proteins down the road and now you have thousands of proteins in this case a nuclear transcription factor that could bind to genes and turn them on and that's how why kinases are so potent in terms of amplifying signals and a recurring Mo signal transduction Pathways okay is that clear now another signal approach besides phosphorilation events that's utilized in transduction Pathways is calcium release so you could basically have a receptor it opens up a calcium channel calcium stores that are sequestered in a certain intracellular location now get released and these calcium ions themselves canbin two proteins and these proteins could alter the confirmational change in this case of calmodulin and now again uh cause it to uh activate another Cascade and again cause amplification to occur now why why are kinases so potent in terms of their ability to transduce a signal and that's because kinases can change the physical structure of a protein so for example if I would be an enzyme and my active sight would be my chest if my three-dimensional structure was one that my arms were wrapped around my chest would I be an active enzyme no however if like my shoulder got phosphorilated and that would change my three-dimensional structure my arms would come out and now my active sight my chest is revealed and now I become an active kise if I wanted to turn my kise activity off I would take the the phosphorus off my shoulder using a phosphotase and now my arms would get back to where they were and we get inactivated and the power of phosphorilation is the ability to rapidly change structures of proteins opening and closing activate active areas and thereby turning them on and turning them off and it's a lot more efficient than going to the nucleus telling a gene to make a new Gene make a new protein have it transported up to the membrane this allows instantaneous changes to occur within seconds that of of when the signal is transmitted it also allows you to turn it off very rapidly well as well so that fast capacity of on and off makes and the capacity for amplification makes kind of is one of the uh transduction molecules of choice for singal transduction okay any questions now in addition to being able to be turned off on as I mentioned you also want to be able to turn off signal transduction Pathways and the way as I had mentioned you would turn off a a Kate activated pathway is you def phosphorate it you bring in a phosphotase you remove the phosphate residues and now what turned off in this case an ship SHP is a well-known phosphotase involved in t- cell signal transduction it comes on it's like sucks off those those phosphorated mues now it becomes an inactivated uh signal transduction molecule the other way of doing it is a little bit more dramatic which is basically just chop the thing up so this is a very humane way of turning off signal transduction no proteins are harmed in this process but sometimes the body just says let's just destroy it the ubiquinated it and now this gets processed through the proteosome and again very dramatic way but you equivalently turn off the signal transduction pathway and alternatively it could also be be destroyed in a losone but for most signal transduction Pathways that are rapidly being used on and off again frequently you clearly would want to use a def phosphorilation approach now in addition to uh the signal transduction molecule itself you also need to have a way of recruiting molecules near the kinases because you need to bring them into an area where they could be acted upon by the kise so in this example here we have a membrane Associated protein kise and down here in the blue are the substrates well if these substrates are down here they really can't be phosphorated by the kise it's too far apart well how would you now allow these kinases to act on the substrate you'd want to have what's called an adapter protein the Adaptive protein has in this case these are phosphorated regions here that the that are going to specifically bind to these substrates it now recruits up the sub the substrates near the membrane and now that they're up in the membrane they're adjacent to the chinise and now the chinise could phosphorate them so these adapted proteins again are a mechanism by which you can recruit substrates near kinases to allow them to be phosphorated and then undergo whatever changes in the function they undergo okay is that clear now this is a um a cartoon of one of the major kinases that's utilized during t- cell signal transduction and this is called a sar family kise and this is first described for obviously SAR but there's a whole family of kinases that have a similar motifs as this paradigmatic sarin and the first thing to realize is that all of these have this unique region right here and this unique region provides specificity in terms of what membrane protein these will bind to so as I had said before if you have a receptor that it itself doesn't have its own kinas domain it has to recruit a kinas to be bound to it this domain provides the ability of the SAR homology kise to bind to a particular protein so for example the one used for Signal transduction has a unique sequence that allows it to bind either to CD4 or or to cd8 or to the T the t- cell receptor that's what gives the specificity in terms of what receptor it binds to in addition it has another sequence called sh3 and sh3 functions as an adapter protein to allow it to bind to other proteins and The Motif that the sh3 domain specifically binds to are to Pi to Proline Rich motifs so if you have a protein it has a lot of prolin then the sh3 domain will bind to that the one that's more commonly used in in in immune signal transduction is the sh2 domain because what makes the sh2 domain so important is it it's specifically binds to phosphorilated tyrosines and the reason that that's important is if you have a tyrosine on a molecule it can't bind this chinise but now a kise phosphorites it now that tyrosine is phosphorilated it's able to bind to this sarc family kise whereas previously it was unable to if you want to stop that molecule from binding to this kisee then you def phosphorate the tyrosine on that protein and it'll come off so this sh2 will only B to the tyrosine if it's phosphorilated okay is that clear and finally you have the actual kyes of the SAR homology kise the sh1 and Sh stands for SAR homology so this is present in all sarc families SAR mology three SAR 2 and SAR mology 1 now this kinas is very interesting because it has the ability to be both positively regulated and negatively regulated as I'll show you a little bit later but the bottom line is that it has within it a tyrosine so here's the kise domain but it also has a tyrosine and when this tyrosine is phosphorilated this allows this kise to be activated but in addition outside of the saramy kise is another tyrosine and this tyrosine when it's phosphorated it turns off the kise and this again allows this C to be very tightly regulated by either being phosphorated or not being popor related and again I'll show you in a little bit in detail in a subsequent slide exactly how this works for the uh immune mediated regulation okay is that clear and what's great about this kind of kise is not only does it have its own activity but it also has ways of binding to other proteins it could specifically bind to some proteins that it recognizes but it also can globally bind to a wide range of proteins as long as that protein has a phosphorilated tyrosine or that protein has a ProLine Rich Motif okay so it has a tremendous amount it can be rather um promiscuous in terms of its binding with a very specific Motif now another Motif that's used by a signal transduction molecules is you know you have to think about signal transduction is is getting a lot of proteins together because when a lot of proteins are together they can interact with each other and let's if we think about our day-to-day activities you know what's the most common place that people go to after work to get together a bar exactly and that's nothing more than uh a scaffolding protein and sure enough this actually resembles what a bar looks like and in fact what happens is is that when you in the absence of phosphorilation events this may be be a be a bar that has absolutely no alcohol so it's a bar before 5:00 and then there's no one sitting here but now there's opening time this gets phosphorated and now you basically have sites that allows a large number of proteins to bind to this specific scaffold protein and now sitting together they can all interact with each other now think about this if this for example were tiine residues when they're un phosphorated sh2 domains don't bind to it however if you now phosphorate a bunch of tyrosine residues in this particular protein now you can have different proteins with different structures different activities all of which share the sh2 domain all binding to these phosphor layer tyrosines if you want them to clear the bar and get rid of these cells what would you do def phosphorate this protein and then they come right off okay so again a rapid way of causing aggregation of proteins that you want to have interact with each other okay so that's just a background of global signal transduction processes so what I want to now kind of give you a background is just the overview of signal transduction I'll go through it rather quickly but then now I'm going to I'll return to it step by step to go into more detail but the reason I'm doing this is just to provide you with the broad overview before we go into the details I think it's important to get the big picture before you now hone down and focus into each of the details so basically to orient you this is the this is is the antigen presenting cell MHC plus peptide t- cell receptor in this case CD4 CDA binding to the MHC molecule that's the beginning of the process it turns out that the CD4 molecule and the cd8 molecule both of them have lck is a sar homology kinase that's associated with it when the t- Cel receptor MHC molecule comes into together it recruits the CD4 protein the CD4 protein or CDA protein now brings l c with it and the lck is able to bind sequences in the in the other portions of the the t- cell receptor I'll discuss in more detail this will phosphorate Tire scenes allow recruitment of another kise zap 70 through sh2 domains this will th Target adapt the proteins activate protein k phosph a uh ultimately protein kise diog glycerol and now these will activate nuclear transcription fact factors in this case n fat NF Capa AP1 for example and then ultimately these transcription factors will get into the nucleus and bind to Target regions on genes and specifically turn on genes so this is basically in a in an overview of how signal transduction at the membrane gets transported across the cytoplasm across the nuclear membrane through these transduction molecules and to ultimately turn on specific genes and specific transcription factors will turn on different genes which gives gives you a high level of specificity okay is that just a just a big picture so now we'll start doing it now step by step so the first thing to look at is that the t- cell receptor is not just the alpha and beta chain the t- cell receptor is a rather complicated large structure that includes multiple other chains in this case what you see is there there's a Delta Epsilon heterodimer and there's a gamma Epsilon dimer and again uh an important thing to appreciate in Immunology is that a lot of multi-chain receptors are given Alpha Beta Gamma Delta as a name but an alpha beta in a MHC Class 2 molecule are totally different proteins from the alphab beta and the t- cell receptor they're just given the same Greek letters but it's a completely different protein so don't get confused when you see Alpha Beta this or Alpha Beta that so the alpha beta t- cell receptor chain is completely different from Alpha Beta MHC molecule just given the same Greek letters within that receptor group and Epsilon has nothing to do with IG just has the same Greek letter okay now if you if you look at this let's look at the t- cell receptor itself this is outside the membrane this is inside the membrane is there any significant intracytoplasmic protein for the t- cell receptor itself what do people think look at the picture is there a lot of pro intracellular protein for the t- cell receptor no right that clear I mean right should be very obvious right okay what is that telling you that's telling you that the t- cell receptor Alpha Beta chains really can't have much to do with signal transduction because if there's nothing sticking into the cytoplasm there's nothing to recruit any kind of kise domains or anything that's going to be doing signal transduction so the alpha beta chains alone are solely going to be involved in recognizing the peptide plus MHC they themselves will not do any signal transduction however that's why they have to be associated with other molecules that are have domains that interact with it in this case the Epsilon Delta the gamma Epsilon and most importantly The Zeta chain if you look at this Zeta chain it's almost all cytoplasmic and not and almost no extracellular and in fact is this Zeta chain turns out to be the most important signal transduction Port of part of this complex and it also has is associated with the alpha beta and this and now if you look at this each one of these has this yellow has yellow domains and these yellow domains are called itams for uh immunologic tyrosine asso activating motifs if you recall when I was talking about FC re ctors I mentioned itims IIM that's inhibitory motifs this is an activating Motif and as the name implies it turns things on so we'll discuss this in a little bit more detail but these are uh tyrosines that when they get phosphorilated they permit the signal to get transduced so you basically have uh we we'll we'll discuss in a little bit more detail you basically have 10 of these and these are the critical location for t- cell transduction to occur okay any questions so therefore the first step in Signal transduction process is bringing the kise near the substrate and the way that this happens for the te- cells is taking advantage of the MHC restriction of CD4 Class 2 cd8 class one because as you recall the CD4 molecule has a sequence in the MHC molecule that it could bind to and when that happens the TCR binds to the MHC molecule but now the CD4 gets dragged alongside of it but when it gets dragged alongside of it it brings with it the saramy Chas that's associated with it in this case lck so now lck which before activation before antigen was seen was distal from these mores now it's broad adjacent to it and when the kindness is brought near these itams these tyrosine activation motives it could phosphorate these itams and activate them once these itams are phosphorated their Tire scenes what type of adapter molecule can it recruit phosphorated tyrosines bind to well sh2 domains and again the way of remembering it is tyrosine and two start with the letter t so sh2 and in fact this is a Kate in this case it's going to be zap 70 but it has on it a sh2 domain that now allows it to specifically be recruited only when these are phosphorated if they're not phosphorated as in here it's not going to interact so again the same concept phosphor tyrosines allow in this case sh2 domains now to bind to it recruit other KES to start the process but everything is started because the CD4 is recruited to the t- cell receptor complex by virtue of CD4 binding to MHC Class 2 or CDA binding to MHC class 1 and bringing with it the lck sarcy kise is that clear does that make sense so this is the saramy kise and again for for immunological signaling lck binds to CD4 and cd8 Finn is one that I'm not going to discuss much because we know a lot more about lck but it seems seems to bind to non phosphorated itams and just turn is is is recruited to the to the signal transduction location sh3 as I'll show you the domains are inactive when the C Terminus is phosphorated and it's activated by dfor sporulation by phosphatases such as cd45 so now to show it in a pictorial fashion this has been rotated 90 Dees so instead of being this way the molecule now is this way but let's walk through the molecule you have the inque region in the case of lck binding CD4 cd8 you have the sarc 3 homology domain going to bind to Proline rich areas the sh2 domain which can be binding to phosphorated tyrosines and now you have the kinas domain within this kinas domain you have the activating tyrosine residue and the inhibitory tyrosine residue when this inhibitory tyrosine residue is is phosphorilated it now becomes a phosphorilated tyrosine what can phosphorated tyrosines bind to what M sh2 guess what what do we have right over here an sh2 domain so now what's going to happen is this gets phosphorated it literally bends itself into like a molecular pretzel to bind on to itself but in so doing what is it covering up it's covering up the KY domain so it can't work and it's also binding to The sh2 Domain so this thing won't bind to any phosphorated tyrosine so in essence it's basically turned off this protein and made it relatively not sticky okay so but now if you want to activate this protein you need to do two things the first thing you need to do is you need to Def phosphorate this end tyrosine once this gets phosphorated it will no longer stick to the sh2 domain it'll flip open revealing this activating residue and this activating residue can get phosphorated by another lck or another adjacent kinase and now this gets activated def phosphorate this phosphorate this and you turn it back off again okay is that clear and it turns out that for the immune response the critical molecule that initiates this process is the cd45 molecule the cd45 molecule many of you are familiar with is the ubiquitous uh protein present on sites but it has a phosphotase domain and when signal transduction occurs the cd45 molecule gets recruited into the local area it def phosphor relates the inhibitory Tyrene of the lck and fin enabling them to be phosphor the activating Motif and now this was what starts the phosphorilation process going now the CD4 bringing the LC that's now been activated can now phosphorate the targets that are present in the t- cell receptor reor complex now as I said before phosphorilation uh of receptors allows them to recruit adaptive proteins so in this case you now have phosphorated tyrosines these phosphorated tyrosines can now bind to sh2 domains and bring in new molecules from the cytoplasm that previously were not adjacent to it and if these molecules have kise activity they could either phorate further residues in this particular molecule or recruit more molecules and phosphorate them through other recruitment adapter motifs and now you could have a situation where again you have new proteins all of which are being recruited into this complex of interacting proteins that are all being phosphorated and all being activated again starting this whole tyrosine process uh this KY process okay so what are the roles of these itams in signal transduction here are the itams present first of all phosphorilation of these itams allow them now to bind to sh2 domains and recruit any proteins that have this s sh2 region uh associated with it there are 10 itams per t- cell receptor complex in this case you have four four items with with the the heterodimers of Al Epsilon Delta and gamma Epsilon and there are three itams per Zeta chain which is six itams so the question there is why so many itams why do you need that many why wouldn't one be good enough and one first answer is well it's great for amplification because one itam you only recruit one group of proteins but by having 10 itams you probably recruit 10 times as many proteins more proteins more phosphorilation more rapid more rapid signal transduction different transduction molec bind to different itams so again you could have a greater mix of potential signal transduction molecules there and also it's been reported to be Associated to allow the TCR complex to associate with the acting cytoskeleton okay so now what's the next step in Signal transduction you're going to need to recruit another kise to that has another group of targets that it could activate and the critical the critical kindness that's going to be recruited is called zap 70 and again this is another demonstration of why immunologists are you friends because zap stands for Zeta Associated protein so what molecule in the t- cell receptor do you think zap is going to be associated with Zeta right that's that's that's pretty straight forward so so therefore it actually gives you the answer so here again MHC peptide t- cell receptor CD4 it's not interacting these are the itams here is lck and here is zap 70 it's nowhere near the t- cell receptor because it's its sh2 domain looks for phosphorated tyrosines none of these tyosin are phosphorilated so zap 70 won't associate with the uh with the itam so I I guess simplistically think of it this way um if somebody uh wants to have a very have a lot of friends that like them they have to have something to offer right so let the World Cup is coming here in June for example right anybody going to the want to go to the World Cup yeah you all want to go right so so let's say someone you knew had a 100 tickets to the World Cup would you all a sudden want to get to know them better right absolutely you know without without the real Cup tickets they were just somebody you kind of casually nodded to and said how you doing you know whatever and walking away but now your friends say hey if you need a real cup ticket this is the person to get to know next thing you know you're baking them brownies baking them cake giving them cookies etc etc the same way too in the absence of phosphorilation this is like having no World Cup tickets no reason for the zap 70 to hang out with it however once this gets phosphorated now it's a whole different story now the zap 70 says oh my gosh phosphorated tyrosine World Cup tickets now there's something I need to to get involved with and now the sh2 domain the Soccer homology Domain now says I'm going to bind to that phosphorate tyrosine and now it all gets uh Associates with it that's how something like phosphorilation could really change a protein from being one that's ignored by one protein to being one that's very actively bound to um by a protein but now zap 70 binds to the phosphorilated itam in this case the data chain and now it's in an environment where there are other kinases so what those other kinases can now do is phosphorate Regions on this zap 70 molecule and thereby activate the zap 70 so the way zap 70 is going to get activated it gets recruited into this High kyes domain location then it itself gets phosphorated and now zap 70 gets turned on once zap 70 gets turned on it now starts being very active it phosphorites all these scaffold proteins which before they got fated before they got ryal Cup tickets no proteins wanted to bind to it now they have rural Cup tickets they become incredibly popular they recruit in this case gads slip 76 L these just um ones you know again clearly the details aren't as critical and now but what it does now recruit is a very important protein called fosol lipac C and what makes fosol liac C so important is that it has the ability to cleave well soide it so this gets activated now and this now has the ability to cleave phosphoinositol B phosphate pip2 into the diog glycerol and ip3 and as many of you know these molecules are very important because these are the molecules that activate protein KY C so ip3 stimulates calcium e flux and Di oog glycerol will directly activate protein KY C so ip3 opens calcium channels and diog glycerol activates protein KY C and now protein KAC is going to help activate this Ras protein and this Ras protein now is going to start a whole critical signal transduction pathway that's going to end up in a signal being transduced into the nucleus and the way this happen is that g proteins again many of you are familiar with G proteins they have kinas domains but they're inactive whenever a small G protein is bound to it so this G protein um GDP is bound to it it's not active however if a guanine nucleotide exchange Factor comes and kicks out the GDP what it now allows is for GTP to bind to it and when GTP binds to it you can see the three-dimensional structure is changing and now this becomes an activated kinas if you basically def phosphorate the GTP to GDP you turn it off so this G proteins play critical role in signal transductions and protein KY C is going to be involved in turning on this Ras protein which now is going to transduce the signal into the into the nucleus okay any questions so far so now we're going to start leaving the nucleus and start Mo sorry leaving the membrane and start a signal transduction process that's going to end up in into the nucleus so again there are multiple Pathways that can transduce the signal from the cytoplasm into the nucleus and the other important concept to realize is that different transcription factors by virtue of their ability to bind to different regulatory regions can activate different genes the genes you get activate is really the summation of the transcription factors that get recruited into the nucleus so in the case of the RAS protein which is over here it triggers a s transduction pathway called the map kyes Cascade and this consists of a like a a relay like a um a relay team that transduces a signal that ultimately ends up into the nucleus and again again this is initially very confusing but when you kind of hone in on it it actually makes tremendous amount of sense so for example let's start with the yellow protein which is a map kinas if you would want to name a protein that phosphorites map kinas what would you call it map KY kinas because it phosphor relates it then you add the kise to it now if you'd want to give a name to the protein this is in red that phosphorites a map kise kinas what would you call it map kise kyes kyes it's that simple so so so here we go we still have this Ras that's been activated it's now going to activate map kise kise kyes each one of these there are different types of map k each one of these have different proteins all which has the same General Motif but have different ultimate Targets in this case for the t- cell receptor it's going to activate a map kyes kyes kise called rap raap now is going to phosphorate map kise kinas and activate that and then Mech this map kinas is going to phosphorate three entire scene on map kyes or irk and thereby activate that so this is the map kyes path again it's very straightforward you have AAS kise that's going to activate a map kise kise kise phosphor map K phosphorites map map Kus k then phosphorites a map Kus it's like this Bucket Brigade each one activates the one Downstream of it but again these have same concept but in they each one is different so raft there can be five or 10 different map kindness kindness KES each one of which has a different substrate in the case of t-o receptors the one that use is WRA to Mech to Earth okay is that clear any questions so it's a kind of a handoff now finally you have to get into the nucleus that's the end goal it's like you know in soccer you have to kick the ball into the goal so to in Signal transduction the nucleus and the genes is the equivalent of the goal you have to get the signal transduction molecule from the cytoplasm across the nuclear membrane into the nucleus and in the case of the map kyes it turns out that this the transduction molecule itself is irk when irk is not phosphorated it can't leave the cytoplasm it can't get into the nucleus when it gets phosphor when it's not when it gets phosphorated now it has the ability to get into the nucleus interact with other transcription factors and turn on genes so this is one mechanism by which signals get transduc from the from the cytoplasm from the membrane into the nucleus map kyes pathway okay is that clear now a second modality for transducing signals is using a calcium pathway using a molecule called nfat and basically as calcium gets uh secreted it activates calci urine calci urine phosphates calmodulin and calmodulin is a is a um phosphatase and fat nor normally is phosphorilated when nfat is phosphorilated it can't leave the cytoplasm it's basically trapped in the cytoplasm however when it gets def phosphorilated then then by by calcium urine calci Def phosphorites and fat now and fat without the phosph the phos the phosphorilation can now migrate into the nucleus look for whatever nuclear transcription Factor binding it binds to and turn on those genes so this is again how the N fat transcription pathway Works in contrast another transcription pathway which many of you are familiar with is the NF Campa B pathway the NF Campa B is regulated through a slightly different way you have diog glycerol activates protein KY where phosphorites another protein called Cara which is actually a pretty cool name and then this phosphates other proteins but the end result is is that it activates this ik KY complex and this ik KY complex has the ability to to phosphorate ikb now this is a complex of nfca and ikb ikb is a protein that's associated with NF CA B and again if you'd want to be simplistic um everybody at one po Point had a friend when they were growing up who was really nerdy and a real loser right and you knew that as long as you were walking with that person and you had to bring that person with you wherever you went you would never be able to have a social life so what you really would want to do would be to kind of ditch that person right because you knew once you ditched that person now everything changed now you'd be able to be you know coolest person on the Block it turned out you were wrong you the nerd but that's another story but the NFC has exactly the same situation the NF Campa B is bound to the ikb and it really is trapped in the cytoplasm it wants to go into the nucleus but it can't as long as ikb is bound but now what happens is that the ik kise can now come it phosphor Ates ikb as I said before you phosphorate a protein you change the threedimensional structure so now ikb lets go of nfca B now that NF Capa B and the ikb gets can get destroyed now the NF capb is unregulated by NF by the ikb it could now migrate into the nucleus and turn on whatever its Target genes are okay another mechanism by which signal transduction could allow a nuclear transcription Factor that's trapped in the cytoplasm to migrate into the nucleus to turn on it its genes okay are you with me so far okay so now we have to answer the question that I asked at the beginning how can you integrate the two signals in order to allow cells to be turned on so just to review dendritic cells for example B cells provide an antigen specific signal but also provide a CO stimulatory signal what how do you those those two signals get integrated at a molecular level and it turns out that the coast imator signal activates a parallel map kyes pathway that results in the um transfer of a factor that requires two P portions in order to be activated and turned on so here you have the raft Mech the the classic antigen specific receptor s signal and that's going to turn on a transcription Factor called elk the map K pathway activated by co- stimulatory signals turns on a protein called June and has shown so basically so these you all know how this KES works okay so map kyes IR activates El key ALK and that turns on Fast Gene transcription so fast is being made by the antigen specific receptor transduction pathway the co imator signal pathway turns on a protein called June it turns out that fos and June together bind to each other form a heterodimer and that heterodimer is a protein called AP1 and AP1 itself is a powerful transcription Factor if you just have Foss it won't bind to the right transcription sites if you just have June it won't bind only when these two molecules form this heterodimer AP1 will it bind to a P1 regulated genes so that's why you need both signals because if you only turned on the antigen receptor signal you only had F that's not enough to turn on AP1 regulated genes if you only had the coor signal and made June that's not enough you need to have both signals each one of which provides one/ half of this heterodimeric transcription factor and the combination of both of them that binds to all the genes that are required to give you a real ampen specific response so that explains at the molecular level how two Independent coost Motor signals can get integrated to turn on genes is that clear it's a very elegant system so as shown in this slide here now you you have Earth you basically now make F the cator signal makes June and F and June now combine form AP1 and AP1 binds to its Target sites in genes and activates the genes okay any questions so now we could look at the same exact picture and whereas at the beginning of the lecture it kind of look like this forign road map hopefully now it'll review it and now things will start to come together and make sense so again MHC plus peptide First Step CD4 cda8 gets recruited depending upon class one or Class 2 MHC what is the CD4 molecule going to bring with it it's going to bring with it lck lck is going to get activated not shown in this picture but cd45 is going to Def phosphorilated it it now gets activated it's now since it's brought adjacent to the Zeta chain shown here it's going to phosphorate the tyrosine residues those tyrosine residues now are able to recruit sh2 domain kinases such as zap 70 zap 70 now is going to activate protein phosph Lipa C protein protein kise this is now going to increase calcium this is going to activate calcium urine calci urine is going to Def phosphorate an fat and now an fat could go into the into the nucleus protein KY C is ultimately going to activate ik K phosphorate ikb allowing NF c b to be released and the two Coast the antigen specific map kyes pathway and the co- stimulatory map kyes pathway are go both going to converge yielding F and June forming the AP1 and this core group of nuclear transcription factors are what going to going to be to turn on so you first have the initiation of TCR mediated signals now you have the biochemical intermediates you have active enzymes and now you have the transcription factors that can migrate into the nucleus and turn on the appropriate genes okay is that clear semi clear okay and again just to kind of this is just showing the same thing in a a a little bit more detail but again the thing I want to focus on in this particular slide are the trans transcription factors because what's very very cool about it is that different techniques are used to sequester proteins in the cytoplasm but then allow them to be released into the nucleus so in the case of nfca B it's physically bound to a repressor protein if that repressor protein gets phosphorated it releases NFC and that nfcp gets into the nucleus and fat when it's phosphorated it stays into the cytoplasm if you def phosphorate it it can get into the nucleus and the map kinas itself has a Target protein that gets phosphorated once these get phosphorated they can get into the nucleus and again in the case of the coor signal they converge in the production of AP1 so these are the transcription factors playing a critical role in for example turning on incin 2 genes and a whole host of immune response genes because as long as you have this regulatory binding region you could turn on the gene so if you have 100 genes and every one of them has transcription Factor binding sites for these factors once these factors get into the nucleus they could turn on those 100 genes so that's how signal transduced at the membrane could turn on hundreds of genes very very rapidly and very very specifically because if you lack these transcription Factor binding sites those genes don't get turned on okay is that any questions okay now you know part of part of you says like who cares you know like okay signal transduction very very complicated you know it's never really going to be relevant and can this do anything for treating diseases well it turns out that there are two drugs that were described cyclosporin and tacus which some of you may be familiar with because they're routinely used for for transplant rejection treatment why what are the effor molecules that are invol cells in in rejecting transplants take a lucky guess CA cells well if you want to protect someone from having their trans transs rejected what cell would you want to block te- cells now normally in in the old days you know like what 10 years ago when we wanted to to turn off the immune system we used extremely blunt instruments so we we put patients on high dose steroids we put patients on um drugs that had very broad imuno supressive activity as well as having a lot of toxicity but the idea is let's identify a signal transduction molecule that's uniquely used by te- cells let's block that molecule and therefore we could specifically block te- cells with minimum side effects ises that makes sense now the reality of it is scientists were nowhere near as as uh as brilliant as doing that way basically you had other pharmaceutical companies that extracted fungi and all sorts of chemicals threw them into some incredibly large screen to try to identify immunosuppressive agents and once they were isolated then we figured out how they worked but sounds good that we designed them in a very very uh proactive manner but cyclosporin and tacus work specifically by interfering with the nfat mediated signal transduction pathway so normally calcium comes in it activates Calin urine which is a phosphatase which def phosphorites n fat allowing an fat to get into nucleus that you all know very well well what happens if you block this process and cyclosporin a and tacus working through different mechanisms but interacting with other cellular proteins somehow block calcium urine and by blocking calcium urine you no longer have the ability to Def phosphorate n fat and therefore n fat can't get into the cytoplasm if it can't get into the cytoplasm you can't get active transcription and therefore the t- cell won't get turned on now simplistically if this is only expressed by te- cells you have a brilliant system to specifically block te- Cell Activation and in fact cypor and tacus have revolutionized transplantation because it has allowed us to have transplant patients and do very very well with actually relatively minimum toxicity however uh there's always a drug company that says well what's the market for transplant is it a large Market very small Market you're not going to make a lot of money making drugs for uh for transplantation let's come up with another immun mediated disease that's relatively common that we could use our drugs to block and in fact a drug company came out with a cream called protopic and again this is the background 1984 fujisawa scientists discovered tacrolimus in a soil sample taken from Mount T sucuba in Japan and then tacus again t- cell t- cell uh inhibitory well eczema I mean anyone here have eczema I mean is it common when you in in Johannesburg yeah yeah so instead of using steroid creams ezem is mediated by te- cells let's use this inhibitory specific for te- cells and in fact it works really really well so for a while protopic was actually a fairly common drug that was used particularly on the face because steroids actually thin the face and cause capillary generation so they get red uh redness but protopic you didn't have that side effect so it looked really really good it made a lot of sense However unfortunately it turns out that the uh n fat is not just plays a role in te- cells it plays a role in other cell transcription and in fact tacrolimus has been shown to cause uh increased risk of cancer well for transplantation you're willing to take the risk because clearly kidney disease liver disease without you're not going to survive but something like ECMO which is not that serious this really put a major crimp in terms of using tmus clinically but again this is a basic Immunology lecture not a clinical lecture just want to underline the fact that by targeting a specific signal transduction molecule you could cure and treat a very very common immune mediated disease okay is that clear okay so all I'm going to say briefly about B cell signal transduction is that it's very similar to T cell signal transduction the same common Motif immunog globular molecule functions like the t- cell receptor when it gets Crosslink it basically recruits other saramy kinases in the case that are associated with adaptive proteins they phosphor tyrosines in the associated proteins and then these recruit sick which is similar to zap 70 and again this also stimulates a Cascade of transduction molecules so basically after recognition I think we've described how Tel receptor activates immune response genes the structural motifs utilize kinases adaptive proteins sh2 domains to allow proteins to bind the hydrogen know what's going on on the outside by virtue of the fact that transcription factors that are sequestered in the cytoplasm in the absence of stimulation are allowed to migrate into the N across the nucleus into the nuclear membrane into the nucleus to activate genes specifically and turn them on and that tells them to make the propr the proteins at the appropriate time again you know I know it's signal transction is a very complicated process but hopefully that giving you the primary uh exposure to allow you to understand it clearer when you review it again thanks a lot for your attention [Applause]
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Medical_Lectures
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Immunology_Lecture_MiniCourse_4_of_14_Antigen_Presentation_to_T_lymphocytes.txt
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so is now the mechanism by which this antigen presentation process occurs so the questions I want to focus on first is what is this with the spear well here you have to think Zoe because Elia has a spare what is the structural basis by which the MHC molecules present peptides to T cell receptors namely basically what is the structure of the MHC how is class 1 MHC different from class 2 MHC and how does that structural difference impact on the function of these MHC molecules how are endogenous peptides targeted to class 1 MHC molecules and exogenous to class 2 this must be some kind of sorting process that goes on inside the cell what exactly is that sorting process also you appreciate that putting a Pharm peptide in a class 1 MHC molecule is a death sentence then therefore it's critical to put the right peptide in the right MHC molecule otherwise you will kill an innocent cell and how does the t-cell receptor see the peptide in the MHC molecule is it seeing just peptide is it seeing some MHC what is what parts of the peptides exactly is it seeing and finally what is the structural basis why how cd4 T cells interact with MHC class 2 and why cd8 T cells interact with MHC class 1 so again this is just you've seen this picture before again to show it again but just to review the kind of core interaction and will now start breaking down and focusing on the structure of the MHC molecules MHC class 1 interface with t cell receptor and CDA now just looking at this cartoon you could immediately start seeing things that will pick up on as the lecture unfolds the first is if you look at class 1 MHC cartoon structure you have a 3 you have basically this long alpha chain having an alpha 1 domain and alpha 2 domain and an alpha 3 domain and this is anchored into the target cell this is associated with the protein called beta 2 microglobulin and look if you look at it beta 2 microglobulin is not interacting with the cell membrane at all it's basically interfacing with the class 1 MHC molecule now the class 1 MHC molecules I'll discuss in a few minutes the Alpha chain there are many many many different sequences for this alpha chain but the beta 2 microglobulin is exactly the same and in fact my beta 2 microglobulin protein is exactly the same as pretty much everyone in the room but my MHC class 1 is going to be different from that of motive the overwhelming majority of people in the room so this is you know one thing we'll focus on in addition you'll see here how the peptide is floating inside this little pocket when it's by floating it means there's no covalent linkages this peptide is not covalently linked to the MHC molecule at all it's only being held in place by hydrogen bond interactions or or other ion-ion interactions it's kind of like floating in this magnetic field but it's not physically being linked to it at all so it could sometimes even be popped out of this cleft and that's an important point that needs to be appreciated if you look at the t-cell receptor this is the hyper variable region and what is it seeing well first of all you see both the Alpha chain and the beta chain are contributing to recognition and you see that each one of these is picking up a piece of the peptide and each one of these is also recognizing the MHC molecule but what this is also underlining is as I mentioned before the same alpha chain with a different beta chain can recognize a different peptide because you can see it's picking up different pieces of the peptide is that kind of clear and if we now go to the class 2 MHC molecule that you see a different structure here you have an alpha chain and a beta chain so it's a heterodimer and again when I was a student I professors will talk about homeland I'm as heterodimers and I would like nod my head like I knew exactly what they were talking about I was clueless I didn't know what a heterodimer would I know what a homodimer was and again it's very simple though a homodimer means that both chains are exactly the same like identical twins that's a homodimer a heterodimer is like fraternal twins that each chain is a complete different structure but the two chains combine together to interact so this would be this is a classic heterodimer two chains they're different in structure but they interact so to heterodimeric change they'll hear that heterodimers and Homa garments but that's what it means but here you now see that the peptide is being held in place a little bit with the alpha and a little bit with the beta so both of them are contributing to it and again the t-cell receptor is seeing some peptide and some MHC molecule but you could you as we'll develop this is being held in place by a single chain and the structure of the cleft is going to be much more rigid whereas this is being held in place by two chains so think about like if you grab on to something with one hand is to be very very tight but if you grab on to something with two hands it's a lot looser and this has tremendous impact in terms of the function of the class 1 MHC peptide interaction in the class 2 again I'll discuss in greater detail again this is just kind of like a little foreshadowing now there are some people who basically think everything can be understood through genetics and there's obviously a lot of truth to that so let's look at what the genetic basis for clip the MHC molecules are so there there are at least 3 MHC molecules of for class 2 and 3 for class 1 class 2 is a little bit more complicated but if we just think in terms of dr dr on DQ and DP each because of the fact that each one has an alpha chain and a beta chain so DPP is the beta chain TPA is the alpha chain what this is telling you is the number of alleles per locus so for the d DP beta chain there are 120 different genes that can encode for that for the Alpha 23 etcetera etc if you go now to class 1 MHC obviously only seeing one bar because you only have one chain the beta 2 is exactly the same for everything but you have HLA a B and C and each one of those has again tremendous number of potential genes that can encode for that locus well what does that mean there are 728 possible HLA B genes and again many of you are studying like b-57b 27 you know that has implications but what this is telling us is in this room the chances of two of us having exactly the same HLA B is very very low it's probably minuscule and similarly for a and C so this is a high level of Pi morphism that occurs within the HLA molecule but again each one of us has two chromosomes which means that we potentially can express two different alleles in coding in a chain a c chain a b chain each one of the Alpha and Beta chains etc etc so that also contributes to our ability to express a wide range of HLA genes now I mean kind of like one L risk difference between HLA a B and C they're all class 1 MHC molecules the answer I don't think we really know yet I mean we're getting some clues I think HIV is really going to be very instructive because as work that that bruce's group and other groups have shown b-57 we know is highly protective for HIV why is that hlaa to doesn't seem to be as protective we don't really know so clearly there's a lot more subtlety to class 1 MHC a B and C then that we know about that we're first learning but anyway the bottom line is is that a B and C all encode proteins that have the capacity to present endogenous peptides and all these different genes have the ability to encode exogenous peptides there are clued variations within so the sequences of a are very different from B and very different from C but all these have exactly the same function and again you can think about this as you know I know strains of flowers or whatever that that egg could be roses for example both in that Rose family you have 50 different roses B can be for example tulip family there are flowers but within each one you have some functionals that are unique to them and again I think this is I think going to be a very exciting area determining differences between these these structures so what does this mean now from a genetic point of view is that is that if you're heterozygous for every HLA locus which means that you could express two different genes for each I'll HLA locus and clearly this is the old days when women used to wear skirts but but clearly you know but clearly I mean clearly by having two pairs of pants on to be a little confusing but they may have set genetics back a long time but but your parents basically combined their genes you have basically four different possible recombination events among the kids but what you're seeing is is that every gene is expressed on the surface at the same time and this is using the genetic concept of Paula Jeannie where you could have a lot of several genes but on the each locus is exactly the same or polymorphism where each locus on the chromosome encodes a different a different gene so that this difference is so this could be HLA a this can be B and this can be C but what is happening is is that your cell is expressing six different HLA molecules at exactly the same time similarly the same thing is happening for class two MHC molecules you're expressing if your parents are if you're heterozygous for every locus at least six class two MHC molecules at exactly the same time so your cells are again expressing at exactly the same time all of the HLA molecules that you have genes to encode and that has implications when we start discussing anchor motifs okay so that's a very important concept to nail down that your cells at any given time are expressing all of the class 1 MHC genes they possibly can Express and if you're heterozygous that's at least six different combinations and if you're a antigen presenting cell you're also expressing all at least six different possible class two MHC molecules so you've a lot of different antigen presenting molecules being expressed on the surface any given time ok is that clear ok so we'll start with class one MHC molecules and um the the class 1 MHC molecules they can present a diverse yet limited number of peptides that are eight to ten amino acids long and well ok so so the point though is what is the structural basis that limits the peptides that the MHC molecule can present so can an MHC molecule present any possible peptide the answer is absolutely positively no they only could present a limited number of peptide but the question is what is the rule that determines what peptide and MHC molecule can present and that's what we're going to start talking about well let's first go to the structure so if we look at the structure of the MHC molecule the first thing as I started talking about before it's a heterodimer with a membrane spanning alpha chain bound to the cell membrane over here and a beta 2 microglobulin that is basically linked directly to the Alpha chain the Alpha chain is polymorphic it means that different genes can encode different alpha 2 chains whereas the beta 2 microglobulin is exactly the same for everybody now where is the peptide sit the peptide is going to sit if you look at the Alpha chain it has three domains the Alpha one domain the Alpha two domain and the Alpha 3 domain and the peptide is going to be sitting in the cleft formed by the alpha 1 and alpha 2 domain and as I had mentioned before the Alpha 3 domain is where the cd8 molecule is going to bind to the beta 2 microglobulin it's unclear exactly what it does the simplistic view is it kind of Shores up the structure of the Alpha chain because in the absence of beta 2 microglobulin it's a very very floppy chain and really can't hold on to peptide well if you knock out beta 2 microglobulin and mice for example they really express extremely low levels of class 1 MHC and functionally they're there their cd8 functional deficient so this clearly plays a critical role but it's kind of maybe like a doorstop or something that you stick in in order to stabilize it now if we this is a view of looking down on the MHC molecule so now think of yourself as the T cell receptor coming into the cell to dock into the molecule and this is exactly what you're seeing this is the MHC molecule if you kind of think three dimensionally these alpha-helix coils kind of look like almost like a shark's teeth and inside of it is the peptide floating inside and you're coming in now to dock onto this peptide and what we'll start discussing is pieces of the peptide point downwards towards this pocket of the MHC molecule and some of the residues in the peptide point upward towards the T cell receptor another point to appreciate if you could see these blue like dotted lines here this is demonstrating that the peptides are bound or within the cleft of the MHC molecule this is almost sealed so that the peptide has to be completely inside this cleft you can't have any peptide sticking outside again the MHC molecule may be a little bit anal it has to make sure that the peptide exactly fits if it doesn't fit it can't deal with it it won't be able to bind it and again this has important implications because this means that the peptide is bound very very tightly once that peptide is in there it's not popping out it doesn't have dangling ends that could get ripped away it's really tightly fixed there and again if you'd want to ask TIA logically why do you think that is why should the MHC class 1 peptide would you want to have it bound so tightly that it can pop out any suggestions but what would be a ramification of putting a wrong peptide into a class 1 MHC molecule what could go wrong hmm the cell can be killed so your your an MHC molecule in molecule and you have a self peptide in it and then all sudden a digested thyroid peptide floats by you pop out yourself peptide the Pharm peptide gets popped in now cd8 cell comes by and you're saying to the cd8 cells oh don't kill me that was an accident back here that's not my peptide you know I'm not infected is this cd8 cell gonna say oh sorry never mind I buy that you know I think it's getting it killed so it's really important for the cell to be totally paranoid about making sure only peptides coming from inside to sell are in its class 1 MHC molecule it doesn't want to be an innocent bystander and get killed so that's why class 1 is really very important to have it tightly bound and just not being able to be exchangeable okay so now if we now do sequence analysis and we ask the question among all those multiple genes for class 1 ma si and what are the constant regions and where is the hyper variability in a different gene genes that are out there and these red lines indicate areas of hyper variability these are relatively constant and when you now mop map it out to the structure of a class 1 MHC molecule lo and behold the location for the variability turns out to be exactly in the peptide groove so in other region the alpha 3 for example there's almost no very very minimal variability the only and in fact now if you focus on we're in the eighth to an alpha 1 and alpha 2 the variability is the variability is within the area's they actually come in contact with the peptides so this is going to be teaching us something incredibly important about the ability of different MHC molecules to bind to different peptides because this is where the variability occurs and this is going to impact on the the range of peptides that any given MHC molecule can can bind to and that is determined by anchor residues that are present in the peptides so if you look and this is always confusing the first time you see these these kind of ball structure diagram but but think of these as basically showing residues from the amino acids and focus on the and this to clarify this this is the outside towards the T cell receptor and this is the bottom of the MHC molecule pointing down towards the cell and if you focus on these residues the the P 2 and the P 5 and P 8 so so these are pointing downward and these are providing what are called anchor motifs and what is happening is is that is that these stay the same but these stay the same but these old change and what's actually binding in the MHC molecule is these three particular residues because they are matching pockets which I'll show you in a few minutes that line up with these residues that allow it to sit inside the MHC molecule and what you now see is for this particular KOB MHC molecule these are all peptides derived from different proteins and viruses that can bind to K of B MHC molecule what they all have in common is the P 2 2 P 5 and P 8 they have pretty much similarly charged peptides in this case who's seen an isoleucine in this case I horse PR that's and wise camphor but this is lysine I see of lysine 3 need I'm blanking on the amino acid code but these are relatively conserved and these have to be the same in order to bind MHC K of B but these all can be totally different and therefore what allows an image and now if you have a different MHC molecule it has a different anchor residue motif in this case the P 2 P 5 and P 9 location of the peptide and again this anchor motif is different from this anchor motif and this means that this particular MHC molecule can bind one range of peptide and this one combined is another one if you're not an egg a residue can be completely different and that's what gives the tremendous variability from what combined to an MHC molecule is that is that clear it may be I'm sure it's a little fuzzy for some people salt I'm going to have more reiterate this again in a second and as I mentioned before the fact that the peptide is limited in terms of how long it is that means it's bound very very tightly you can't stick out of the CLEP so the ends are like almost sea down by hydrogen bonds to make sure it's really staying in there extremely tightly different MHC molecules have different structures and therefore it allows different groups of peptides to be contained within this cleft so now looking at a very two-dimensional point of view what you have are one group of residues in the peptide point down towards the MHC molecule these are the contact residues these are the anchor motifs that allow this peptide to bind to the MHC molecule these have to be relatively conserved at these locations however the other residues in the peptide it can be any amino acid you want any of the whatever 22 amino acids they can be which means therefore if you have y yl in these locations you could be presenting hundreds and hundreds of different peptides all which have different sequence residues at these different times and what is recognizing these particular residues what molecule the T cell receptor and what that means is is that any given MHC molecule can express thousands of different peptides that can be recognized by thousands of different t-cell receptors that detect clear and this is this is just another way of showing it basically showing that the t-cell epitopes are pointing up and the anchor residues are pointing down and you can actually see there they have these klux inside of the MHC molecules that they're bound to different MHC molecules have different kinds of cleft and therefore they'll use different anchor motifs so we just so now to put together all physiologically what that means though is is that if i would just have one MHC molecule that's all i would have i would really be limited with how many peptides i could present because i'm limited to whatever anchor protein that particular MHC molecule has right so how do we get around that we get around that by having not one MHC molecule not two not three not four not five but six possible MHC molecules with six different angle Chiefs which means that when among all six of those MHC molecules if you're heterozygous you can now present a very wide range of peptides and therefore hopefully present almost any possible peptide to a t-cell receptor and now you understand why we have so many MHC molecules why we have so many genes for MHC molecules because I guarantee you there are some peptides I can't present but you could present and again if you think in terms of humanity if I could not present peptides from a virus what do you think would happen to me when I get infected with that virus I'm going to die so so what humanity has done is basically it said let's make hundreds of different MHC molecules so if some new virus comes maybe we'll wipe out you know ninety percent of the population because they can't present it but we're guaranteed that somewhere in humanity is somebody that potentially could be able to present the peptide and be able to survive so maybe the be fifty seven eight HIV story may be relevant in terms of that being super able to present it again continue now being worked on but now I think understanding of why we have some so many MHC molecules and why all that we're always being presented any given time on the cell ok is that clear ok so now if we shift gears to MHC class 2 molecules an MHC class ii have similarities but also have differences from class one MHC so they also can present a diverse yet limited number of peptides but in contrast to MHC class 1 which are relatively limited in the size of the amino acid eight to nine amino acids MHC class 2 are very liberal they say you know thirteen fifteen sixteen no problem I'm more than happy to present it it's not anal compulsive at all well why is that what is the structural difference between class to it and class one MHC molecules and well first of all as again as a to talk alluded to before it basically consists of two dimers an alpha dimer and a beta dimer both of them are anchored in the member but both them probably have a little bit more wiggle room in terms of of interaction and also it turns out when you look at the peptide binding cleft instead of these two kind of being sealed together these are like how to opened up and the example people talk about it's more like kind of a frankfootas here in South Africa hot dogs right to think of a hot dog bun you know the hot dog is always sticking out the ends I mean imagine if you had a hot dog bun you got to get exactly the right size hot dog you know and otherwise it wouldn't fit you know wouldn't go over very well so this is pretty much what the cleft looked like it's a little bit different and now if you look at the peptide the peptide sitting inside of it but it's wide open and you can actually have peptides dangling out in the breeze um and again as I'll show you in a second this is important implications in terms of anchor motif binding okay now so the possibility exists there for that class two MHC peptides can fall out and new peptides can pop in does that bother you why doesn't it bother you right right did not kill ourselves so the worst thing that could happen is you know you could have a different t-cell receptor but you yourself are not going to get damaged and actually that may be advantageous for you because let's say this farm peptides floating around you have in phagocytose that that particular bacterium processed it but now you want to just stick those peptides into your MHC to present it that's fine to you maximizing your ability to present peptides which is good for class 2 MHC the more peptide presentation the more help the better so in fact that's why the MHC class 2 does not have to be anal compulsive about making sure the peptide that's in it has to come from inside and therefore the rules of binding or a lot looser than they are for class 1 MHC molecules and in fact now this has important implications in terms of anchor motifs because class 1 and class 2 MHC both use anchor motifs so in this case your position one is hydrophobic position four is negatively charged and position nine is a hydrophobic these are the anchor motifs but look what's happening what's happening is is that all of these peptides can be presented by exactly the same MHC molecule but what you do with the peptide is you move the peptide back and forth in the pocket to enable those anchor residues to bind so if it's if it's located at the end at the I guess this would be the C terminal end over here so that's fine and then it sticks out over here or if it's here you stick it out over there you have a lot of movement you can do back and forth with the MHC molecule since the structure is limited you can't move it back and forth so you again you have less capacity to present peptide but also it explains why you don't have to be so concerned how big the peptide is because there's a lot of room to pack I mean if you go on a trip and you have a giant suitcase you just throw everything into it because you know there's plenty of room so two class two MHC molecule doesn't have to be as concerned how big the peptide is is that clear okay so again to kind of reiterate class 1 MHC cd8 alpha beta class two MHC here a heterodimer of the alpha and beta chain again being presented cd4 so what basically now provides you with cd4 or cd8 specificity as I had mentioned before is the ability of the class one MHC molecule or the class two to interface with cd8 or cd4 so so here will you see a cd4 again pointing to the beta two region and here but the cd8 is binding mostly to the Alpha 3 region and that's what's provided specificity and again what's really interesting is you could take a t-cell receptor that recognizes a cd4 peptide swap out the cd4 for the cd8 and change it into a cd8 cell and now we'll only be able to recognize it if a CD one molecule is presenting so the specificity is not determined by the peptide it's solely determined by cd4 vs cd8 okay is that clear so now kind of just to summarize and a class 1 MHC molecule in terms of its structure alpha chain beta 2 microglobulin whereas class 2 is alpha chain beta chain peptide size limited to a two-night amino acids whereas class 2 is much broader range of size it could be the cleft of the MHC the peptide must be within the cleft class to the ends of the peptide could dangle behind the infinity class 1 is tightly bound because you know if you tighten find the wrong peptides of death sentence where it's class 2 it's loosely bound and t-cell interaction class 1 to CDA in class 2 is cd4 any questions okay so now the next question is how do peptides get into those clefts and what are the functional ramifications of this process and again as we've said over and over again presentation of a foreign peptide in class 1 is a death sentence so it's really critical to make sure that you've put the right peptide in the right MHC molecule so how does this happen so it's cells have to cellular compartments one which is the vesicular communicates with the extracellular fluid and is continuously basically bringing in fluid from the outside processing and sending it outside again whereas the cytosol is basically is not in communication with the outside environment it's kind of basically sealed environment and therefore what that's telling you is that any peptide that's coming from the cytosol is an endogenous peptide whereas any peptide that's coming from the lysosomes or from the endosome our exogenously FS are coming from outside so again this is now going to be telling the cell whether it's been permanently infected because once the infections in the cytosol is permanent or is it just bringing something from the outside and there's no need to worry about killing the cell and this is again just a slide I showed you the previous lecture showing again cytosol pathogens therefore are going to be presented class one seen by cd8 cells and uphill whereas intracellular pathogens which means that is not a terminal infection class 2c for help to induce killing or B cells will be presenting internalized antigen in order to recruit T cell help to stimulate B sub class switching so what compartment is located and determines what it sees so peptides that are presented by class one MHC molecules are derived from intracellular proteins and to basically take the process you basically start by having intracellular antigen it comes from inside of the cell it gets processed as I'll show you in a minute in protein called the proteosome and the proteome basically slices this protein into the appropriately sized peptides these peptides are going to be loaded in class one MHC molecule and the class one MHC molecule cannot leave the endoplasmic reticulum unless it has a peptide inside once a peptide is loaded it can now get cycled to leave and again rust on the surface but this peptide ultimately is coming from inside the cell if you're not infected all these peptides are self peptides nothing will happen to the cell because there's no cd8 that recognize cell peptides hopefully in the outside environment freshman is the photo soul intelligent cutter okay excellent question so pretty song anyway ever seen egg slicer no okay so so it we're not when I was up my kids were growing up you know you want to you want to give your kids food that's as easy to make as possible right because you want to quickly throw food on the table so what's better than a bit of boil hard-boiled egg right just throw it in number three minutes boil so but then you got to slice it so these exercises are really cool because you put the egg in it and there's these wires here they're exactly spaced at exactly the same distance and then you teach the kids that the only way they get to use this is if they eat the egg and they love doing this so therefore they're more they say please can you make me an egg okay if you insist so then now you take you take this thing you push it down and now voila you have exactly the same size slices of egg so addressing your question protostome I mean you know intelligence is a relative concept but but they apparently have the capacity to really slice peptides in relatively equivalent sizes to give you that eight to nine amino acid difference size that you need to efficiently a low class one MHC molecules when they're eight to nine doesn't sort of things p3 five and eight don't they have to be doesn't have to be 35 amazing those yes so the proteome has no idea what your anchor motifs are so the possibility absolutely exists that it may be off so if it had sliced it over to me no acids over maybe the anchor residue being the right location and again clearly you know there's some produce om is basically probably Wiggles the peptide up and the protein up and down a little bit to have some leeway so again I'm not enough of an entomologist to give you the specific answer but I would I would assume that yes not as rigid as this that it's like you know eight to nine slice eight to nine slice but the size is the same so whether it is moving back and forth from the sequence that it probably is but the size is going to be eight to nine amino acids because if it's thirteen amino acids it's worthless for class one MHC loading because you just can't load it okay is that makes sense so now the peptides have to get loaded into the MHC molecule and in order to be loaded the molecule that's utilized to transport it is cold is a tap protein and is actually there's two chains the tap one and it have to they bridge the endoplasmic reticulum and they're playing a role in terms of pumping the peptides into an area adjacent to the MHC molecule and again it's amazing the immunology how you go from a simple concept to a complicated way of getting things done but it's actually just to walk you through the slide the first thing that happens is you make the MHC class 1 molecule well this MHC class 1 molecule needs to bind what protein beta to my globulin so therefore it basically gets anchored to this protein called calnexin until the beta 2 microglobulin binds to it once the beta 2 microglobulin bind to it the calnexin lets go of it the class 1 MHC beta 2 heterodimer now moves next to the tap protein and this is kind of like you know anyone ever go to the airport to pick up luggage right you have the conveyor belt right and there are two kinds of people there's the people that they just stand anywhere but there are the kinds of people they have to stand right where the luggage is coming out you know right bill and like the first people there and usually the luggage is last I mean but anyway but and they only they have to be right there because that would get the luggage right away and running at the first cab and get whatever they have to go well the same thing is happening think of the tap molecule as where the luggage is coming out because the peptides are being chopped up by the ribosomes into all these different peptides to being pumped up by the tab and these class 1 MHC molecules are sitting right by the tab and exit they're waiting for the peptide come out if it's the wrong anchor motif it's not their luggage they have to wait for the next one to come but if they do get the one that has the right anchor motif now they could leave the endoplasmic reticulum migrate and ultimately be expressed in the surface of the cell okay is that clear now where the implication is very clear if you knock out tap for example if you have a mutation and tap are you going to be able to express MHC class 1 molecules and the answer is no we'll discuss an immunodeficiency that is actually an immunodeficiency they lack tap and they don't have class 1 MHC molecules and in fact it turns out they have very few cd8 cells because there's no class 1 MHC for them to mature off of in Islamic environment ok so this is basically how the loading occurs and the appropriate sized peptides if the if the MHC molecule has no peptide can it be expressed on the surface no because you don't want an empty pepper clock to be there that a Pharm peptide collage into to basically allow the cell to be killed as an innocent bystander so there's a very very tightly regulated process in terms of this transport okay any questions ok so now if we go to class 2 MHC molecule here the antigen is coming from outside of the cell and this gets basically taken up into the lysosome and digested into peptides so it's basically going to be digested by proteolytic enzymes that are nowhere near as intelligent as as the proteosome well we'll see in a second what the implications of that are but like in parallel and it's like kind of like coordinated ballet the class 2 MHC molecule is being synthesized it doesn't peptide inside its moves leaves the endoplasmic reticulum again as opposed to class one MHC which you can't leave until other peptide class two as you said helper who cares you know you're not as paranoid about what peptide gets into it it now comes into this fabulous assault and now it basically gets lowered with peptide is drawn to the surface and now is expressed on the surface with the peptide that it got loaded with now in contrast to the kind of egg slicer where you're going to get exactly the same size peptides the proteolytic enzymes in the in the FATA lysosome are not as strict in terms of how big the peptides they are a lot of them basically need to look for the appropriate amino acids in the next to each other in order to slice it so think about this is as slicing bread free in you basically you know fixed slices thin slices there's a lot of variability and the end result is that you have a large group of slices that have different thicknesses and the same way with the peptides that are generated for the class two MHC molecule but does it matter no because class two MHC is very flexible in terms of peptide sides the same way you don't care how thick the bread is you know you just put some marmalade or jelly on it and it tastes fine as opposed to class one MHC in the eggs but it has to be exactly the same thickness now clearly all loading the peptide is is very important and you do want to make sure that you don't lose sight of solid peptides into it you want to make sure you only load peptides that come from the fabuloso song and the way that you do that is you basically use a peptide as a placeholder and the peptide that's used as a placeholder is called the clip peptide and the basic mechanism is is that you have this invariant chain it now falls into the cleft as a placeholder this invariant chain gets digested leaving this clip peptide that builds this clap but prevents anything else a peptide from being put into there as long as it's being transported this is you know when they ship things they always stick something in the middle so so to protect it they put a or something over so nothing gets in it and this is just showing a different figure again how it ends up performing the clip peptide and now what happens is it gets placed in the phago lysosome here is the clip peptide here are farm protein gets that jessyca peptide and now HLA DM which is an HLA molecule gets recruited it helps pop open the class 2 MHC molecule also the acidic environment also helps the clip peptide gets popped now the peptide and get loaded inside of it and now and now and now will be expressed on the surface for t-cells to see it so again a little bit different approach but again because plaque to MHC is going to be helper you don't have this high level of paranoia you don't have this tight control of how the peptide gets in there and where it comes from okay is there any questions yeah I always thought it is the same as Mary see ya the answer yes HLA is the same as MHC but it turns out that this is a you know one of the interesting surprises that in addition to being able to this particular HLA molecule is not is its function is not as much to present antigen but it to help facilitate this clip from coming out you know if you knock out this HLA DM they basically have clip that stays in there's a lot more so it's um prostitute on sprocket as efficiently okay so so now again class 1 and class 2 cd4 cd8 well now the question is what is the T cell receptor seen is it only seeing peptide is it seeing some MHC how much MHC is a seen what determines whether the t-cell receptor recognizes the antigen or not and it turns out that this is this important concept called MHC restriction and what MHC restriction means is that in a normal antigen presenting cell this is the MHC molecule and this is the peptide that's being presented here's a t-cell receptor what you see is that the t-cell receptor is seeing some MHC sequences it's also seeing some peptide sequences and the summation of this interaction is generating enough affinity to cause recognition to occur okay is that clear the summation of all these the t-cell receptor just sees residues pointing up at it it doesn't know whether the residues coming from peptide a residence coming from the MHC molecule it just knows that it has a molecule that binds to that residue you know hydrogen bonds right click so can't see it just sees this topology of residues pointing up it doesn't know where it's coming from but let's say for example here this t-cell receptor is only recognized as MHC in this case a for example if you take exactly the same peptide and you presented an MHC be true it could recognize this peptide region but it doesn't recognize the MHC B region and therefore it lacks sufficient affinity to bind to that peptide and therefore this t-cell receptor will only recognize peptide in the context of a specific MHC molecule and that concept is called MHC restriction which is a critical concept in T cell immunity so the same exact peptide that my T cells could recognize when my MHC molecules presented will not be recognized when your MHC molecules presented by my T cells okay and the flip side is if you take the MHC alone but instead of having peptide X your peptide Y even though this region of the t-cell receptor can recognize this region but this region doesn't recognize the peptide and again you don't have enough affinity you need a combination of MHC plus the specific peptide to get that level infinity and again you don't get recognition and this is peptide restriction you need to recognize the appropriate peptide obviously except in a row okay is that clear now the question is well um why is it that that farm sometimes I could recognize far MHC molecules even though did not want I've been pre-programmed to recognize and this is like when you start getting into some of the hand waving explanations that immunologist tend to do but this is basically kind of providing a potential scenario by which this could occur and we know that because for example if if I would get transplanted with somebody else's cells for example I would reject them and the question is well how can I reject that person cells if they have a different MHC and my T cells only see my MHC right and that's like you know clearly what are the major observations so you can't argue with that observation so you have to come up with a explanation for an explanation that they generated was this is a classic situation t-cell receptor recognizes self MHC t plus self peptide because you get enough affinity from this summation of these interactions to get binding however it may be that a non self MHC can present the self peptide that I recognize but it does it in such a way that there's enough interaction between the peptide and the TCR to generate enough affinity that you don't need contributory affinity from the MHC molecule for binding to occur so it's almost like there's some peptides that are super antigenic maybe because they have very very strong charges on them that they alone could cause t-cell recognition independent of MHC or it may be that some non self MHC residues are so polar for example that they may generate a large enough binding affinity that even in the absence of a peptide that's being recognized that alone is enough to tell the t cell you've bounced you recognize and therefore mounted immune response so this again even though the classic situation you need summation of peptide plus MHC it may be that peptide alone may be enough some peptides or some non self MHC s may be enough to get binding and again to the t-cell to be activated a mountain immune response and this may be the explanation for why you can mount a very potent immune response against non self MHC molecules okay any questions okay so now just to summarize the the differences between peptide processing class 1 and class 2 class 1 MHC endogenous peptide class 2 is exogenous peptide loading and class 1 is an endoplasmic reticulum the endosome for class 2 peptide for folding is the antigen derived peptide for class 1 for class 2 its clip t-cell CDA class 1 cd4 and cellulose Aquila presentation dead for class 1 and activation for class 2 now one of the really kind of exciting things that just emerged in immunology in the past few years is the ability to quantify antigen specific t-cells so for example if I wanted to ask the question how many t-cells does someone have in their circulation that recognizes an HIV peptide how would I answer that question up until a few years ago I wouldn't be able to tell so however this is um Altmann Altmann and davis came up with this concept that if you basically take synthetic MHC class 1 molecules that are soluble and you basically biotin elate them and then you add streptavidin and now you form a tetra of these MHC molecules you could load them with any peptide you want provided this peptide has the appropriate anchor motifs to be bound by that peptide so now you have a molecule that has mhz plus peptide for them you add a fluorescent dye to this and now you mix them with t-cells if the t-cell receptor just recognizes that ma c-plus peptide it will now bind to all - - to this cat remember you need to have for at least three MHC + peptides to have enough affinity for the soluble of factor to be bound but now you could but virtually the fact that it's expressing a pleura Chrome you can do flow site of metric analysis and now you can identify what fraction of any population of t-cells recognizes any peptide MHC molecule so you basically could develop a panel of of peptides from any pathogen you want in a whole panel of HLA or groups and then ask the question what fraction and you could demonstrate beautifully as infection happens what goes up what goes down and really start to get an understanding of what's going on at the antigen specific T cell responsive level so this is really surf to revolutionize our ability to analyze t-cell responses and especially in HIV has really allowed two dramatic advances in our understanding of how a t-cell response and which was a report and which ones aren't important or less important okay any questions okay so now this is just going to end with really uh how things can go wrong so so now clearly in order for t-cell to get activated it has to bind only the peptide that it's specific for right that's very straightforward now about in about now over 30 years ago in the United States there was an incidence of 55 cases it's called toxic shock syndrome and I'll show you a picture of what it looks like and 95 percent of the cases were in women 95 had onset during menses 73 percent had staph aureus and the case fatality rate was 13% which is huge previously healthy woman also just died from toxic shock syndrome for no reason at all and these are also young women which makes it even more dead stating and uh so in apparently uh in other states this is high as 25% fatality but the question what was going on no one had a clue and this is what it looked like I mean these says this is basically have toxic shock where the skin basically starts forming these blisters and tremendously tremendous level of necrosis and obviously going to have significant infection very dramatic disease what was going on what was the cause and it turned out that the cause was due to the production of a super antigen by the bacteria and what the super antigen does is it binds to MHC and applies to the t-cell receptor in that sense it's like Krazy Glue so that instead of only having T cells that are antigen specific responding it made all t-cells think that they were responding to an infection all those help with T cells were making all these factors it was like the cytokine storm revving up the immune system causing tremendous amount of autoimmune destruction and this was the etiology of the disease who's super antigen mediated and it turned out it turned out that this was due to having a super absorbent tampon called the rely tampon which allowed bacteria to grow very efficiently because all this absorbed in blood bacteria said this is a great place to grow and then then since they removed this tampon from the market this was resolved but again it's an example of how an over exuberant new response due to a super antigen that is able to to drive the activation and proliferation of t-cells independent of antigen can cause devastating effects now in order for antigen presentation to occur it actually uses what's called an immunological synapse and what this immunological synapse does is it focuses these are the t-cell receptors and MHC plus peptide but now you have a large number of these focused in a very very focused small area why is that important when we talk about T cell signal transduction you'll hear about how the interaction the signal transduction molecules occurs and by focusing multiple TC RS in a very very small area that allows amplification of signaling to occur and the way you do this is by having the ican molecule and the LF a molecule adhesion molecule on the outside forming this kind of three-dimensional like doughnut and the center of the doughnut are the T cell receptors focused and the outer ring are the adhesion molecules that now basically push them it's a lipid bilayer so they're movable and push them into focus into this very tiny area and this now allows very much more efficient signal transduction to occur and that's important in the activation if you don't have this T cell synapse formation you have a markedly less efficient capacity to mount in a an antigen T cell response that's sensory I mean but what's very kind of devastating about this is that HIV is always out there trying to figure out how it could take advantage of an immune function to its own nefarious purposes and what HIV has done is it actually calls placed co-op to this process so here you have ikm LFA TC o flip this around okay so so so i flip the top and not the bottom so i messed this up a little bit apologize so we're basically on visited normal t cell receptor synapse where it's supposed to be what happens with HIV is you have the target cell the effector cell but instead of focusing on the t-cell receptor you now focus HIV and when you see here on this kind of magnification here you have your cd4 molecule here you have okay oh no I take it back I didn't oh okay this is what happened this ended up that stuff okay follows you have MHC class 1 c d3 normally lots of synapse here you have i cam 1 but now what it's focusing is the HIV so BP 120 and gagra folks over here against cd4 and cxcr4 or ccr5 and this now allows HIV to zip across and infect cell through cell to cell interaction so again it's taking this kind of tight membrane interaction that's utilized for signal transduction and as tis used it as a way of allowing HIV to get transmitted from cell to cell okay so so basically in this lecture the question we started out with was one of the structural basis by which MHC molecules present peptides to the t-cell receptor and I think that we talked about the alpha chain the beta to my globulin class 1 alpha and beta chain for class 2 and again the implications in terms of flexibility of the heterodimer of the alpha of class 1 and the alpha beta of class 2 determines what size peptides and how tightly the peptides are bound we discussed how endogenous peptides are targeted to class one proteome endoplasmic reticulum class one binding and then only then could it migrate to the periphery and MHC class 2 lines clip and the peptide transfer occurs inside the vacuole a has T cell receptor suited peptide MHC it sees a combination of both but there are exceptions where some peptides can be very high affinity some MHC can be very high affinity and that may account for ability to generate a graft rejection and the structural base of cd4 T cell MHC class 2 again as I mentioned before a cd4 binds to the piece of the class 2 MHC molecule and cd8 binds to the Alpha 3 region of the MHC class 1 molecule so again thank you very much for your attention I hope you know a lot more immune ology now than you do this morning thank you very much and see you tomorrow you
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Medical_Lectures
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Review_Session_for_Exam_2_for_Kevin_Aherns_BB_450550.txt
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Kevin Ahern: Good evening, everyone. Student: Good evening. Kevin Ahern: [high voice] Good evening! Student: Are you going to put another example Exam 2 up? Kevin Ahern: I am not putting an example Exam 2 up, no. I think people study those too much. There was accidentally one up there and I found it and I yanked it because I think people waste their time doing that. It's way better to spend the time studying the material than I know you've heard this before, but I'll say it againóthan studying old exams. I spend more time dealing with questions that I haven't talked about this term on old exams than I spend on questions that would be relevant. So it's not worth your time. The format will be exactly the same as the first exam. The only reason I show you the first one is to give you format things and that's basically it. Blah, blah. How are you guys doing? Student: Pretty good. Kevin Ahern: How's the studying coming along, it's a little early? Kevin Ahern: Studying, what are your approaches? How are you studying for this material? Student: Note cards. Kevin Ahern: Note cards, I really recommend note cards. Writing down note cards really is a good way to go, and, especially the further you get into metabolism, you're going to find that note cards are really your best friend. Putting things on note cards helps you because writing things down really helps put it in your brain. I discovered that when I was in graduate school. I waited until I was in graduate school before I discovered it, but I discovered it and it was really good. Writing it down really is a very, very useful thing for you to do. I haven't written the exam yet. I haven't even thought about the exam yet, to be honest with you, so I'm pretty much an open mind. You guys can convince me what to put on this exam, I suppose. The format, as I said, will be exactly like the last exam. The point distribution may be slightly different. They vary sometimes from one exam to the next. But for the most part, the exam will be, not "for the most part," the format will be the same. Like I said, point values may change slightly, but other than that, there shouldn't be any change. Content will change, of course. But I do, as I've said before, work from my highlights as a way of writing my exam. So look at my highlights. Look at my lectures. If I talk about it in class, it's fair game. If I don't talk about it in class, then I'm not going to ask you. I will challenge anybody to find a single question on my exams where I have not talked about it in class. I try to be very careful to do that. I don't set out to trick you. I really have no intention of tricking you. My end aim is in determining your level of knowledge, so I really want to make sure that I do that. All right, so this is going to be like the last review session. I will be available for questions and let you guys have at it. So what are your questions? Neil? Student: In glycolysis, you talked about aldolase, the reaction that it catalyzes, as having a very positive Delta G zero prime. So I was wondering, just to clarify, the way that it still makes a forward reaction is because the enzymes that precede it vary the concentrations up? Kevin Ahern: So Neil's question has to do with the aldolase reaction, and basically, how does the cell manage to make that reaction go forward, in view of the fact that it has a very positive Delta G zero prime? Let's take a look at that reaction. Here's the reaction that's relevant. The Delta G zero prime for this reaction is something like plus 20 kilojoules per mole, and that's a very large positive energy barrier and yet this reaction goes forwards. I think that when we think about Delta G, it's always important to remember that Delta G consists of two components: a constant, which is Delta G zero prime, which in this case is plus 20, and a log term relating to the concentrations of reactants and products. So the log term is the concentration of products divided by the concentration of reactants. For Delta G to be negative, and given the fact that we have a Delta G zero prime that's very positive, the only way Delta G can be negative is if we have that log term be negative. For that log term to be negative, the only way that can happen is if we have small amounts of products and we have large amounts of reactants. Somebody in class said, "Well, those Delta G zero primes of the earlier reactions help to push that." In a sense, they do, because they do favor making a lot more of reactant, which is this guy, right here, and the reactions after this are very efficient at taking away products. So when we decrease the numerator and we increase the denominator, that makes that ratio become much smaller. The smaller that ratio is, the more negative that log term is, and we get it negative enough and the reaction goes forwards. Now, I didn't get a chance to talk to you in class about the tricks that cells use to accomplish that. You're not responsible for it on this exam. You are responsible for the general nature of making that term be negative. After, in fact, probably during the lecture tomorrow, I will talk actually about how that happens, but you're not responsible for it on the exam. The exam only covers through Reaction 8 that I talked about. Somebody said, "Well, you showed Reaction "9 and 10 on the screen. "Are we responsible for those?" The answer is, "No, only the material through which I talked about and the material through which I gave you highlights about." So I'm not sure if that answers your question, but I hope it does. Okay, let's go home then, right? Other questions, yes Connie? Student: For sugars, what type of ring structures do we need to know? Kevin Ahern: For sugars, what type of ring structures do we need to know? Any ring structures that I told you in class you're responsible for. So you're responsible for the straight-chain structures of glucose, fructose, ribose, galactose. You're also responsible for the ring structures of those. Specifically, you're responsible for the six-membered ring of glucose, the five-membered ring of fructose, the six-membered ring of galactose, and the five-membered ring of ribose. Student: So we don't need to know like the furanose form of glucose, then? Kevin Ahern: You do not need to know the furanose form of glucose, nor do you need to know the pyranose form of fructose. Yes, sir? Student: What about sucrose? Kevin Ahern: And you should also know sucrose, that's correct. Sucrose only exists in the ring form... but, yes, sucrose, you're right. Student: To clarify, you said "six-membered ring of galactose"? Kevin Ahern: Six of galactose, yep. Yes, sir? Student: In recitation, in the first week of recitation after the first exam, they were talking about [unintelligible] Do we have to know how to calculate that or get that? Kevin Ahern: That was on the last exam. That was material on the last exam. Student: Right, but [unintelligible]. Kevin Ahern: It'll be on the final, but it's not comprehensive since the last. That was material we covered for the last exam. Unless you want me to put it on there? Student: Could you speak about proto-oncogenes and oncogenes? Kevin Ahern: Yes, certainly. The question has to do with what's the difference between proto-oncogenes and oncogenes? The proto-oncogene is the one I always like to describe first. The proto-oncogene is a normal gene that exists in our cells and it plays a very critical role in controlling cells' decisions about division, for example. And I say, A, that it turns out that there are about several hundred of these that play very, very critical roles. Epidermal growth factor receptor, for example, is a proto-oncogene. Now, if the epidermal growth factor receptor doesn't communicate information properly let's say it gets left in the "on" state, where it's constantly telling the cell to divide then that proto-oncogene is no longer functioning normally and that proto-oncogene that does not function normally is known as an oncogene. And the reason it's known as an oncogene is the term "oncogene" means "cancer gene." By far, the most common way in which a proto-oncogene is converted into an oncogene is by mutation. So it takes mutation to convert a proto-oncogene into an oncogene, and there are hundreds of examples where that can happen. It's because there are so many oncogenes that there are so many different kinds of cancer. There are so many different ways that we can screw up the system that we don't have one type of cancer. We don't have one cure for cancer because there are just so many ways in which the signal or the information can be screwed up. Does that help? Student: Yeah, thanks. Kevin Ahern: Neil? Student: So when you speak of a mutation, you're saying that, once the cell manufactures the receptor, it's a messed up receptor, basically the genetic material that it's made from is mutated? Kevin Ahern: This is related to the proto-oncogene that you're talking about? Kevin Ahern: Okay, so what I said was, if the proto-oncogene is mutated such that it's not communicating the signal properly, and the example I gave was where it was communicating a constant signal to divide, like constantly, for example, the receptor might mutate to the point where it is always acting as if it's bound to epidermal growth factor. So if it's always in that same conformation that it would be in as if it had a epidermal growth factor, then that's part of that signal that normally would tell the cell, "Now is the time to divide." But if it's stuck in that mode, it's always telling the cell to divide, and I'm telling you that will happen as a result of mutation. So I'm not sure if that's answering your question again, but I hope that's what.. Student: In the aspartyl proteases, what causes the water to detach and act as a nucleophile? Kevin Ahern: What causes, in the aspartyl proteases, what causes the water to detach and act as a nucleophile? Okay, so let's go back and take a look at those, and I'll show you the aspartyl proteases. There's the aspartylóoh, that wasn't what I was looking for. The mechanism that they use to do their strategy is here. So his question is, right here, this water, how does it become a nucleophile? The answer is that it has to be activated, just like every other nucleophile that we saw. So when I showed you the serine protease, we had that hydroxyl group hanging off of serine. The hydroxyl group of serine is not active. Only when that proton gets pulled off and we make an alkoxide ion, then it's a nucleophile, and then it attacks the carbonyl group. We saw a similar thing when I had the cysteine proteases. We had the SH and that H got pulled off by the histidine and made an S-minus, and that was a nucleophile, and that attacked the carbonyl bond. Well, here, water itself is not a nucleophile. It has to be activated, and the way it gets activated is exactly what's happening right here. This carboxyl side chain of one of the carboxyl groups is abstracting or taking this proton away from the water. That leaves a negatively charged hydroxide behind, and that negatively charged hydroxide is a nucleophile. It goes right straight for the carbonyl group, just as the other ones did, and attacks it. One of the big differences between this and the serine proteases and the cysteine proteases is that this OH is not attached to anything else. It's not attached to the enzyme. In the case of the alkoxide ion, the oxygen was attached to the enzyme. In the case of the cysteine proteases, the sulfur was attached to the enzyme. This guy is just floating freely in space. So this was the example I gave you where I said this activation and this attack does not take a fast step and a slow step. The slow step in the other ones required water to come in and release that thing. This water is actually doing the whole thing right here. So basically it's a one-step process that is breaking that peptide bond. Make sense? But activation, that's the key to your answer... the answer to your question, I should say. Connie? Student: So it breaks the peptide bond by the oxygen's electrons come back and just destroy that bond itself? Kevin Ahern: The oxygen here is a nucleophile, meaning it has extra electrons and it's seeking the nucleus. The nucleus is right there. That causes electronic rearrangement that breaks that peptide bond. Do we need to know the further mechanisms of how exactly that bond is broken? Kevin Ahern: Well, I haven't shown you in class anything about that, beyond the fact that it attacks that bond. If you recall, when I talked about the serine proteases, I talked about making the unstable intermediate that gets stabilized by the oxyanion hole? Kevin Ahern: That's as much as I've said about the mechanism. Student: Is activation always from removing a proton? Kevin Ahern: Is activation what? Student: Is it always caused by removing a proton? Kevin Ahern: For all the processes I've talked about in the proteases, yes. There are other ways of activating molecules, but for the ones that I've talked about here, in each case it has been removal of a proton, whether it was from a serine side chain, a cysteine side chain, in this case, removal from a water. You've got a good stack of note cards there, Jerrod. Student: Yeah, I do. I do have a question. I'm just trying to remember all the points before I embarrass myself. Kevin Ahern: I won't embarrass you. The ATCase, on your high lights it says you want us to know how ATP, CTP and aspartate affect it? Student: Okay, So how does aspartate affect it? Because I was reading that to say it starts to favor the relaxed state? Is that just when it's by itself? Kevin Ahern: His question is a point that I made in the highlights about, for ATCase you should understand the effects that ATP, CTP and aspartate have on the enzyme. Basically, what I talked about in class was that, first of all, this enzyme was a great example of allosterically regulated enzyme, and this allosterically regulated enzyme is regulated by those three compounds in the cell. So when we look at the structure of ATCase, we discover it has 12 subunits 6 regulatory and 6 catalytic subunits. And we describe two possible structures that the enzyme can exist in: an R, or relaxed state; or a T, or a tight state. These R and T forms that we talk about here correspond exactly to the R and T we saw with hemoglobin. In the case of hemoglobin, the R state favored the binding of oxygen. The T state favored the release of oxygen. In the case of this enzyme, the R state favors the binding of substrate, which is necessary for the reaction to occur. The T state disfavors the binding of the substrate which is necessary for the reaction to occur. Now, getting back more specifically to your question, ATP and CTP both affect the enzyme by binding to the regulatory subunits. ATP favors the formation of the R, or the relaxed, state, which will favor the activation of the enzyme. CTP favors the T state, which is the tight state, which disfavors the binding of substrate. Remember, again, these are not on/off switches, but up or down. And then the more specific question you had was aspartate, how does aspartate affect it? Aspartate does not bind to the regulatory subunits. Aspartate is a substrate of the enzyme, and it also favors the formation of the R state. So sufficient aspartate will flip it into the R state. It's independent of anything else that's there. It's independent of anything else that's there. It doesn't require anything else to be there to flip it into the R state. Student: Can you go over PALA? Kevin Ahern: PALA, yeah, P-A-L-A. PALA is another molecule I talked about relative to aspartic acid... I'm sorry... relative to ATCase. PALA is basically a suicide inhibitor of the enzyme that looks like aspartate. It sort of looks like aspartate. The enzyme will bind it, but in the process of binding it, it becomes covalently linked to the enzyme. So PALA, as I emphasized in class, is not a natural substrate of the enzyme. It's a man-made molecule, and this man-made molecule, when it binds to the enzyme will lock the enzyme in the R state. Now, that's interesting because it's covalently bound. It's locked there. You say, "Well, if it's in the R state, and it's bound, and it's suicide, the enzyme's not active, what does that mean?" Remember that R state and T state refer to structures, R being the relaxed, T being the tight. So I can still talk about the R state structure, even if the enzyme is dead in the water. Let's imagine the R state's relaxed, it's all nice and big. the T state, it's all compact. So if I have a PALA that binds to the enzyme and flips it into the R state, the enzyme is in this big, relaxed form. It just can't do anything. So I can distinguish the structure of the enzyme from the thing that the enzyme is actually doing. It turns out that PALA actually was very valuable for distinguishing these R states and T states. When we see PALA working the way that it does, it tells us, "Ah, now I understand why aspartate has the effect on the enzyme that it does." Does that answer your question? Other questions? Student: In the insulin receptor, what has the SH2 domains? Does IRS-1 have an SH2 domain or...? Kevin Ahern: Let's go to the figure and we'll answer that directly. You can actually see it in the figure, but I will show you that. So, let's see, insulin, it's right here. Signaling, right here. Anything that's bound to a phosphotyrosine, as you see on here, has an SH2 domain. That's an SH2 domain, right there, because that's a phosphotyrosine. That's also a phosphotyrosine and that's an SH2 domain, right there. Your book confuses things a little bit because they talk about an SH3 domain. An SH3 domain is just another binding domain, but it doesn't bind to phosphotyrosines. It binds to something else. Student: I was under the impression [unintelligible]. Kevin Ahern: I'm sorry, I can't hear you. Student: I was under the impression [inaudible] PIP-2 to PIP-3. Why is IRS-1 also bound to PIP-2? Kevin Ahern: Why is IRS-1 bound to PIP-2? It just happens to bind to it. Student: It doesn't [unintelligible]? Kevin Ahern: And there's plenty of PIP-2 here in the membrane. So you can think about it as an anchoring thing. You can think about it as anchoring, helping to anchor it. Student: Can you talk about calcium's role...? Kevin Ahern: In signaling? In signaling, you're talking about? Student: I guess with the prothrombins and how they're related? Kevin Ahern: Okay, in prothrombin. That's the other place I've talked about calcium. So, yes, I'll be happy to. I think it was right here. Student: [unintelligible] Kevin Ahern: I'm sorry? Student: [unintelligible] Kevin Ahern: I can't hear you. Student: [unintelligible] Kevin Ahern: Say it again. I still can't hear what you're saying. Allostery in regulation? if we go to the blood clotting scheme and I talk about prothrombin, prothrombinóokay, let's look at the big picture and then we'll come to prothrombin. The bigger picture is that we have this scheme of blood clotting that I showed. People asked me, "Do I have to know what the intrinsic pathway is versus extrinsic pathway?" No, I haven't talked about that in class. What I focused on was everything down here. Basically, these two pathways can be activated by different processes that are happening inside of our body... damage, bruising, cutting, whatever. So these processes get activated. We have two ways of starting this cascade, and the cascade terminates down, in here. So the aim of these two pathways is to, first of all, convert prothrombin to thrombin, which, in turn, converts fibrinogen to fibrin. Now, the relevance of prothrombin to calcium is that, in order for prothrombin to be held at the site of the woundóremember, we've got prothrombin floating through our bloodstream all the time. We cut ourselves, that's where we want our prothrombin to certainly be. We want it to accumulate. We want it to do its thing at that place because that's the place where we want the clot to occur. So we want to have something that ó kind of like the 2, 3 - BPG tells the body where the metabolism is happening rapidly. we want to have a signal to tell the prothrombin where to go. In order to do that, we have to modify prothrombin. So prothrombin gets modified by an enzyme that uses Vitamin K. That enzyme that uses Vitamin K causes prothrombin to get an extra carboxyl group on it. I'm going to come back to this figure in a second, but first I want to show you what's happening in this modification. In this modification, here is the side chain. Here's prothrombin. Here's a side chain of a glutamate. Prothrombin has several glutamates and they can each be modified with this addition of a carboxyl group on the end of the side chain. The normal carboxyl of glutamate is just the stuff in black. The additional molecule is this guy, right here. So the addition of this guy requires an enzyme that uses Vitamin K. So Vitamin K is going toóit's called a "pro clotter" because of this modification. We'll see the importance of this modification in just a second. Now, the addition of this second carboxyl group causes this end of prothrombin to recognize and bind to calcium. Calcium is abundant at the site of the wound. Now we've got a way of attracting and holding onto prothrombin at the site of the wound. So when we've got prothrombin at the site of the wound, the place where this conversion is going to occur is at the site of the wound. So now we've got thrombin active. To get thrombin active, we convert fibrinogen into fibrin, and fibrin, of course, is the material that makes the clot. So that activation of prothrombin is a very necessary step. If we inhibit the action of Vitamin K by using a blood thinner like warfarin or something like that, then prothrombin doesn't gain a carboxyl group. Therefore it never gets attracted to the site of the wound and our blood has a hard time clotting. So that's how a blood thinner works, or how one blood thinner works, anyway. Does that answer your question? Student: Are both hard and soft clots watertight? Kevin Ahern: Are both hard and soft clots watertight? I've never tried to blow water through one, so I don't know. I would guess that a soft clot would probably be less watertight than a hard clot would, but I honestly don't know the answer to your question. They're fairly watertight. I mean, they've got to stop that flow fairly quickly, so I would say yes, but they're not going to be as good as a hard one. Student: How are hard clots formed, again? Kevin Ahern: How are hard clots formed, again? Hard clots, okay, the difference between a hard clot and a soft clot, I'm going to have to show you the figure for the polymerization. the difference between a hard clot and a soft clot is that the soft clot simply involves inserting these betas into the Bs and the alphas into the gamma sites on the fibrin. So we see this network of polymers that form. But these are simply quaternary interactions. They're not covalent. Remember, quaternary interactions are things that generally involve hydrogen bonds. They may involve hydrophobic interactions and so forth. But in this case, there's nothing here that involves covalent interaction. Covalent bond formation will nail those guys together. So to put these guys together, there's an enzyme called glutaminase that will covalently link the side excuse me, the side chains of glutamine and I'm going to forget it off the top of my headóand lysine. I thought it was lysine. Put glutamine together with lysine to make a covalent bond. Now, these are happening in all kinds of places throughout this network. So it's not just here, it's not just here. But anytime these guys are in close proximity, if there's a lysine next to a glutamine, they're going to get covalently bonded together, and that covalent bonding of them together now forms the hard clot. Clear as mud? Clear as mud. Student: Just a question about the format of this exam. Will there be more questions on this exam than the last one? Kevin Ahern: So the question is, on the format of the exam, will there be more questions on this exam than the last exam? I would say, probably not. What are your thoughts? Student: It just seems like there's a lot more material for this exam. Kevin Ahern: It seems like more material than the last exam? Do you want more questions? Kevin Ahern: No, I mean, seriously, because if I have more questions, then each one's worth fewer points. So, I mean, I'm not asking the question to be silly. Student: We still have only 50 minutes. Kevin Ahern: You still have 50 minutes. What I was very pleased about on the last exam was it's one of the few first exams where I had very few, relatively few people, saying they had too little time. Usually the first exam in 450 is the only one of the exams I give in 450 and 451 where people complain about the time. So I don't think time will be a factor for you on this exam. I hope not. Student: So back to [unintelligible] maybe I just missed it. Student: But does the calcium activate the prothrombin or..? Kevin Ahern: Does calcium activate the prothrombin? No, It's just there to attract it to the site of the wound. Student: And it just activates itself? Kevin Ahern: No, remember those other pathways are converging to activate the prothrombin. So that's what's activating the prothrombin. Student: So the converging of the pathways activates prothrombin? Kevin Ahern: Thats right, it's not the calcium. Calcium doesn't activate the prothrombin, no. And keep in mind, when I was talking about this material and the blood clotting, we're talking about activation of zymogens, and every zymogen I've shown you is activated by breaking peptide bonds. That's what's happening all the way down that scheme. Yes, Connie? Student: For the catalytic triad, how does aspartate make histidine more negative [unintelligible] to tear that proton off the serine? Kevin Ahern: So She's asking about the catalytic triad and the role of aspartic acid in making histidine more negative, is the way I described it. So let me show you what happens with that. if I can pull it down here. So, catalytic mechanisms, and catalytic triad. Here's the catalytic triad in the active site of chymotrypsin or as it would look in the active site of any of the serine proteases. How is this guy over here affecting things that are happening all the way over here, is basically what you're asking. As I noted in class, what happens is the binding of the proper substrate... I'm having more trouble with this thing. That's too bad, i like this. There we go. The binding of the proper substrate to the enzyme causes a slight conformational change. Everything we've talked about with respect to proteins this term has been slight changes that happen, and these slight changes are very subtle. You saw in the case of hemoglobin how that very subtle change caused the oxygen-binding affinity to go way up or way down, depending upon which direction it went. The slight change in shape that happens here is seen in the active site. In the case of the catalytic triad, this aspartic acid is moved closer to the histidine. Aspartic acid is negatively charged. Histidine is what I like to describe as a "sink" of electrons. These electrons are resonant, almost. And so you put something negatively charged on this side of the sink, what's going to happen to the electrons? We're going to be much more likely to find them on this side than we are on this side. Negatives repel negatives. So when that happens, this side of the histidine becomes relatively more negative. That makes it much easier to pull off this positively charged proton from the side chain of serine. That's what's happening in this process. It's actually the proximity, the closeness of this guy that's affecting this overall movement of electrons. Student: Can you go over, just you talked about why this is affected when you remove, when you disable all three instead of just one? Ahern: Okay, sure. Let me answer this question, I'll come back to that. So did that answer your question okay? Yeah, that experiment's an interesting one I think. And it may be a little hard for students to understand. I'm happy to explain that to you. Here, so this is an enzyme know as subtilisin. Someone asked me the other day, 'do I have to know subtilisin?'. No, it's just another example. It could be just serine protease in general. So this would be true for essentially any serine protease that we happen to examine. What this experiment concerned was trying to determine the relative importance of each of the members in the catalytic triad. Serine, histidine, aspartic acid. So using genetic techniques, it's possible to make mutations that affect only one of those amino acids. Then you can collect that protein and see, 'hey, how active is this protein?' Alright, so you start with wild type protein, unmutated. You measure a certain Kcat. So this guy's got a pretty good Kcat over here. We see there is the Kcat of this wild type protein. We get over here and we examine the first mutant protein. This is a mutant protein where the serine has been mutated to aspartic acid. We noticed that in every mutation they make, the amino acid got changed to aspartic acid. So again, we're reducing the number of variables, we're always making the same relative change. We're making everyone into an aspartic acid. If we examine the activity of this enzyme, and the only mutation that's happened is we converted the serine in the active site to an aspartic acid, we see the activity go down by one, two, three, four, five, six, almost seven orders of magnitude. That's a log scale. That means that this guy right here is one ten millionth as active as this guy. That tells us that aspartic acid is pretty darn important. The same thing happens if we mutate a histidine to an aspartic acid. The only mutation in this enzyme is this histidine to this aspartic acid. We see it also goes down about 7 orders of magnitude. The serine and histidine play essentially equal roles in that catalytic process. If we mutate only aspartic acid, we still see a pretty good drop, but we don't see a drop as deep as this. And the only point of this one is to show us that, while aspartic acid is important, certainly we see a difference from here to here, that's about 1, 2, 3, 4, 5 orders of magnitude maybe, maybe 100,000 fold reduction in activity, but not as much as these guys. It says that aspartic acid isn't as critical in that process or in that catalytic action as these two are. Now we look at over here and we say okay, here's the mutation of all three of these, how come it's not even any lower than this? And the answer is that these guys, any one of these is deadly, essentially as far as the enzyme is concerned. It doesn't take it all the way down to the uncatalyzed. And why did I say that happens in class? Anybody remember? What else are factors in this catalytic process? Student: I mean the overall structure is... Ahern: The overall structure of the enzyme is still pretty much there. That tells us there's something in the structure that's still playing a bit of a role in that catalytic process. We're not destroying the structure of the enzyme in doing this. But essentially, these are all pretty much dead in the water. This is one ten millionth of activity of the wild type. Does that answer your question? Student: Yeah, I guess I just thought it would be a little bit lower, serine and histidine are the same. Ahern: Well it may in fact be a bit lower. So it's a little hard to look on a bar graph like that and see, and remember we're looking at a log scale, so a little bit lower would be hard to see on there. We're not going to see 7 orders of magnitude again, no. Neil? Student: Does that mean that the most important aspect of the catalytic role is serine? Ahern: I'm saying this tells us that serine and histidine have equal roles. Right? Either one knocks it down as low as the other, as all three do. Yes, Elliot? Student: Are all of these changed to alanine? Ahern: In this case, each one is being changed to alanine. Keep in mind this is one enzyme, this is a different enzyme, this is a different enzyme. So only in this enzyme do we change all of them to alanine. Yes? Student: I have a question about epidermal growth factor. Ahern: Epidermal growth factor, okay. Student: [inaudible] Ahern: Okay, let me pull that up before I get going to your question. So, epidermal growth factor. Right here. And EGF is, oh, wrong one. I do that every time. How about...no. How about, where did I do that? Is it here? No. Well now, come on. Not there. It will be the last one I get. Da Dum. All right. Back to your question. Student: [inaudible] Ahern: I'm sorry, the SOS you're talking about? Student: The SOS, yeah, is that attracted to Grb-2 or does Grb-2 have an attachment to tyrosine or [inaudible]? Ahern: I'm not sure I understand your question. Are you asking if this is coming in as a group into this? Student: Yeah, is it coming in as one big group or does Grb-2 come first and [inaudible]? Ahern: I can't answer that question. I don't know the order. It would be safe to assume for our purposes it's sequential. But I mean it's possible you could get 2 coming in at once. I don't know the answer to that question. Student: And the other question was [inaudible], how does that cause the internal section to become [inaudible]? Ahern: Okay, so the question is the dimerization of this, how does this favor the activation of the tyrosine kinase activity I think is what you're asking, right? So this is very much like what we saw in the case of the insulin receptor. The only difference was the insulin receptors start out as a dimer and the two ends were basically stuck in each other's active site, blocking them. The slight shape change allowed one of them to get a little bit further out and get phosphorylated. It's a very similar mechanism that's happening here. When these two guys come together, this guy's face gets stuck in over here and it gets phosphorylated which in turn causes it to start phosphorylating this. So it's very similar what's happening with the insulin receptor. One activating the other and then multiple phosphorylations happening as a result of that. Okay? Student: How is the Grb-2 attaching to a [inaudible] changing SOS [inaudible]? Ahern: Yeah, okay, as you can imagine, there's a lot of complexity in here. So his question is how can I explain how this binding of this thing causes this thing to change shape which causes this thing to dump its GDP and get to GTP? It's really, it's a schematic diagram is what it is. It's not unlike the schematic diagram we used to show the activation of the GDP with the adrenergic receptor. I showed you the sort of black box and I said then this dumps its GDP and puts a GTP in there. The same thing is happening here. That's really beyond what we can cover here. But it is, all these things are happening as a result of these slight shape changes. And that's the theme we come back to over, and over, and over. Slight changes in the shape of the protein really affect there. Yeah? Student: Other than the dimerization, is there any significance in the fact that two EGF have to bind? Ahern: Her question is, she said other than the dimerization, is there any significance to the fact that 2 EGFs have to bind? The significance is these don't exist as a dimer in the cell so obviously in order to make that dimer, we have to have EGFs bind. One to each one. Student: I was just wondering if concentration mattered. Instead of together, one molecule binds? Ahern: Her question has to do if concentration matters. It turns out it's actually a very good question. Since you asked it, I'll very briefly tell you a story. Your book doesn't cover it this time which is why I didn't go into it. You're not responsible for this, I'll just tell you, but it's kind of a cool story. There's a related receptor that's very similar to the epidermal growth factor receptor called HER, H-E-R. And it resembles this EGF. And in fact, it resembles it enough that HER can bind to EGF without needing epidermal growth factor. Normally in the cell, probably at a low rate, HER occasionally binds to EGF, it stimulates the response and the cell divides and everybody's fine and happy. Mutation of the promoter of HER causes HER to be overproduced in some cells. So when this happens, you've got way too much HER and HER goes around grabbing all the epidermal growth factors and starts stimulating all of them and starts stimulating cells to divide uncontrollably. So this is an example of an oncogene. HER is an oncogene that will cause that to happen. So we're just making too much HER. It's kind of like when me made too much bcr-abl. Making too much HER and the cell is stimulated to divide. Well it turns out HER is commonly stimulated to divide in breast cancer. And there is a very interesting and a very good treatment for HER related tumors and there's a monoclonal antibody called Herceptin that is targeted specifically to bind to HER and prevent that stimulation from happening. Herceptin is a very, very effective anti-cancer drug. It has very, very few side effects and for HER related tumors, it's very effective at knocking it down and in fact knocking it out. And that relates directly to what you're asking about which is concentration. Concentration of these could be a factor. So if we had too much epidermal growth factor for example, we would probably have a similar problem. Let's see, Neil? Student: To clarify, you said there was a difference between the structure of the insulin receptor and the EGF receptor? Ahern: Yes. Student: They're both dimers in effect, correct? Ahern: The EGF normally does not exist as a dimer. When it doesn't find to EGF, it's a monomer. It's floating around in the membrane, not bound to the other one. That's the thing I talked about when this guy binds to EGF, this little loop flips out. And so if it doesn't bind EGF, then there's no loop and there's nothing to basically attract to the other one. Student: Do you call it induced dimer? Ahern: I'm sorry? Student: Do you call it induced dimer? Ahern: Okay. I can call it Kevin. [laughing] Yeah. Student: I'm kind of confused about the rad's GDP, how it goes to GTP. Do the Rads drop the GDP? Ahern: Yes. It's just like we saw in the beta adrenergic receptor when we had the G protein that dropped the GDP and replaced it with GTP. Exactly the same thing is happening. That's why it's showing you that GTP is going in and GDP is going out. So we're not putting a phosphate in there, we're actually kicking out the GDP and putting a GTP in there. Okay? Connie. Student: How come it's not a G protein? Ahern: It's a G like protein. So it's different enough from a G protein we don't call it a G protein but we do call it a G like protein. It's just nomenclature. Yeah? Liz. Student: G proteins always have those three units like the alpha unit, the beta unit and the gamma unit? Ahern: Liz's question has to do with do G proteins have an alpha, beta and gamma sub unit? As far as I know, they do have all three. This one as you notice, Ras doesn't have the other two so that's why we call it a G like protein. This would be more like the alpha unit that we saw on the G protein. Other questions? Yes? Student: Can you explain restriction modification system again? Ahern: Certainly. Restriction modification systems. I think restriction modification systems are pretty cool, pretty interesting stuff. I talked about them relative to catalytic mechanisms and they are, most of what I had to say about the enzymes themselves were more the importance to them relative to bacteria than about mechanism. I remember very briefly talking about mechanism. But I think they're most interesting from the perspective of what they do. So it was related to the mechanism that I talked about this because when I talked about mechanisms of actions of enzymes, I pointed out to the need to activate various things. We saw in the case of the proteases, we had to activate hydroxyls or serines or waters and those activated nucleophiles then attacked specific, in the case of protein, carbonyl groups, and caused peptide bonds to break. In the case of restriction enzymes, we also see activation. We have activation of water again. And that activated hydroxyl that arises, attacks the phosphodiester bond. It's not attacking a peptide bond. The phosphodiester bond is the bond that's between adjection nucleotides in DNA. So a restriction enzyme has a catalytic site that favors the activation of water to break phosphodiester bonds. And restriction enzymes themselves have the ability to recognize specific nucleotide sequences in DNA, just the same way that a protease has the ability to recognize specific sequences of amino acids in a protein. So in the case of the restriction enzyme, what's happening is when the restriction enzyme finds the proper sequence, it changes shape, and the change of shape causes the DNA it's holding onto to bend. And the bending of that DNA allows a nice little pocket for magnesium and water to be located for water to be activated by the enzyme. So that's the activation part of that process. To answer the bigger question about restriction enzymes relative to bacteria, restriction enzymes are important as a protected mechanism for bacteria against invasions of viruses known as bacteriophages. So restriction enzymes are always paired in a cell with what's called a methylase. So they're called restriction modification systems. The restriction part is the restriction enzyme. The modification part is the methylase. And what a methylase does is it recognizes exactly the same sequence as the restriction enzyme does. But instead of cutting it, it favors the addition of a methyl group at a specific place in that site. And what that single methyl group does is it prevents the restriction enzyme from recognizing that, therefore it doesn't bend it, it doesn't allow that site to be cut. So the modification system is there to protect the cellular DNA from destruction by the restriction enzyme, and the restriction enzyme is there to attack invading viral DNA. As I noted in class, it's not a perfect system. It's not a perfect system because yes, sometimes a virus will get methylated first before the restriction enzyme gets there. You can imagine situations where the restriction enzyme might occasionally cut the bacterial DNA and that can happen, too. But the system is better than no protection at all. That's what we see throughout biology. Yeah? Student: Does this mean the modification goes in front of the restriction? Ahern: His question is the modification going in front of the restriction? The modification is on a separate enzyme. So it's a chance thing. It's really a chance thing. Connie. Student: Does each bacterium have its own restriction enzymes? Ahern: Good question. Her question was does each bacterium have its own specific restriction enzymes? We see a lot of variation in restriction enzymes from one bacterium to another. Within a given species, we see limited numbers. So in the case of the one I showed you in class, that was EcoR5, that is found in e coli, and e coli, across all e coli, we find maybe 3 or 4, 5 restriction enzymes. But if I went to something like salmonella or I went to pseudomonas or something like that, I would find restriction enzymes that recognize different things. Student: But if I just isolated one e coli bacterium, would it just have one restriction enzyme? Ahern: Her question is if I isolated an e coli bacterium, would it only have one enzyme in it? It might have as many as 2 or 3. So it could have several within a given bacterium. It's a small number, it's not a giant number. Jerry? Student: Are restriction enzymes only on the DNA or will they be out in the cytoplasm...? Ahern: His question is basically where are restriction enzymes located. So remember that we don't have a nucleus in bacteria so the DNA is in the cytoplasm all the time, but I think more specifically your question is "is the restriction enzyme found only on the DNA?" And the answer is no it's not. It's soluble in the cytoplasm and it's floating all around. And that's important. Because we don't want it just sitting there on the cellular DNA. Because if it were when a virus comes in, the restriction enzyme would never see it if it were staying only with the DNA. Connie. Student: For feedback inhibition, do you still call it that if the end product doesn't knock out the first pathway, like the second or third? Ahern: Connie's asking me a semantic question here. The question is, I describe feedback inhibition as the end product of a pathway inhibiting the first enzyme in a pathway, what if it knocks out the second enzyme in the pathway? Well it's really a semantic argument. The answer is in a sense, I suppose it is. Because remember pathways, what we define as a pathway is random. I'm not going to split hairs like that. Because again, what's the last molecule in a pathway? That's also random what I define as that, right? Yes, sir? Student: In recitation today, it was mentioned that they have anabolic and catabolic [inaudible]. Ahern: Yes. Student: [inaudible] Ahern: So his question has to do with anabolic and catabolic electron carriers. That's not something I talked about yet in class but since you asked about it, I will answer it for you. So catabolic processes of course we remember are breaking down of large molecules into smaller molecules. Anabolic processes take small molecules and build them into larger molecules. So cells get energy be catabolism, they use the energy in anabolism, making things that they want. So catabolic processes, the one you've seen so far is glycolysis. We'll talk later in the term about glycogen breakdown. We'll talk next term about the citric acid cycle. We'll talk about fatty acid breakdown. And all of these are catabolic processes. Larger molecules being broken into smaller ones. And when that happens, there's frequently oxidation that has to occur. Oxidation, you recall, involves a loss of electrons and loss of electrons, when those electrons are lost, as I said, they don't just disappear into thin air. Something has got to happen to them and cells use electron carriers. There are two electron carriers we commonly see in catabolic processes. NAD and FAD both accept electrons commonly in catabolic processes. In anabolic processes, that it making things, like the best example I can give you is in making fatty acids that we'll talk about next term, we see a different electron carrier that is used primarily, not exclusively, but primarily in anabolic processes. And this is NADPH. So that's what we see NADP vs. NADPH. The NADP carriers tend to be more involved in anabolic processes. The NAD and FAD carrier's more involved in catabolic process that's what they're referring to. Yes? Student: Can you define allostery? Ahern: Yeah. Can I define allostery? I'm glad that you asked. Allostery is the, when a small molecule binds to a protein. Actually, I defined it in class as an enzyme where a small molecule binds to an enzyme and affects the enzyme's activity. That effect can be positive. That effect can be negative. Depends on the molecule, depends on the enzyme. Okay? Connie. Student: A follow up to that question, does allostery ever tell you where it binds on the enzyme? Like a regulatory sub unit or active site? Ahern: Connie's question is actually a complicated one. It is, "does allostery ever tell us where it binds?" The answer is just by the kinetic data itself, we don't really get that much information out of it, no. We need other structural changes and other structural information to know are there regulatory sub units, are they separate from catalytic sub units, and that's a more involved process. Yes? Student: What's the first messenger in the angiotensin system? Ahern: What's the first messenger in the angiotensin system? Angiotensin. Yep. And you're not responsible for that. I just give that as an example. So I just showed you that phospholipase C pathway saying we've got this going through here. And I give angiotensin of an example of a type of pathway that does that but I didn't talk specifically about the first messenger in that. Yes? Student: Can you go over the synthesis N link, like glycoprotein? Ahern: The synthesis of the N linked glycoprotein. Certainly. Let's go to carbohydrates. Okay. When we go to synthesize N linked glycoproteins, first of all, N linked glycoprotein, their synthesis starts endoplasmic reticulum. And in the endoplasmic reticulum, some of the common things are added that you see on the screen here. So all N linked glycoproteins will have a common core of 5 modified sugar residues. I'm not asking to know which ones are there but you should know that there's a common core that's there. I talked about how the synthesis of N linked glycoproteins starts. I'll talk to you about that in a second. N linked glycoproteins, their synthesis starts in the endoplasmic reticulum but it's completed in the Golgi apparatus. So N linked are made in both places in essence. The O linked glycoproteins are made only in the Golgi apparatus. Now, to answer your question I think, which is how do these guys get made and... that's not what I wanted. Dolichol phosphate, okay. I talked in class about the role of this molecule in the initial synthesis of that carbohydrate Christmas tree on glycoproteins. I talked about how that's made. So dolichol phosphate is a molecule that plays an important role in that process. Dolichol phosphate is a membrane lipid. It's found in the membranes of the endoplasm reticulum. And to start, it looks like this. So we can think of the lipid bilayer, this non polar portion is stuck in the lipid bilayer, and this phosphate is sticking out because the lipid bilayer on the inner portion is very non-polar. This guy doesn't fit very well. So this is sticking out into the cytoplasm. The phosphate sticking out into the cytoplasm. And when we go to make an N linked glycoprotein, we've got to start making that core. And we start making it on here. So there are enzymes that start putting those phosphates on this phosphate sticking out on the cytoplasm. And then, so I talked about magic, and then something magical happens and this molecule inverts. So the thing that was on the phosphate on the outside flips and now it becomes on the inside of the endoplasmic reticulum. It's crossing that lipid bilayer in doing this and carrying with it those modified sugar residues on there that will ultimately get put onto that N linked glycoprotein. Once it gets inside, additional residues may be added and then that beginning of the Christmas tree will be transfered to a target protein, making it a glycol protein. Yes? Student: I'm sorry if you already said this, but the ones that are added to it prior to it flipping, is it just the 5 core...? Ahern: I haven't specifically said that. I said we start with the 5. It doesn't matter for our purposes if there are 3 or 4 or 6. The important thing is that the process is starting out there in the cytoplasm. So anyway, once it gets inside, once that flip has happened, then as I say, additional residues may be added to that, and then it'll be transferred off of this phosphate onto the target protein and then this guy will turn around and flip back out and be ready to do another one. I forget who asked me the question. Does that answer your question? Student: Yeah, so it ends up flipping it out to the other side? Ahern: Yeah. Student: And it just goes to the Golgi? Ahern: Well, the protein goes to the Golgi. Remember this molecule is stuck in the membrane. This is not attached to the protein. Student: Which membrane is that located in? Ahern: This is in the membrane of the endoplasmic reticulum. Yeah? Student: Going back to the EGF notes, the table on there that has like the definitions of like ras growth, raf, we haven't talked about that in class. Are we supposed to know...? Ahern: As I said in class, you're only responsible for what I've talked about in class. So I haven't talked about that in class. You're talking about this guy here, right? I like arf, it's my favorite. Yes, sir? Student: [inaudible] Ahern: About what? Student: [inaudible] Ahern: His question has to do with the three different pathways that can happen after pyruvate and how much detail do I want you to know. Whatever I said in class. Yes? Student: Does protein kinase A phosphorylate phosphodiesterase? Ahern: Does protein kinase A phosphorylate phosphodiesterase? No. Student: [inaudible] Ahern: So let me answer your question. You're wondering how phosphodiesterase itself is controlled basically. So phosphodiesterase just to remind everybody, phosphodiesterase is an enzyme that breaks down cyclic AMP. It converts cyclic AMP to AMP. And when it's converted to AMP, it no longer behaves like cyclic AMP. It can't affect protein kinase A anymore at all. So it's generally on. So more importantly, we're concerned with phosphodiesterase, how do we turn it off? And that's where caffeine comes in. Caffeine is an allosteric inhibitor of phosphodiesterase and stops the enzyme from breaking down cyclic AMP. So when the enzyme stops breaking down cyclic AMP, cellular concentrations of cyclic AMP increase, and that then favors that cascade that ultimately breaks down glycogen to make glucose. But no, it does not phosphorylate that, no. Connie? Student: What's the difference between suicide inhibition and competitive inhibition besides the covalent bonding? Is there anything...? Ahern: What's the difference between suicide inhibition and competitive inhibition besides the covalent bonding? None. Student: None, okay. Ahern: Alright, so you guys look like you're a little worn out here. What do you say we call it an evening? I will be available. My schedule as I noted is lightening up this week so I should be around if you have questions. Please feel free to come by and see me. I probably will announce in class tomorrow that I will allow you to suggest a question for the exam. So if you want to send me, email me a question for consideration for the exam, I will use one student submitted question for the exam. Student: I was going to ask you a question but I was busy writing. Do we need to know how bicarbonate comes off of the carbonic anhydrase? Because I have an idea of how carbonic anhydrase creates carbonic acid from carbon dioxide. I'm not sure how exactly it comes off. Ahern: It's just released just like any other product of an enzyme would be released. Student: Okay. [END]
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Medical_Lectures
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Breathing_Wheezing_and_Gasping_for_Air_Our_Respiratory_System.txt
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Stanford University well good evening everyone great to see you again tonight I have to say I've appeared before you in various attire over the course of the last weeks couple of weeks ago I was rushing out because I was on clinical service tonight I'm leaving for Washington so if you want to see what I wear on the red-eye this is it so I will of course - out or as soon as this is ending because I'm gonna be heading off to the airport around 8:30 or so that said here you are tonight along with me sitting here doing something that just comes natural and you're not even thinking about it we're just taking breaths in and out and we assume that our our functions and life goes on but pause for a moment and think about that process which is so passive that it doesn't necessarily compute in in our minds and yet without it we wouldn't be here at all last week we knew we'd be hearing things in the week before that seeing them but without our lungs and oxygen getting into our system nothing would really function so it's fascinating to to think about this process and I'm very pleased tonight to introduce really a wonderful Authority but also a very close colleague and friend dr. norm risk who's going to be leading the discussion in fact unlike the last two weeks he's not only going to be leading it but doing it solo which is going to be a great tour de force norm like a number of the other speakers shares a certain functionality he came to Stanford only from having first been on the East Coast where he did his undergraduate work at Harvard and then moved down to New Haven to do his medical degree at Yale and then did his sort of mandatory time at UCSF before seeing the light and coming down to Stanford and his background is in both pulmonary medicine really the lungs and how they functioned in health and disease and in critical care medicine and in our parlance around Stanford and certainly in my own view he's one of the finest doctors in the entire Stanford community if there is a problem that needed attention he'd be someone that I would definitely go to and he plays a critical role in leading the intensive care unit in training the individuals who come there for additional education and of course and also consulting and seeing people with pulmonary disease which is really the focus of tonight's discussion he is the Guggenheim professor of medicine at Stanford and in my own realm works very closely with me because since I've been here nine years ago he's been serving as our senior associate dean for clinical affairs so in addition to working in the ICU leading the ICU doing pulmonary medicine and lots of other assorted activities he also spends a lot of time helping to coordinate our clinical programs in relationship to the medical center so many functions many hats um but tonight distill down to a single purpose some telling us about our respiratory system and how it works and when it goes awry so norm risk thanks Phil for that overly generous introduction and I never really realized you wanted to be a pulmonologist but now I know a new secret about you okay tonight we're going to talk about the respiratory tract and both in health and disease and what I hope to do is give you some real feel for it intuitively and then also teach you a little bit of physiology about it and then we'll kind of shuttle back it and the last half of the talk to talk about common lung diseases and how they required what they mean so the goals of the discussion are really to introduce the structure of the respiratory system and how it interacts with the environment because it is a very interactive organ give you a sense of the function of it and how it works in exercise that at altitude and then the common symptoms that people are afflicted with breathlessness wheezing and coughing and what they portend and then I'm going to describe some of the common disorders some of which derived from exposure to the environment and some of which are intrinsic to the lung biology so first I'm going to talk about the complex and fascinating structure of the lung and the respiratory tract so if you're going to design a lung and we're looking for optimal design characteristics you'd start by something that would provide a very large surface area because we need to absorb oxygen from the environment and excrete carbon body dioxide back into the environment where's the oxygen with nutritional substrates like glucose are this the source of energy and/or metabolize themselves to produce carbon dioxide just and excreted but you need to have this very large surface area to interact with the environment a very compact space that it can fit in your lungs your chest cavity and then you need to have an internal transport system to move that oxygen around you're absorbing from the environment and to excrete the carbon dioxide and pulmonologists like myself think that party vascular systems simply serving the beneficence of the one and they're just moving around the things the Lord does and it has to be mechanically very efficient because you wouldn't want to use a lot of your caloric needs expend a lot of caloric needs on just the simple act of breathing which you think of as being a rather effortless activity and then because it does interact with the environment and is exposed to the atmosphere you have to protect the lung you'd have to protect lung from toxins and infecting organisms in the environment and then because your lungs connect your digestive tract you have to insulate yourself from saliva and food and things that could get into your lung so I'm going to start with I'm going to use this pointer my little mouse I'm going to start with just some simple overall descriptions of the lung and become more complex as we move along so normally air moves preferentially through your nose behind your nose an area you can't see are turbinates or conca these 100% humidify the atmosphere so that air that moves into your lung is a hundred percent humidified and they also strain out particulate matter that might be noxious to your lung then after the air moves through the nose it moves behind your tongue this muscle and then behind your tongue into the larynx behind the epiglottis the epiglottis is a flap on your larynx that closes to keep food and saliva out but is open when you're trying to breathe so air moves from behind your nose through the bit beyond the epiglottis into the larynx and down into the trachea and your Airways behind the epiglottis is the esophagus where food would move down behind your trachea into your stomach so there's a little apparatus there your larynx that allows you to both phone eight and protect your airway from things that otherwise would be going down as is commonly said the wrong tude now if we look first at the bronchial tree and I've lifted this picture from your textbook now I'm at the top of the bronchial tree is the larynx air moves down through a rigid trachea it's reinforced by cartilage the same kind of material in your nose keeps your nose having shape and your ears shapely so air moves down the trachea and then into the left or right mainstem bronchus which and then into your lungs and an ever dividing bronchial tree see I might be want to use this if you look at the bronchial tree knot schematically but actually and this is a photograph of the bronchial tree of a human which all of this small air sacs the alveoli have been digested away and you're just left with the profusion of airways that make up our lungs so here's the trachea and the right mainstem bronchus left mainstem bronchus is hidden but you can see how many branches and how find those branches become and I'm going to give you more of a feeling with that with microscopic views of it in a minute but a lot of fairly large fraction of the lung is actually made up by these conducting Airways that gas moves in and out of your lungs from so the Airways branch repetitively about 23 to 27 times and they become extremely small in fact microscopic and then at the end of airways things that we call terminal bronchioles bronchioles or small bronchi at the end of these terminal bronchioles the airways break up into small distance ibill or sacs called alveoli alveoli of the business end of the organization where the gas exchange occurs and they're microscopic in size so if you look at that schematically I want to give you some of the attributes of these Airways first they're conducting your ways in which gas air from the atmosphere moves by bulk flow that is it yet moves in a discernable fashion with a fair amount of speed in and out of your airway then and these as I mentioned already these Airways are buttress by cartilage so they're fairly stiff and they also have bronchial glands that making mucus in your central Airways your larger Airways the conducting Airways and then eventually after about 19 or 20 divisions they begin to develop little outpouching x' which are alveolar ducts smaller Airways and when they get down to about 25 or 27 divisions they break up entirely into the small air sacs in which gas exchange occurs so perfusion of your ways and of course they start as a single trachea and become extremely numerous if you actually cut across a long and inflated long you'll say that it has a texture that's somewhat like a sponge now the scale of this is half a millimeter so you know there's 25 millimeters in an inch so we're looking at a fiftieth of an inch there so these are really quite small and many of these are alveolar ducts and the very smallest one's barely visible at this magnification our alveoli and the airways themselves as they divide more and more become embedded in the lung part of the substance itself and these Airways are extremely flexible and gets smaller and larger as the lung expands and contracts with each breath very small stairways are truly embedded in the lung and become indistinguishable from the alveoli themselves so this is a scanning electron micrograph and now we're looking at 50 millionth of an meter as the scale 50 microns very very small and this is the the where the airway the respiratory bronchials sup into an alveolar duct and then becomes the little air sacs alveoli where gas exchange actually occurs so they they literally divide so many times that they have faced themselves and become these little air sacs well if one thinks about this kind of in terms of the surface areas involved with Airways as you begin at the trachea and divide progressively I'm plotting here airway generation that is numbers of divisions against the total cross-sectional area of all the Airways so you start with just one little trachea then as it divides and divides it never really cuts in half it cuts maybe in two branches that are two thirds that size so the total surface area augments with each division and as you get up to about ten your way divisions then the surface area really takes off and becomes extremely large the cross-sectional surface area of the many Airways this the airway size becomes smaller but the number so overwhelms the size that the surface area gets larger the trans trans cross-sectional surface area so air moving it moving down the first Airways the larger Airways moves pretty rapidly but as it begins to approach Airways of the giant cross-sectional area it becomes progressively slower as it moves and then as it gets past about 15 airway divisions it hardly moves at all it almost moves by diffusion it is still refreshing the most distant Airways in the alveoli but it's not moving rapidly in and out with each breath it's moving rapidly in and out in these Airways so a consequence of that and this is important in asthma which I'm going to talk about in a little while is that as the airway generation gets more and more numerous as the cross-sectional area gets larger there's very little resistance to airflow not much flow in terms of velocity whereas in the upper Airways like your trachea and your left and right mainstem bronchus and the larger Airways more centrally in your lung those Airways have a fair amount of airway resistance and move air in and out of your lung there's some frictional work that you have to overcome to move that air in and out so most of that frictional work that airway resistance resides in your bigger Airways or smaller Airways contribute to it very little so that asthma and other disorders that constrict your Airways and make that resistance worse so that you feel as if it's difficult to take a breath those occur largely in the larger Airways the smaller Airways don't participate very much and that difficulty but I'll have a lot more to say about that a bit later so if you go back to what the actually looks like and I'm going to shuttle between kind of diagrams that make a point or you know schematic representations and the real thing as it exists in nature so this is a the real thing it's a photo micrograph of alveoli and capillaries so this is an alveolar sac cut across and in its wall are small little capillaries that is small blood vessels that contain red cells and this is where gas exchange actually occurs now I've shown you some scanning black-and-white electron micrographs that showed you those sacks this is what it looks and like in the most common kinds of pathologic specimens that we deal with in medicine all the time so little air sacs about 200 microns across very thin walls lacy really with red cells coursing around through this network Avira sacks picking up oxygen and dumping carbon dioxide so so far I've talked about the Airways and how they gradually divide become eventually tiny little alveoli which gas exchange occurs but of course there's that other organ that's necessary for the function of the lung the heart which is main function is probably symbolized by the fact that it's in between the two lungs it's well situated to service the lung and venous blood from the rest of your body which is enriched with carbon dioxide because of cellular metabolism courses back to the the heart that blood is also depleted of oxygen because it's been used up in cellular metabolism so venous blood comes from all portions of the body to the heart again I've lifted this illustration from your book and it comes to the right side of the heart the right ventricle and is pumped out through the pulmonary artery to the lungs and so there's a main pulmonary artery here and I left and right pulmonary artery and those of course break up in a similar fashion arbor izing sort of like bronchial tree does the tiny little blood vessels eventually that surround those capillaries so that venous blood depleted of oxygen but too much carbon dioxide in it goes out to the lung dumps its carbon dioxide picks up oxygen and returns to the heart through pulmonary veins and behind the front part of the heart is the left ventricle which then pumps all of that oxygenated blood out to your tissues with every beat every beat has about a fifty or fifty-five percent of the volume of the blood and the heart is ejected with each beat normally the heart pumps about five or six liters of blood containing this oxygen out to your tissues if you exercise it can get up to roughly 20 liters of blood flow so one thing I want to point out here is that the lung here these are cut ribs it resides in a cavity that cavity is called the pleural space and the lung actually is lying loose in that and it's only attached right here on each side or which we call the hilum that's where their Airways and the arteries and veins enter the lung and the rest of the lung is not really attached around the edges it's inflated maximally and fills the space but isn't attached and so really in a sense the lung is hanging from the hilum when you're upright just from that one spot but it's fully expanded into the space around it so to get to the business end again where the gas exchange occurs and then try and give you some feeling slightly more quantitatively for what actually happens so each alveolus is wrapped by very small blood vessels the capillaries which transport the carbon dioxide to the alveolus to be removed and the oxygen away from the alveolus to be consumed the space across which it occurs is very small to about half a micron I'll give you some feel for that and some other photographs in a moment now venous blood that's returning from the body and pumped eventually into the capillary bed has a certain amount of carbon dioxide in it you'll notice I have a number of 46 that number is the partial pressure of carbon dioxide now gases when we measure their concentration we do them with reference to barometric pressure so there's at sea level where we were pretty close to right now this is roughly 760 millimeters of mercury of barometric pressure when they're storms is slightly lower and when there's highs and the meteorologist points out to you it's slightly higher but it's in that range 21% of the atmosphere is oxygen so 21 of that 7 60 millimeters of mercury of barometric pressures occupied by oxygen molecules so when you inhale a breath you're inhaling mostly nitrogen that you're inhaling 21% oxygen now remember I said that when gas enters your nose it's wholly humidified that actually adds 47 millimeters of mercury of water vapor to the gaseous mixture so that dilutes out slightly the amount of oxygen that comes down in jarabulus and because you're dumping carbon dioxide in Doral feels it also dilutes out the oxygen a little bit more the pressure and the overall pressure in the alveolus is atmospheric it's that 760 the amount of of gas in that alveus the partial pressure of oxygen is about a hundred and four so it's 760 minus the water vapor times 0.2 one the amount of oxygen that's in the environment and then diluted out some by the carbon dioxide so there's about a level of a hundred and four partial pressure of oxygen in your alveolus oxygen returning in the blood fluid from your your veins is to pleat it down to a level of about 40 as it makes the circuit and equilibrates by diffusion with the oxygen in your alveolus it rises to roughly a hundred not quite the hundred and four across the membrane but pretty close and the carbon dioxide started at forty six or so maybe fifty depending on how much activity you're doing and it equilibrate to the level of very close to the 40 that's in the alveolus so you have d saturated blood coming in that his blood doesn't have enough oxygen too much carbon dioxide makes the route it diffuses across a very tiny membrane and leaves approaching the same gas mixture that's in the alveolus and that's how we do gas exchange so I promise to show you the tiny little barrier across which diffusion occurs this is an actual electron micrograph a higher power view of barrier so this is the capillary it's got a lot of fluid plasma in it it has these red blood cells cut across and these are all in the capillary and so you can see the wall of that alveolar capillary interface it's not very thick but it's mostly capillary so this is all capillary this is all blood vessel and then there's these very thin pancake lining cells type one numa sites so yes diffusion occurs right across that little membrane it can be extremely narrow in some areas and the blood leaving the capillaries looks very much like the alveolar partial pressure of gases so another point that is not intuitively obvious is the function of red blood cells is to transport oxygen oxygen and carbon dioxide most of the partial pressure of oxygen and carbon dioxide that were measuring the actual content of that blood in terms of the oxygen and the carbon dioxide is all attached to red cells there's some dissolved in the fluid the plasma but that doesn't amount really to anything substantive it's all attached to the red blood cells so they're the transport mechanism so this brings me to hemoglobin hemoglobin is the compound that fills those red blood cells it's a complex compound the first thing to say about it is it's about 250 million hemoglobin molecules in every red cell that's coursing through in those walls and the compound itself has four chains of polypeptides here - blue - red those are strings of amino acids and they fit together with these things in green called porphyrin rings porphyrin rings are the actual site that oxygen binds to on the hemoglobin and if I could show it to you in 3d you would see that when oxygen binds to the in rings actually the shape of the hemoglobin molecule changes a little bit it accommodates it and of course these porphyrin rings also require iron which is why anemic patients that is patients without enough hemoglobin require iron in their diet produce more hemoglobin so each gram of hemoglobin is can carry about one point three six CCS or milliliters of oxygen so it's a very efficient mechanism one point three six milliliters of oxygen per gram means that in every hundred cc's of blood most of which of course is just fluid not hemoglobin but in every hundred cc's of blood there's about 20 milliliters of oxygen so this is a very efficient compound floating around in your red cells packed in it's the entire content of red cells and all it does is bind our oxygen and carbon dioxide and move it around so I've shown you this before and pretty soon I'm going to get out of this sort of light math and go back to pictures that are more intuitive I've shown you this before but I want to make the point that oxygen dissolved in blood rather than attached that oxygen that's attached to the hemoglobin really doesn't contribute much to blood content it's all the amount that is attached to the hemoglobin so that raises the question and of how does it attach or what is that light quantitatively so there's something called the oxygen hemoglobin dissociation curve on the bottom I'm plotting those po2 s that are in the alveolar space that's the concentration of oxygen in the alveoli and then on this axis the y-axis I'm plotting the saturation of blood that is how much of the blood is carrying compared to the amount that it could carry so normally I told you that the alveolar po2 the content of the alveolar gas has a light level of about a hundred mm-hmm and so it totally saturates all of the hemoglobin the good news is that even if your lung doesn't work perfectly and you only have a level in your blood of about 55 or 60 you're still above 90 percent saturated in your hemoglobin so even if your gas exchange isn't perfect if you achieve a level of about 55 or 60 then mostly you've saturated your hemoglobin and you have plenty of oxygen being pumped out to your periphery so there are clinical devices that we use in medicine that measure very simply how much your blood is saturated with oxygen these are called oximeters they do it by colorimetric lights by bouncing a light beam off of your fingertip and they just look at your hemoglobin and tell us how much saturated it is there are very simple little devices they cost about a hundred bucks here is one you put your finger in it as in the reading it starts counting your pulse and then after about thirty seconds or so maybe a little bit less it tells me that my saturation is 98% and I'm going to pass these around I'm not really looking for business better but if your saturation is very low hide it from your neighbor and see me afterwards so low is when your hemoglobin is saturated less than 90 percent why am i defining at that level because long-term studies in patients that have oxygens hemoglobin saturations below 90% don't live as long they have more hospitalizations and it shortens their life and this is the threshold for use of oxygen as a medicinal agent and is what medicare recognizes as threshold and is commonly accepted in clinical medicine so if your saturation falls below 90 and it's habitual not just extremely episodic and you definitely need oxygen if it's episodic it probably occurs when you exercise and you should use oxygen when you exercise yes yes well that's interesting question so because oxygen binds to hemoglobin one point three six milliliters per gram of hemoglobin if you have less hemoglobin you have less arterial oxygen content usually there's an adaptive response and your heart pumps more blood around to make up for it partially so it mitigates the anemia but at a certain level it can't keep up with it and there are no long term studies in humans showing that at any particular level of hemoglobin that you should be given oxygen there are some studies that have looked at whether you should be given more hemoglobin a blood transfusion those indicate that normally when your hemoglobin is reduced by about half because in 100 cc's of blood you usually have 14 or 15 grams of hemoglobin if you could get your hemoglobin down to seven grams of hemoglobin that a transfusion is warranted but there are no studies to say that if your heme Elvin is six you should be on oxygen and it doesn't actually affect the arterial po2 the amount dissolved in the liquid it does affect the amount of oxygen in your blood that's being transported yes no and I'll come to that in a moment on average they they're about no the question was sorry yes I'll do that routinely the question was are all the alveoli the same size and shape they vary somewhat in shape they vary somewhat more in size and fashion that I'll show in a moment in general there are about 200 microns across and red blood cells about 8 microns across so they're about 25 times as wide as the red sauce yes it also binds to oh repeat the question question was short memory low oxygen the question was what carries the co2 carboxyhemoglobin it binds to hemoglobin as well I didn't show that dissociation curve there's a lot that's interesting and more sophisticated as you can imagine about that dissociation curve for example if you're febrile if you have a fever and you need more oxygen because you have more oxygen consumption when you have a fever then it shifts to the right it makes delivery that is unbinding of the oxygen from the hemoglobin to your tissues much easier and then if your pH is too low invariant when people are very sick their pH sometimes drops it also unloads the oxygen more readily film yes so the Dean is pointing out that there's this long bank as I said I think above a po2 of about 55 your night you're more than 90 percent saturated and so although we normally most of you in the audience probably of the po2 between 90 and 100 and your oxygen saturations probably 95 to 100 but even if you fell down to a po2 of 60 it'd still be above 90 percent saturated so it does provide some reserve but of course when you start falling this way there's a very steep fall so you have a lot of reserve you do fairly well to your po2 is somewhere around 55 then you really start dropping how much oxygen you're transporting to your body and I'll come back to some of that as well okay so just to look once again these are alveolar capillary beds wrapped around alveoli the capillaries wrapped around the alveoli they're about 300 million alveoli and their diminutive size of course increases vastly the surface area compared to this it compared to it if it comparing it to if it were a single sac the total volume of both of our lungs is about 5 to 7 liters so let's say somewhere shy of 2 gallons but the total surface area of the alveolar space is about 80 square meters not feet meters so the size of let's say a tennis court and if it were the size of a single sac surface area would be less than a square meter so this I think underlines what I said in the beginning the design characteristics of the lung lots of surface area and a very small surface a very small volume okay I mentioned before again using the picture from your text that the lung is attached mainly at the hilum where the vessels in the Airways in red and otherwise it's kind laying kind of loose in the pleural space there are no muscles to ventilate the lung within the lawn the only muscles in the lung are in the if airway walls and so they can constrict your Airways but there are no muscles in the lung itself that helps you ventilate move air in and out it's all external to the lung it's in the chest wall and the diaphragm and now I'm going to try and explain that a bit so if you do something rather grisly cut somebody in half like this hopefully at autopsy then what you see is a lawn resting in a pleural space attached only in the hilum on both sides and here you can see the lungs pulled away a little bit from the interior of the chest wall there's a potential space between the lung and the chest wall called the pleural space normally the lung is fully expanded and up against the chest wall but in pathological States it might fall away from it what really is happening is that the lung is very dispensable and it expands into a vacuum behind a that surrounds it the vacuum around it is in the pleural space so the pressure within alveoli is atmospheric because it connects through the Airways to the atmosphere but the pressure around the lung is sub-atmospheric it's a vacuum and the way the lung is ventilated primarily is by manipulation of that vacuum around the lung so our respiratory muscles that reside in our chest walls move our ribs out to create more of a vacuum around the lung and the lung moves into the vacuum that's how we take a breath and then when we relax our chest contour falls back the vacuum is not as intense and the lung retreats so it's just responding to the vacuum around it now besides the muscles in the chest wall there's also the diaphragm a muscle at the bottom of the lungs when you take a breath that descends in that direction when it descends it creates more of a vacuum around the lung the lung moves into the space so the diaphragms are very powerful muscles for ventilation the muscles in the chest wall are less powerful but it's still important so if we think of this schematically and this is a little bit shocking to think of that lung doesn't actually ventilate itself it's not really a muscular organ what really is the muscular organs or the diaphragms and the muscles in the chest wall and the force inflates the lung is is the pressure across it it's the difference between barometric pressure and alveoli and the vacuum and the pleural space now I should have pointed out one other thing so the one is residing in the pleural space the hearts in between with the major bronchi that are just beginning to divide and the esophagus runs behind Moo coursing down to the stomach these are three separate compartments their airs they're airtight there's no movement of air between them only through the airways but this vacuum of course has to be airtight or it'll is this vacuum and humans are quite lucky that we have two separate pleural spaces that don't connect because penetrating injuries through the chest wall that puncture the lung and release air into the vacuum ruin the vacuum and the lung collapses and thankfully that we have two lungs of one lung collapses we can ventilate on the other until we can repair the first this is how Ronald Reagan's survived being shot in the chest he dropped one lung because the vacuum was destroyed by the penetrating bullet let air leak out of his lung it's also but his other lung got him to the hospital where I understand he asked if the surgeon was a Republican I'm coming to that great the question was what happens in an asthma attack we're getting to this because the Dean asked me to talk about wheezing and gasping for breath okay another interesting feature is not all animals have sealed pleural spaces that don't communicate between both so probably the reason that buffalos are largely extinct is because they have one pleural space so when they were shot by a Buffalo hunter and a penetrating injury entered one pleural space it ruined the vacuum for both lungs and the animal promptly suffocated or a bow and arrow so count your plural spaces and be blessed so if you can think about this schematically pretty soon I promised you all go back to pictures of the real world as the one being attached to the airway and residing in a closed space here a bottle so if you take a pump and suck air out of the space out of the bottle the lung consents it can get through air yet to air will just inflate and fill in the vacuum that's what's happening is that your muscles in essence are changing the pressure around the lung the lung keeps moving in and out as that pressure changes and you can measure this and figure out how the sensible the lung actually is by connecting a measuring device to it and you can plot how you drop the pressure around the lung the lung volume goes up yes well I'm not sure we'd have to see the design the question was what was wrong what would be wrong with muscles being attached directly to the lung wall I guess we'd have to envision some system in which the lung was harder than it is you know had more ability to be directly pulled upon I mean as it is it kind of passively moves in and out for protected space enclosed by a membrane the pleura that's moist and it has to do this of course every breath of your life through many years so there's no articular surfaces like in joints that wear out I'm not sure why that the wisdom is the way it is but it definitely appears to work I can imagine what other design methods but this is one that seems to have little wear and tear yes and most vertebrates they work similarly when you get to in vertebrates it becomes much difficult much more much different like for example gills but this is true for dogs cats horses rats mice elephants okay so how do these respiratory muscles work I've introduced it already when you're in expiration these are the ribs there's muscles between them and when you take a breath those muscles elevate your breastbone your sternum and they increase the front-to-rear dimension of your chest so they make the volume bigger when they do that also your diaphragm descends pulls down so there's two things increasing the volume of your chest front to back and having your diaphragm descend or what create the vacuum around your one now actually when you exhale you don't normally have to use any muscular activity at all you store the pressure that you generate the force in your expanded chest and your decended diaphragm by doing muscular activity then when you relax they just recoil to their prior position so most exhalation is passive however if you're ventilating very rapidly then you can do active exhalation as well and compress your chest wall and make you exhale more rapidly so you can ventilate more rapidly so now I'm going to show some more clinical images even up how some of these things look to show you that how these things play out in clinical medicine to some degree and give you a feel for what they actually look like so this is a CT scan the last slide was I borrowed from Jeff Rubens presentation of CT scanning showing a scanner and how people are moved through that scanner to put somebody through scanner and you collect images of how the radioactivity of radiation was absorbed then you can use those create any kind of image you want out of that data bank so you can create cross-sectional cuts of the lung to look at the lung or you can look at the heart you can look at the bones or whatever you'd like so if you look at the bones it looks like this the sternum the bones the heart behind it I've left out the lungs here to give you some feeling for the way the ribcage actually works this happens to be my rib cage when I was in a bicycle wreck I had a CT scan and this is the 3d reconstruction of that looks a little scalable if he has moving and I had some fractured ribs so here's a jagged rib fracture capable of puncturing the lung and in a normal CT your lungs are all the way out this is not me so I'll be ahead of normal CT scan lines all the way out to the periphery all of this material is long and you can see an airway cut across and blood vessels and the heart and the mediastinum and it's separate compartment and these two compartments for the lung fully expanded this is my lung after I punctured it with a fractured rib so this is what happens in any penetrating injury and of course when you have something like that in your lung is partially collapsed thankfully a little on your other alarm but you have to do something to return that vacuum so that your lung can re-expand and doctors aren't always the most clever people so they do something pretty crude they put a tube in make a hole in your chest wall stick a little tube in there and hook up to a vacuum suck out the air re-establish the vacuum it's called chest tube that's how you get your lung to re-expand again so we do this all the time and people that have for example surgeries on their long with their pleural space is violated lose the vacuum comes ambient pressure its ticket just to Bennett look at this suction get the lung back up now I also want to point out if I have a couple of times that your lungs hang from their attachments right there in the hollow so I'm coming back to the question someone asked about our D alveoli always the same dimension and shape and they're not exactly the same shape but they vary a lot and dimension so at the top of your lung the alveoli are actually quite distended and at the bottom of the lung during particularly exhalation they're quite collapsed and when you take a breath this doesn't expand very much but this expands a lot so when you take a breath most of the air goes to the bottom of your lung and why is that it's because the lung is hanging from the hilum the top of it is moving into the vacuum but there's nothing weighing down on at the bottom of the lime you've got the whole weight of the lung lying on the one below it and so this is relatively compressed now what I'm really saying or another way of saying it is that the pressure around the lung is more negative up here than it is down there and that's true and we know that that's due to gravity because John West a physiologist at UCSD took some hapless medical students but chest tubes in them this is before we treated medical students well and dr. pizza wasn't the Dean then and I was UCSD anyway he put them in YouTube planes and made them weightless and then measured the pleural pressure gradient and there wasn't any so that demonstrated that the real reason at the top of the lung which we'd known for a long time is more expanded than the bottom is because of the effects of gravity and it's due to the pore pressure gradient so far I've talked a lot about loops yes yes they can in fact up the question is can you survive with a portion of your lung removed and the answer is you can survive with only one of your two lungs and for example when Ronald Reagan was shot he had a pneumonectomy he had one of his lungs removed and more commonly people will develop lung cancer and have a portion of their lung removed or even a whole lung removed and it impairs you but you have enough reserve so that you can get by usually without oxygen you won't be an Olympic skier but you can play a you know mild to moderate set of tennis with one one okay now I'm going to talk about oops yes no penetrating injuries the question is when you say puncture the lung do you really mean punctured or do you mean the pleural space is entered and the answer is if you make a hole in the chest wall and let air in from barometric pressure in your pleural space your lung will collapse even if you do nothing to DeLonge but sometimes what happens is in a penetrating injury or a rib fracture or something else is that there's a puncture that goes through the chest wall into the lung tears the lining of the lung and the lung leaks there then even if you sealed the chest wall you have a continual leak from your lung into the pleural space which is ruining the in the vacuum and so that also causes a lung collapse so either one opening the pleural space to barometric pressure by a hole in your chest a lot of holes that their small seal up muscles just move around them but if you puncture the lung the leaked internal here you usually just have to let it heal and if it has a persistent leak and you can't then when it doesn't heal then you have to an operation and over so the terror of the lung yes good question the question is if you can live without one lung then I think you're implying maybe a single one might work and a lung transplant yes this is for other people to donate to you so let me answer both questions my question in your question so in certain kinds of lung disease like emphysema the most common kind of lung transplants a single one transplant that is you can get by with one lung so if you have one good lung it works well and the other lung is in feminists it doesn't matter you're so much better than you were before the single lung transplant now the other question is that our donors with lung sort of like donors with kidneys we you know we many living related donors give away a kidney to their loved one or even someone they don't know the the only case in which that occurs in lung transplants is parents to children when they don't have to give the whole lung only a lobe of the one because it does it does confer some disability to only have one one to the donor which a single kidney having a single kidney does not confer any disability on the donor and they have a normal lifespan so we don't do whole lung living related donor transplants but rarely very rarely a single lobe that's given to a child the trouble is crystal grow out of it but then question is you know you're buying them some number of years yes it is the question is is the saturation for adults and children threshold for needing oxygen the same and it is it's about 90 percent saturated okay I'm going to move on to some remember I said first we're gonna talk about the structure and now I'm going to talk about the defense of the lung from things in the environment which will play into other issues like how we acquire diseases so normally I've told you that the epiglottis is a little flap right here that keeps food and saliva from your mouth out of your lung and it's part of the larynx and this is a picture of an actual larynx with two vocal cords these vocal cords come together and you blow air through them you phone a door make noises music or speech and this little flap the epiglottis can flop over and keep food out of it so that you protect your airway people that have strokes and have muscle weakness of the muscles of their lyrics have trouble with that and aspirate secretions and the lung get repetitive pneumonia so I'm going to show you now an actual larynx this is a little tube you're looking at from the inside through the mouth all the way down to the larynx to allow a look at the larynx we're going down the tube and here's the end of the tube and there's the learns and this lyrics oops this larynx is a little unusual because it has a polyp which is why they're undergoing the procedure here our vocal cords the flap of the epiglottis arytenoid and it's right there behind your Adam's apple and you can see it's pretty vascular there's blood vessels and as you go through you plunge into the trachea itself and this trachea has some polyps in it that's why they're undergoing the procedure but you're right in the airway with the cartilage in the airway and I'm going to show you in a little bit a normal foreign cost pieces you can see this in more detail so it's also true I've said that the epiglottis sometimes lays over your airway and protects it this is a little video of an epiglottis doing that you can see there's some air the vocal cords are back in there bubbling and the epiglottis is closing it off so you do this all day long every time you swallow your epiglottis flops back you protect your airway food goes down the right tube okay so the first line of defense of these yes question Sassy's ring is actually in the esophagus and so it's in the other tube behind your larynx right it's partially down that's the tube that connects your mouth to your stomach yeah so the question was can you explain a Satsuki ring and Schatz keyring is a little ring that sometimes occurs peculiarly in people with iron deficiency further down in the esophagus yes yes so the question is what happens if food goes down the wrong tube you start coughing a lot so if you aspirate a grape or a peanut children do this a lot they coughed a lot or adults it's much more commonly pieces of dental work and not infrequently we get a call from a dentist terrified that a piece of dental work went down the wrong - they lost control of it so you coff coff coff and you attempt to expel it out of your larynx back into the your throat and then usually you don't expel it into your mouth it's too far down below and you swallow it so if that were a tooth that was lost that's great you swallow will come out the other end but if it's a tooth that's lodged in your lung and you can't get it out then you have to take a bronchoscope a flexible fiber-optic instrument go down into your lung and fish it out and we all have lots of stories about things we've fished out you know I'll tell you briefly two of mine one of them was a Stanford student he was using a blow gun and put it up to his mouth and inhaled the next time so he had a metal blow dart down his life far another one is a carpenter on the roof that his mouth full of nails cause he was milling somebody came by and slapped him on the back pretty bad it's like fished out you know the roofing nail or the most elegant one was a person on a motorcycle with a tie tack and the wind blew it it flipped up hiked aquent when the chest x-ray you could see his initials and his gold typing all of those were fished out of the bronchoscope okay we're moving on on defenses so in addition to these gross defenses you also have in the lining of your lung these little things called cilia these hairlike projections very fine projections this is the wall of an airway these are the cells that line your way these fine little projections that move in unison and move small particulate matter and toxins that you might inhale out of your lung back into your throat so you can swallow them and they move all day long they're very amazing organism or organelles so this is the van scanning electron micrograph and they move in kind of waves to move mucus out of your lung always being produced in your lung and with the mucus particulate matter so that's another level of defense if you absorb if you inhale for example microorganisms that might otherwise cause an infection they're usually moved out by what we call the mucosa ciliary escalator and this is a very important function if you have anything wrong with your cilia you will get repetitive infections so there is such a syndrome a experiment of nature called the dissolute Ilia syndrome if you cut across those cilia they have an elaborate apparatus and the centromeres that are called there's always nine doublets and two singlets some people have some abnormality of their cilia it doesn't beat normally all of those patients have problems silly or not confined your Airways they're also in the linings of fallopian tubes and so that eggs from your oval end up in your uterus where they belong and they're also on sperm so patients that have problems with their cilia get lung infections cilia line your sinuses so they get chronic sinusitis and they tend to be infertile and in addition they're in embryology the normal rotation of organs is dependent upon cilia and patients have what we call situs inversus sister Cecilia don't work their organs are on the wrong side so there are parts on the right so the left their livers on the left so the right and so fili are very elegant about very important for lung defense now mucus in the lung itself is produced by these cells called Clara cells or goblet cells and they're interspersed with the cells that have those little hair like protrusions the cilia so they're always producing a material that lines your Airways that's the protective coating of your lung and if you look in glands little pits in your Airways where these secretions are formed you'll find both ciliated cells and these cells that have this little bulbous element those cells are the cells that are producing mucus continually it turns out that if you put a little if you inhale some fine dust and it goes into your lungs it clears up a half-life of about two hours so the mucociliary escalator brings it up you don't notice it if you swallow it we're doing this all day long some patients have trouble with this apparatus and in particular cystic fibrosis the devastating disease course its genetic its occurs in about one the care years are about 1 in 25 white people there's less common in non-caucasians races but so if it's 1 in 25 in the general Caucasian population and you marry a Caucasian and your chances are one in 25 times 1 in 25 or one in 625 that you'll marry somebody then you're both carrying the gene and if your child gets a double dose of the gene which occurs one time and four then that child has cystic fibrosis so one in 2500 live births are Caucasians the United States have cystic fibrosis and those common genetic fatal disease and how does it kill you eventually thankfully a lot later than it used to now they have adjust lifespans about 35 it kills you by the mucociliary escalator not working very well and that in particular so there is a particular genetic defect in chromosome 7 that regulates how fluid and ions mainly ions are pumped out of that ciliary and mucus escalator so that in cystic fibrosis is too much of a water and the salt is resorbed and the liquid lining level is too thick doesn't move normally and you can't clear organisms and so you get repetitive infections and eventually these cause serious airway damage and entry and usually death around the age of 35 and then the last the last defense if it gets beyond your cilia because it only gets the cilia only go down to the terminal bronchioles there's no silly in your alveoli if the particles so small it gets beyond your cilia then your last defense is a cell that lives in the alveolar macrophages there's a 3d version of it these ingest microorganisms and toxins try to protect you from things that get beyond mucociliary escalator okay so I'm going to summarize the structure of the lawn it provides a giant surface area about 85 square meters it's provided by 200 and 300 alveoli we've talked about how blood carries oxygen and carbon dioxide back and forth and the lung hangs within a closed pleural space and it's kept inflated by the lung around it I've by the vacuum around it and respiratory muscles influence that vacuum and then you have these anatomic and biologic defenses and I'm going to talk to some about function and I'll be relatively brief about this and then we're going to talk about diseases of the line so when you're inflating your lawn what are the forces that have to be overcome to inflate the lawn I told you create a vacuum around it but the lung itself has some elasticity so you stretch the lung you have to overcome that there's elastic forces but more importantly because there is that lit our liquid interface there is a fine coating of fluid in all of your spaces your alveoli and your Airways there is something called surface tension that occurs at that air liquid interface and you're used to surface tension in other settings surface tension is what's responsible if you wax a car and your water that and you throw some water on the card all the water beads up it doesn't spread out in the thin mono layer that's surface tension and it turns out that two-thirds of the work of breathing does not actually due to stretching the lung it's due to overcoming surface tension what is surface tension it's due to the attraction of water molecules and this schematic alveolus so they're attracted to each other and because they're not on a flat surface there's a net vector that moves towards the center so you're attracted to each other as they do that they ball up they round up that's why water rounds up on your waxed car when you dump it on the car it all beads up it's beating up because of the attraction of water molecules and intermolecular forces so that's the main thing you have to overcome why do I weigh my yes yeah so that's the question is why do why do people feel more comfortable when they're breathing highly humidified air rather than dry air if surfaced instance the main issue whether or not you read dry air or wet air doesn't matter by the time the air gets to your lung by the time it gets to your lung it's fully humidified because as long as your nose breathing if your mouth breathing you ventilate very rapidly you might be able to get some dry air in but for ordinary circumstances it's just more comfortable in your upper airway really doesn't influence your surface tension at all okay so why am I talking about surface tension because we have an elegant mechanism to reduce surface tension to allow us to breathe more easily and that is we have cells called type 2 cells that secrete a substance into that liquid in all of your alveoli that diminish the surface tension and in the absence of that it's too hard to take a breath and you asphyxiate from being exhausted this occurs early in childhood it's called infant respiratory distress syndrome primarily occurs in premature babies who haven't had full maturation of the system don't produce enough surfactant their surface tension is too high in their lung they can't breathe and for those of you that are near my age if you'll remember that during the Kennedy years Jacqueline Kennedy had a child Patrick Bouvier Kennedy who died of infant respiratory distress syndrome after 39 hours in the White House as a national tragedy and because that child died because was a premature child that didn't have surfactant production in the modern area we have artificial surfactant and in neonatal ICU like Stanford's if someone's born prematurely we dump surfactant into their airway spreads around on their lung and they can breathe more easily so it's it's an interesting system now I'm going to talk about air moving in and out of your lung because we're warming up for talking about asthma and COPD because that's the problem in asthma difficult to move here in and out so to ventilate you have to overcome some of the silastic work due to the tensile elements in your lung plus the surface tension of referred to but you also have to move air in and out that causes resistance work to be done because air has friction as it moves through your Airways so if you think about pattern of air flow and tubes in some tubes it might be very orderly which we call laminar air flow and in other tubes it might be turbulent and then in all tubes that branch there's a transition point where it's turbulent right at the transition point this is a vast oversimplification because Airways are not rigid tubes and I'll come to that too but for the first type laminar air flow the resistance that's encountered as you move air through it is exquisitely sensitive to the radius of the size of the tube so in particular resistance is inversely related to the fourth power of the radius so very small changes and the radius of the airway cause large changes and how much resistance the patient experiences as they breathe in or out so this is why modest amounts of bronchospasm are deeply felt and remember that I said that most airway resistance is in the larger Airways which is where bronchospasm occurs and for turbulent flow it's a kind of orderly but I'm sorry a disordered chaotic airflow and it's mainly occurs during rapid breathing as an exercise and so when your exercise not only you have to move more air but your moving air that's turbulent as it moves through your Airways because you're moving at a high velocity and resistance is proportional to the velocity of the air movement so exercise high levels of exercise have relatively more work than just the amount of additional ventilation but as I said real Airways are not rigid tubes and they look a lot more like this so this is the view bronchoscope looking down there's a soft membranous portion which does not have the cartilaginous armoring you can see what looks like cardio gated steel or something corrugated tubing those are a cartilaginous plates while looking down the trachea this is the right mainstem this is the left mainstem and I'll give you a little feel for what it looks like as you move down so right there moving down the right mainstem looking up in the right upper lobe there's three divisions and then there's the middle lobe right in here with medial and lateral segments and we're looking down the bottom and you can see the glistening of the secretions and a little pit there where mucus is made right there and so this is a very healthy looking airway in a normal individual a little bit of mucus stranding being moved up by the mucociliary escalator look down the other side this is the left mainstem working our way down there's the Airways to the left upper lobe you can see vessels and the walls and again the glistening of that thin layer of secretions so I don't want to pretend that they're rigid tubes but it is true when you look at these areas that you can imagine that if there was some airway constriction right here you'd have a lot of trouble moving air and out of that airway and that's what happens and people have bronchospasm okay so now I'm going to talk about symptoms and diseases dis Nia is the medical term comes from Latin and Greek similar words for breathlessness so the most common form of breathlessness that we all experiences when we exercise I'm going to give you some feeling for how that occurs I'm plotting here the sensation of breathlessness against how much oxygen the subjects consuming and how much air they have to move in and out of their lungs so ventilations in liters per minute oxygen consumption and liters per minute and how breathless the patient feel so if you have normal ones normally it rescue you consume about 250 CCS of oxygen then as you begin to exercise you might move up to as much as 2 liters of oxygen consumption so Eightfold at that same time you're moving from about 5 liters of ventilation up to about 25 or 30 liters of ventilation so you're you know you're proportionally or ventilation is kind of tracking your oxygen consumption and you don't have to really shoot up to very high levels of ventilation until you get to about half maximal oxygen consumption above that your pH begins to change because you're using oxygen and the fashions not optimally efficient some of your muscles have some oxygen debt and you become slightly acidotic and then that stimulates your ventilation more and it really takes off and you can get up to about a hundred liters a minute of ventilation and a maximal oxygen consumption of about four liters for 16 fold higher than basal oxygen consumption so most people begin to feel breathlessness after they get above about 20 liters of ventilation so you can increase it six-fold or let's say four or five fold anyway without feeling breathless when you're exercising start getting in here begins to bother you so that's the most common cause of course of breathlessness but there are other causes - so altitudes one you all recognize so this is a shot of my Sun at 6,000 meters that I pick and it's Everest Makalu behind it and altitude obviously influences the sensation of breathlessness and there's a lot of been work that has been done in Nepal about altitude respiratory physiology this is from the New England Journal Medicine this this year so base camp is about 5,300 meters I go back down to London these were British climbers the barometric pressure in London 754 pretty close to my 700 see as you climb up to about 6,000 meters it's essentially halved and as you climb up to the summit it's about two-thirds of the barometric pressures disappeared and of course what this means is that the alveolar po2 is much lower because remember 21% of the atmosphere is oxygen but if you don't have much barometric pressure than the po2 and the alveolus is going to be quite low it's about 45 instead of 104 and your arterial po2 is about 43 so this is simply due to changes in barometric pressure and if you look at it again at our familiar oxygen hemoglobin dissociation curve this is 6000 meters po2 s 43 saturation is about seventy two or three percent so you're begin to be on that steep part of the curve and you can tell this when you exercise there at all you feel breathless you've lost about a quarter of the amount of your arterial oxygen content but when you get to the summit of Everest po2 is only about 20 to 22 in the alveolus and 19 to 21 on the summit that's near death levels which is why many climbers of course use oxygen but the most common cause of breathlessness is excessive work of breathing it's not a low po2 or oxygen saturation it's that it takes too much work to breathe that seems rather astonishing but I'm going to show you why that's true so there's two main kinds of mechanical problems with the lung that make it difficult to breathe one is that you have a restrictive disease it's due to a lack of normal distance ability of the lung so or lack of this density bility of the chest wall so if you have deformities of your chest wall and your chest wall can't move normally it's too much work to breathe and you feel breathless the more common one is obstructive diseases and that's when there's too much airway resistance so that moving air in and out of your lung is difficult and you feel that you're going to suffocate and most of the the most common disorders that cause breathlessness and are not related to your heart not functioning that is lung causes of it are obstructive diseases so now I'm going to talk about some of these so if we go back to my diagram of exercise and lung disease and discontent lung disease when you have injury to your lung gas exchange is not as efficient you don't have good matching of the blood and and the gas in your alveoli capillary interface and you begin to need to ventilate even at relative much more at low levels of oxygen consumption so you may be stiffer or you may be more obstructed and in addition you're much less you need much more ventilation to adequately oxygenate your blood so very high levels of ventilation are required for levels of oxygen consumption that are much lower so you have a high load and you also have abnormal mechanics and how you breathe so if this is a normal CT of the chest and again this is a cross-sectional cut blood vessel cut across the airway now they're cut across the airway heart in the middle the lung is kind of filling it up and you get out towards the periphery of the lung yet you mainly see kind of gray tissue I mean those are just alveoli cut across but if you have pulmonary fibrosis a scarring disease of the lung then your lung is very difficult to stretch it's not distensible scarring of the spaces between the alveoli causes the stiffness there's a variety of causes of it they can be genetic occupational or even dude all autoimmune diseases and it results in progressive respiratory muscle fatigue and destruction of the alveolar capillary interface so you have trouble moving air in and out because your lungs stiff and you have inefficient gas transfer and the mean survival is only two to five years after onset the disease so this is a devastating disease if you look again at our normal capillary bed that I showed you when we're talking about structure this is pulmonary fibrosis we're still have some alveoli but in between you have big bands of scar and this is a very inefficient way to try and transfer gas now the other main cause is obstructive yes question I think there is thank you I'm clearing my mucociliary escalator try to keep it pristine okay yeah so black lung is a term that was coined to describe the lung disease that coal miners had and it's a disease that causes some scarring of the lung but it also causes some destruction of lung tissue that is emphysema and really it's most the scarring of it is most common in coal miners have worked in coal that had hard coal in it soft bituminous coal city kinds of coal don't really cause much long injuries make your lung black and make you look bad but they don't really cause lung injury but hard coal does and but it's a mixed obstructive restrictive disease so asthma asthma is a disorder in which your Airways constrict due to inflammation in the walls of the airway that inflammation leads to the muscles in the airway wall narrowing the radius of the airway and part of this disorder is also mucus hyper secretions so in inflamed airways the cells that produce the mucous coat react by producing more mucus so no matter what the cause of the inflammation there's always more mucus production inflamed but those inflamed airways set these set the context for airway constriction this is a very common disorder in the u.s. the prevalence is about 6% in adults and 3 to 5 I'm sorry about 3 to 5% adults and 6% in children which is why you're used to hearing people say they quote grew out of asthma there actually is an asthma epidemic worldwide and some countries like Australia as many as thirty five percent of the population have asthma and in the United Kingdom in some areas it's 25 to 30 percent it seems to a bit increasing and the reasons for that are not entirely clear but they probably relate in some areas at least to environmental air pollution so I've said it's an inflammatory process this is a normal airway it has scanty amounts of muscle so you know photo micrograph this has a small airway like two millimeters across so it has small amounts of muscle but inflamed asthmatic Airways have a lot more muscle and constrict much more readily and I have on my handy-dandy iPod or iPhone an example I think it will come up I recorded it earlier here it comes we gonna do it so that's soft wheezing of small amounts of air whistling through tiny Airways in an asthmatic attack and literally people feel as if they're suffocating when this occurs now I asked McCann be allergic and when it's allergic it's usually due to white cells that are producing antibodies that react against things that you inhale and trigger other kinds of cells mast cells release substances that make your airway constrict but there are many insiders of asthma it's not just allergies the most common cause actually of an asthma attack is a viral infection and usually people experience this is I had a cold and it quote went into my chest and then we started wheezing so went into your chest you probably did have epithelium in your Airways infected with the organism but it sets off an asthma attack that's been much much longer so inhalation or ingestion of things that cause allergies like sulfites and wine strawberries peanut oil crustaceans those can also cause it toxic fumes as in air pollution like sulfur dioxide and ozone trigger asthma attacks and maybe less intuitively airway cooling for any reason causes asthma attacks so you can air wait you can cool your airway when you exercise if you mouth breathe and ventilate a lot ambient temperature starts cooling your airway when you're skiing in cold air or mouth breathing at night and the mouth breathing at night is why the most common time for asthma it's three or four o'clock in the morning so ambient temperatures in the house fall your mouth breathe you cool your airway and you have an asthma attack if you look around the world there are many causes of asthma and Eastern European and developing countries it's in its infections its parasites its urbanization and pollution and farming environments it's fungi and mold and exposed to microbial organisms and farming enterprise some kinds of industrial things cause asthma something called diisocyanate sin paint particularly car paint causes asthma very commonly so there's a whole variety of inciting problems with it so related to asthma but somewhat different this term that you've heard on television COPD so COPD is chronic obstructive pulmonary disease it's a obstructive disorder too much resistance to moving air in and out of your lung it's nor on normal airway wide-open and it's also kind of held open by the alveolar walls that radially tether it so it's tethered open and it's remember this smaller areas are quite small and COPD you have disrupted alveolar walls so it's not quite so tethered open you also have inflammation in the wall much as an asthma but worse mucus hyperkinetic recién & bronchoconstriction there's kind of two clinical types which I'll come to in a minute but the causation is largely in the United States cigarette smoking so cigarette smoking induces the macrophages that live in alveoli remember the mop-up cells and things get beyond the ciliated epithelium induces them to release various kinds of factors that make white cells that is neutrophils release things that did just part of your one and cause lots of inflammation and they reduce the release things that are known as proteases that chew up proteins and they actually make holes in your lung and in addition that causes a lot of inflammation in the airway lining cells so there's kind of two clinical spectrums and some people have some of each the term chronic bronchitis is defined rather ecumenically as a call for more than three months of the year for at least two years so cough means airway inflammation anybody that coughs all the time is having airway inflammation and these patients have wheezing sputum production and they can't exhale normally so they can't ventilate their alveoli as well and eventually they'd can't get rid of all of their carbon dioxide because they can't exhale normally they develop an abnormal chest wall that we refer to as barrel-chested they have too much inflation of their lungs because they can't exhale and get it out the other version besides chronic bronchitis is in fuzzy my emphysema doesn't have much golf associated with it it's just destruction of your line chewing holes in your lung by white cells stimulated by cigarette smoke and it causes a thin wasted appearance and some of these patients also have trouble exhaling and get somewhat barrel-chested and it progresses very reliably with continued smoking it doesn't have real flare-ups in the way that chronic bronchitis is chronic bronchitis more closely resembles asthma emphysema is just a progressive destruction of lung tissue so if we look at a chronic bronchitis here way I'm showing getting the normal airway and then we have this very inflamed airway has lots of muscle around it because it's always constricting and it has secretions in the airway emphysema instead of that nice homogeneous appearance to the lung tissue on a CT Scott you have these big holes all throughout the lung and that is the definition of an szema holes in your lung evident on the CT scan and it's matched by this picture of just progressive breathlessness and if you look at emphysema in a cross-section of the lung instead of a nice sponge like substance you have tons of holes in your life so not a good thing here's the microscopic view of normal alveoli here's an emphysema design where the alveolar walls are all destroyed when you do that you destroy the radial tethering of Airways and Airways partially collapse and can't move air in and out normally there are some causes of airway constriction that are not related to cigarette smoke and air pollution and asthma so there's some diseases that just diminish your central Airways but they're actually quite rare one example of this is an inflammatory disease of cartilage and remember I said in your old major Airways cartilage just resides and kind of keeps your Airways open so this is a disorder called relapsing polycon dryness in which all the cartilage throughout your body gets inflamed and it constricts your Airways there's a patient that has Airways in their lung much bigger than their trachea and she was suffocating because of an obstructive disease I put three stents in here metal stents expanded them and held her Airways open and you look down in her airway you can see sort of a quilt from that stent that's in the airway and this immediately relieved the symptoms and she's 15 years out there are ways still paitent but for the vast majority of people that have instructor disease it's COPD it's progressive if you continue to smoke about 14 million citizens have it and interesting there is a genetic component to it about 14 or 15 percent of Caucasian smokers have airflow obstruction compared with about 3 percent of non-smokers those non-smokers all have asthma if they had early obstruction and if you compare that the asian smokers three times as common in Caucasians as Asians and the World Health Organization is now predicting because of cigarette smoking in the third world that it will rise the sixth most common worldwide cause of death to the third and another decade so this is truly an epidemic and in case you think it's only in China these are the statistics for Santa Clara County there's about 38,000 cases of pediatric asthma and a hundred thousand cases of adult asthma and a population of 1.7 million but there's 44 thousand cases of chronic bronchitis and 20 thousand cases of emphysema so this is a very common disorder very common killer in the United States so in some ways you can begin to think of lung diseases as things that are due to our exposure to the environment and hailing things or things that are intrinsic to the lung like malignant diseases or genetic disorders or diseases of the blood vessels but a major cause of morbidity is ambient air pollution not so much in America but in the third world but even in America we have unhealthy air in many regions so this is the EPA's counties designated as not attaining the standards of the Clean Air Act and you can see that it's along the eastern seaboard and it's in California and in the Central Valley and if you look more closely oops I didn't mean to do that I shouldn't have done that No maybe I can sort of reboot it no don't restore it I'm afraid it went to its home in the EPA okay if you look at our local area ozone standards in California are exceeded in these regions non-attainment areas doesn't show wellness well it shows better on the projector it's yellow in the San Francisco area which means that we have marginal non-attainment of the standard now the standard for ozone in the United States is 0.08 parts per million concentration or lower actually a scientific panel of experts in air pollution and lung health recommended of the Bush administration that it should be 0.060 but they set it at 0.08 oh and this is responsible for many deaths every year in the United States the EPA now is proposing a new standard 0.060 back to where the scientific panel advised and if you look at how many unnecessary deaths and days lost from flares of chronic lung disease would be saved by going to 0.06 so the answer is four to twelve thousand deaths a year in the United States and about two and a half million days and people miss work or school 58,000 aggravated attacks of asthma and about eight million days when people have to restrict their activities because of air pollution according to current standards so I urge you to support Clean Air Act's speaking of clean air besides gases like ozone and sulfur dioxide there are also particulate air pollutions particulate air pollution is suit for example and where are things deposit in your lung depends on their sides so this is shows on the right a graph that at about 10 microns in size most things deposited in your nose and your throat they do so by inertial impaction you inhale it they have enough mass that they just impact on the back of your throat it's sticking swallowing they get down to about one micron in size or smaller and they make the bend and go down into your lung if you land on your ciliated epithelium they're transported out in the vehicle ciliary escalator if they get beyond that you're stuck it's or macrophages that can help and asbestosis is a example of a terrible disease caused by particulate matter in the environment that people inhale particularly they're working with insulation so the green arrows show a little asbestos fibers and this is supposed to be a lacy alveolar capillary wall but instead it's all scarred down in reaction to this today asbestos so it's caught this has a about 15 to 20 years until it occurs after you inhale it so most people get it starting 15 to 20 years and it gets worse until maybe at 40 years it's devastating it's also associated with malignant tumors so cigarette smoking though by any standard is the biggest problem and us cigarette smoking has declined since 1965 to 2005 by about half so this is that decline and the percentage of patients will older than 18 that smoke now there's some amazing statistics associated with this which I'm sure you've heard before there are annually 438 thousand Americans that die prematurely because of cigarette smoke that's the same number of people that died in on the US side in all years of World War two and Vietnam so just take all of the people that died in those two wars on the US side and kill them every year and that's what cigarette smoking is doing it's pretty amazing now the tobacco industry spends about thirteen billion dollars defending it mainly and discounts on cigarettes and coupons to get around state laws that are progressively taxing cigarettes and so the cigarette companies release coupons you can buy them at the old rate to a dict you to cigarettes and you'll continue to smoke them well of course the big Bugaboo cigarette smoking besides chronic obstructive lung disease is lung cancer there are 1.2 million new cases of lung key in 2008 and it's expected to rise by about 25 percent by the year 2010 and sorry 20 here's something amazing half of regular smokers die of diseases attributable cigarette smoke 1/4 of regular smokers die of attributable diseases between the ages of 35 and 69 so a lot of people are dying in what is sort of the modern middle age because of cigarette smoking and in the last century 100 million people died because of cigarette smoking so if you look at new cases in the United States the most common cancers you probably can't read this slide and men.there prostate cancer and women their breast cancer but if you look at the death rate because prostate cancer and breast cancer don't always kill you the death rates of lung cancer far exceed the death rates of all other cancers so this is in men lung cancer deaths peaked around 1990 and have been descending along with cigarette smoking in men women have not been declining in their cigarette usage as much and so in about 1985 lung cancer exceeded breast cancer for the first time as in the most common cause of cancer death and it's way about breast cancer now it's almost twice what breast cancer is so these are sobering statistics how does lung cancer present it presents usually as a small nodule somewhere in the lung lung has no pain fibers so you can't feel that you have a nodule a malignant tumor growing in your lung it's completely asymptomatic until it invades something else or obstructs something yeah so we really need a way to screen for these of course the best thing to cut lung cancers is cessation of all smoking the United States short of that people want to devise methods to screen people that smoke regularly and that has been in so far an unsuccessful enterprise so the perfect test for screening would be inexpensive sensitive that is not miss case is specific that is not confused cancer with other diagnoses that cause passivity on imaging studies safe which of course radiation imaging studies are not and acceptable to physicians and patients so the main methods are chest x-rays or CT scans or examining sputum for malignant cells that's called sputum cytology none of them are inexpensive CT scanning is relatively sensitive the other two methods are not only sputum cytology is specific and this is one of the biggest problems so if you do a CT scan normal 60 year old the US population 30% of them have some nodule on their chest and in their CT scan and most the overwhelming majority those nodules are not malignant so to prove it's not malignant you have to biopsy the nodule which means putting a hole in you and so there's a fair amount of expense and risk to doing that in addition to the radiation incurred by the screening techniques so there are a number of problems with the perfect screening tests and in addition there's more sophisticated problems so if you screen for a diagnosis and you make it at an early time you don't have any effective treatment then the patient dies at the same time as if you hadn't screened for it except that you think you caught it early and that you think therefore through lead time bias that they had a longer life span because you just knew they had it earlier so the group that you identified seems to have a longer life span than the group that was unexamined but it's not really true it's just a lead time bias or if you you can have length time bias too so very indolent tumors that don't grow rapidly you have a high probability of discovering with a random screening test because they hang around for a long time and so but that population of tumors causes a lot less damage because it grows much more slowly and you might die of something else before it does anything B and then in addition there are some tumors that grow so slowly that they're just essentially non progress and again overdiagnosis means that you yes you discover that you have lung cancer but you have five other comorbid diseases and they kill you first and it's not really valuable to make the diagnosis other than it worries you to death so the current organizations that issue proclamations about this like the American College of Chest Physicians American Cancer Society US Preventive Services Task Force none of them have endorsed screening for lung cancer and I think the best thing to say about it is stop smoking so if it doesn't appear as a nodule in your lung that slowly grows and then becomes symptomatic when it invades your chest wall or some other area or spreads to another organ it can occur in your airway then it blocks your airway you can't clear your secretions and you get pneumonia distal to it so it's the mucus clearance problem again and so this is what tumors look like that hair in your airway again using a bronchoscope so here's a malignant tumor growing in an airway it's actually in both ear ways and obviously no secretions are going to come out of those Airways and the lung behind it is going to be collapsed because it can't get any air and these things look much like this they look like kind of variegated abnormal looking tissue growing in your Airways so how do you know if you get lung cancer whether it's going to kill you the main question is how rapidly is it growing because ones that grow rapidly tend to spread early and have a malign natural history and then also one way of knowing whether it's growing rapidly and spreading soon is whether it's already spread 75% of lung cancers when you first discover them you cannot remove surgically and cure the patient and the remaining 25% half of them live five years so you're looking at overall mortality rate of about 85 to 87 percent of lung cancer and so for most patients we don't have again therapy and the most common reason for it to be spread as it spreads to regional lymph nodes so lung in addition to having arteries and veins and Airways also has a lymphatic system it's fine lymph fluid that drains organisms and into lymph nodes which are the sites of white cells that can kill organisms and the lymph nodes though also can trap tumor cells they're very complicated maps that are made about where lymph nodes spread has occurred that all cancer oncologists and surgeons used it estimate prognosis and patients so they develop a tumor in their long and it spreads to a lymph node and if it's a regional if there's no lymph node that's spread at all you know about a 60% survival if your tumors removed wholly entirely removed of course that's only a very small fraction of the patients that have lung cancer because to most of them it has spread already two lumps nodes if it's just a lymph node right next to the tumor then you have about a twenty five to thirty percent five-year survival but if it's spread into the mediastinum where the heart is and you have an eight to twenty percent survival and if it's spread to a distant lymph node then you only have a two to five percent five-year survival so lymph node spreads very important but of course it can spread to other organs as well now use PET scans for that PET scans are scans that rely on localization of glucose that is tagged with a radioactive element fluorine-18 formed in a cyclotron so cyclotron generates the fluorine-18 we tagged it on to glucose injected intravenously then it goes to hypermetabolic areas in the body so tumors are hypermetabolic and it tends to label them this is an example of a label on the liver a person who had liver metastases from one cancer so the way we stage lung cancers is we first see how big is the tumor because big tumors have a worse prognosis so there's a T in staging system t1 means less than two center meters in size and t3 is large then there's where the lymph nodes spread has occurred that's n0 2 in 3 and then there's the presence or absence of distant metastasis you can group these factors together and make survival charts but overall lung cancer is about 85% fatal for the current detection and treatment modalities and does have a simple method to eliminate 85% of it in United States which is not smoking I'm going to use the last 10 minutes to talk about some infectious diseases first pneumonia it's acquired by failure of our defenses against aspirated or inhaled organisms they're about 450 million cases a year and it causes about 3.9 million deaths a lot of these are in the sub-saharan region of Africa where a million die each year and about 700,000 in South Asia half of the people that die of pneumonia are under the age of five in the US there's only about 50,000 deaths attributable to it that would be 1/8 the number of deaths people that die from cigarette smoking in about 1/4 of the number of people that die of lung cancer so bacterial pneumonia you inhale an organism and a sufficiently large number overcomes your defenses and your macrophages you're trying desperately to kill it you have villi and then it causes an infection that spreads and so this is an example of such an inspection this area of opacity is due to pneumonia this person recovered completely sometimes ammonia is eat a hole in your lawn a lung abscess those are more severe pneumonia and certain organisms are more prone to do that if you look at the alveoli this is one alveolus stuffed with pus white cells and that is what pneumonia looks like it's just your lung drowned and a collection of inflammatory cells plus and you cough up pus and thankfully in the antibiotic era most ammonia is promptly treated to do relatively well if you look in the sputum of people there call hang up this bus to see these red things that are white cells that's the pus and the organisms that are causing the pneumonia unfortunately sometimes pneumonia is fatal this is a flu-like illness that occurred in a graduate of Stanford B School came back here for a reunion his kid five years old at a group a strep throat he acquired the organism from his child and 36 hours after he had his first symptoms he was desperately ill by 48 hours he's coughing up blood and pus a helicopter him from Monterrey where he was dying down there and he expired in one hour of arrival of Stanford and you can see that virtually all areas of both lungs filled with pus so it can be a devastating illness even in the modern antibiotic era so in the old days people died of a different kind of pneumonia and that was tuberculosis brought under the map by Robert Cox he worked in the 19th century largely on anthrax cholera and TB and made up many of the postulates by which we now judge things to be clearly due to an infectious disease so his postulates were that you have to find the organism in the disease you have to isolate it and grow it and you have to inoculate it into another animal and cause the same disease than isolate it again and that stood the test of time and of course he had a Nobel laureate for it in 2005 so these are his original pictures of tuberculosis what the organisms all arranged in the kind of circle that we call a granuloma and it may surprise you that until 1943 for as far back as recorded history the most common cause of death was tuberculosis in fact in total 1943 in the United States it was the leading cause of death tuberculosis and in 1943 there were more sanatorium beds there were all other kinds of hospital beds which is pretty shocking since tuberculosis is so well controlled now so these sanatoria this is the Trudeau sanatorium most famous of all in Saranac New York for just packed with people they had no real therapy except fresh air and they all lived together and a large number of them died together today it's still a major healthcare problem that's why Gates Foundation is interested in it more than 2 billion people one third of the world's population are ineffective at TV one in 10 infected people actually become ill with TB and in 2008 1.8 million people died of it five of the percent of the cases are multi drug-resistant and the former Soviet Union 22 percent are and they're now emerging organisms which are untreatable which we call XDR extensively resistant TB 2 percent of the isolates in part of the Soviet Union are xdr-tb and the most appalling thing is that most of the multiplet resistant drug TB multidrug-resistant TB is not being treated adequately which is creating new reservoirs of more resistant organisms so tuberculosis worldwide looks like this it's mainly in Africa and the Soviet Union India China Micronesia Indonesia not very common the United States here are the purported TB cases in the United States less than 10,000 currently a TB case rate is in California relatively high higher than the national average and in the southeast that's because these are areas are for immigrant populations to enter the United States and reside if you look at the number of TB cases in US vs. foreign-born persons the pink is the farm borne it's still it exceeds the number that are u.s. born so slightly more than half come from foreign-born people that are immigrated and they come from Mexico in the Philippines in India and Vietnam and China and Guam Haidee and there is thankfully just about half a percent of multiplied resistant TB in the United States because we have an extensive apparatus and the Public Health Service to protect ourselves against it the last thing I'm going to talk about diseases is the modern plague the h1n1 virus so this is a scanning electron micrograph of the h1n1 virus it has these surface cell projections hemagglutinin and neuraminidase 'as the hemagglutinin allow the virus to attach to cells in your airways and in your mouth and the neuraminidase is allow progeny of very ions and it reproduces and makes a lot more virus to break a well from those cell surfaces and spread h1n1 viruses have been around since 1957 off and on but they were more common before 1957 so that older people tend to have seen them before and be more resistant the current strain the swine so-called swine influenza is really a recombinant RNA genome that exchange some parts of its segment a genome between birds pigs and humans what really causes a giant epidemic is when there's a massive shift in the H in the neuraminidase is in hemagglutinin that accounts for the large pandemics so the virus now this is from the CDC two days ago is not very widespread the brown indicates not widespread sporadic and the epidemic is basically over for this period it may come back again in the fall or in the summer but this is the peak of the cases there were 57 million Americans infected by the sea the estimates it's roughly one out of six there were 250 7000 hospitalizations but only 11,000 deaths wasn't very mortal normally when we have an antigenic drift in a little epidemic every year our seasonal pandemics usually carry 15 to 30,000 deaths we actually had fewer deaths than we would normally expect in this epidemic than in a normal flu year and the number of visits to emergency rooms is back to the baseline about three percent of people have something that's a flu-like illness they're still devastating cases this is ones in the hospital right now the 50 year old man no risk factors on a ventilator now for a month most likely going to die the other people that we lost in our intensive care units were people that had impaired immunity from bone marrow transplants chemotherapy we had three pregnant women on a ventilator for over a month he's sectioned them early their kids are okay in there they're all survivors that's his CT scan the last thing I'm going to say is them great mysteries and the new frontier is why do some people get pneumonia and recover completely and others have a fulminant course and die with the same organism and this has to do with hosts response and you can think about why is it that only 15% of heavy smokers develop COPD or only five to ten percent of people infected with tuberculosis get active disease and the answer is host response the most common cause of coughing up a lot of blood in the United States is a cousin of T V Mycobacterium avium and it's ubiquitous in the environment we all inhale it but only Caucasian women that are middle-aged acquired the disease this is obviously a genetic issue and we've learned some about the genes involved with that problem but another place to look is ten thousand Americans died of second-hand smoke why only them the answer is that they of a gene defect and their glutathione s transferase gene release all of them do so the big new frontier is so-called personal medicine what does your genome mean when you inhale things from the environment or aspirate something that kills some and others escaped scot-free so I conclude I talked a lot about the structure of the lung maybe more than you wanted to know about the function of one a bit about the transport systems and some of the more common diseases I've shown you that their lungs are living things and I hope you think of yours now as a living thing and I'm gonna conclude with some words from some poets about breathing so it's breathing as close to my heart literally so Shakespeare eighteen of William Shakespeare sonnet 18 and William Shakespeare but thy eternal summer shall not fade so long as men can breathe or eyes can see so long lives this and this gives life to B or William Butler Yeats near the end of his life thinking about his impending death and what his legacy would be I balanced all brought all the mine the years to come seemed waste of breath a waste of breath the years behind imbalance with this life this death so we all think of breathing as poetic normal embedded deeply in our consciousness John Donne is virtuous men past mildly away and whisper to their souls to go while some of their sad friends do say the breath goes now and some say no so I'm hoping that your breath doesn't go for more please visit us at stanford.edu
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Medical_Lectures
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Vasculitis_An_Overview.txt
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FEMA this is Eric Strong from Stanford University and today are we talking about vasculitits. After watching this video you should be able to define vasculitis. Describe typical features suggesting vasculitis describe the modern classification system of vasculitis and most importantly identify a probable vasculitis. in a patient with atypical presentation 0 this video will not review the specific criteria the individual forms a vasculitis that's because this will take an hour to do would be incredibly boring and its unnecessary since the criteria is easy to look up when needed. Instead I'll be focusing on vasculitis as the general category a disease particularly how to recognize a patient who probably has vasculitis since the varied multisystem presentation can be easy to confuse with other diseases So first what is vasculitis? Vasculitis is that they burst category of inflammatory diseases of the blood vessels these diseases range in severity from self-limited dermatological additions to acute and rapidly paid almost a system diseases All forms of vasculitis are characterized by endothelial damage into more proliferation thrombosis and eventual vascular occlusion last letters can affect every organ system typically in back in as both patterns based on the size of the affected vessels and the underlying pathologic mechanisms all so what do those recognizable patterns look like the first two layer a pattern recognition has to do with the size of the involved bustles and the knowledge that basket latest tends to affect by organ systems the most which other skin gastrointestinal system can these no system and the muscles it can also affect the lungs which I've left of here because it's not common other when it does happen its involvement is usually very prominent when a patient has a basket like this affecting the small blood vessels here she can get a skin condition called palpable purpura which show a picture up in a minute GI involvement leads to me go siders and minor GI bleeding the GLA merry Lai the kidneys are affected which most commonly manifest as he maturity a without red blood cell casts and proteinuria patients can get a pollyanna Robert P and the muscle involvement result in my algiers which is a fancy word for muscle pain a medium vessel vasculitis a result into skin conditions called earthy Manado some and livid over to kill aris patients get abdominal pain and rarely pal preparations in the kidneys damage is not limited to the commercial I'm so now there may be mature your with RBC casts along with flank pain from ischemia in the brain patients can get a wide variety of mental status changes from somnolence psychosis along with strokes and in the muscles they get my own status which differs from I'll just and that there is a usually lab and pathologic evidence of muscle damage such as an elevated creatine kindness a large vessel vasculitis can result in cyanosis and discoloration of the extremities balan park Shin hypertension from involvement of the aorta or renal arteries but no key Macharia strokes from involvement other crowded and claudication which is pain in the extremities due to lack of blood flow unfortunately the presentation a basket by this is much more complicated than this because no one individual vasculitis disease leads to all the findings in anyone column all there are however certain clinical features that are highly suggest to the basket like this the first to something called Mon underwriters multi-plex which is a simultaneous or sequential dysfunction a individual non-contiguous peripheral nerves in a seemingly random pattern this typically presents as loss of sensory and/or motor function with an individual nerves over days to weeks for example this patient pictured involvement up the owner might result in weakness of the fourth and fifth fingers involvement of the lateral cutaneous from raw nerve might result in past teachers over the lateral thigh and involvement other common peroneal nerve could result in full drop all in addition to basket by this other causes upon and rightist multi-plex include diabetes HIV and alloy dosis another feature that is highly suggested the vasculitis occurs when an unknown multisystem disease as very prominent involvement up the long and kidneys especially the combination of pulmonary hemorrhage and I the renal failure and/or he Macharia as the only other Donbass tonight this diagnosis typically does this is anti GBM antibody disease also known as good pastors disease all livid over to kill aris which is a place like reddish depart role discoloration of the skin is frequently seen in medium and occasionally small vessel vasculitis it certainly not Pathan demonic however as an idiopathic form seen young women is the most common etiology this finding finally is palpable purpura which is the development up numerous raised non blanching purplish lesions on the skin which accused the most prominent in the lower legs ankles and Pete as shown here in addition to these poor clinical features there are a variety of other features which are less specific but are also commonly seen a basket like this they include headache hypertension pulmonary hemorrhage without being on Parliament Obama pain abnormal urinary sediment a skin condition called earthy Manado some which consists up multiple red tender not short on the shins arthralgia is and claudication all here's a picture apparently min ago some all although I said that I wouldn't be reviewing individual diagnostic criteria of vasculitis it's important to be familiar with the general classification scheme unfortunately the classification of vasculitis is confusing and unsatisfying due to many reasons including overlapping presentations it makes a primary and secondary forms a vasculitis evolving understanding of the underlying pathophysiology is an evolving consensus regarding use of eponymous names and varied opinion on whether classification should be based primarily on the size of the affected vessels or underlying pathologic process as a consequence different sources may present slightly different classification schemes the following classification is largely based on the most recent recommendations from the American College of Rheumatology the first order division within vasculitis is based on that size would target muscles large muscle vascular disease include talk to ya issues arteritis which predominantly affects the great vessels of the aortic arch and as a predilection for young asian women and giant cell arteritis also known as temporal arteritis which primarily affects branches at the crowded arteries and which is seen almost exclusively among the elderly medium vessel vasculitis includes polyarteritis todos a which is the most widely distributed within the body the vascular disease as it can affect any organ callous sake disease affects almost solely young children and is best known for causing coronary artery aneurysms and primary CNS vasculitis into mystery mimicking all kinds of primary nor logic and psychiatric disease small vessel vasculitis is further subdivided into that which is associated with an auto antibody called anti new to fill cytoplasmic antibody and that which is associated with immune complexes thank associated vasculitis includes three diseases with long and frustratingly similar names granulomatosis with poly and gids which is a particularly dangerous vasculitis primarily affecting the Kines longs upper airway nose and eyes houston affiliate granulomatosis with poly and gids which is characterized by the combination asthma sinus problems drop at the and peripheral your sin filial and microscopic poly and gids which is clinically very similar to granulomatosis poly and gids the exception to the locker upper airway involvement and the lack of granulomas on biopsy all the final category opinion complex associated vasculitis includes I G a basket by this it typically self-limited disease see mostly in children which is characterized by pop opera abdominal pain astrologers and renal involvement usually limited to Macharia while globulin the mic basket lettuce which is caused by antibodies called while globulins which precipitate when the temperature drops below body temperature most cases are triggered by chronic hepatitis C infection and last hypersensitivity vasculitis which is usually limited to the skin but can also cause Bieber arthralgia stand accused limp and not the it can be triggered by medications particularly penicillins and cephalosporins along with a variety of chronic infections a quick word about some %uh those names some types a basket like this previously had different opinion based names for example granulomatosis with poly any ideas was known as Wagoner's granulomatosis until just a few years ago houston affiliate granulomatosis with poly and gids was known as church Strauss syndrome and I G a basket by this was known as he knocked shoreline purpura impact most positions they were paired to these diseases by the eponymous even if the scientific literature shying away from them journals and professional societies aren't just changing the names for the sake of avoiding opinions and general instead Wegener's granulomatosis was named after feat Rick Wagoner was a Nazi doctor was speculated to participate in experiments on concentration camp prisoners but it seems that once one opinion was intentionally removed from the formal ex-con all the others are following it seems likely that copy of his arteritis and Kawasaki disease will both be renamed in the near future in addition to the types of primary vasculitis just listed some systemic diseases can trigger a secondary vasculitis for example connective tissue diseases malignancy chronic infection sarcoidosis and bishops disease the last which is often considered a primary basket latest itself there are also many diseases which can mimic basket like this including PIC Berger's disease house people axis cholesterol Emeli Sande rum anti GBM antibody disease bacterial endocarditis and amyloidosis some sources actually list Berger's disease and anti GBM and the body disease as true masculinities highlighting the overlapping any incompletely understood pathophysiology evolved these disorders so what are the general steps to diagnosing a basket latest first the Commission must identify a collection of clinical findings which I this is just about the latest in general or optimally wanted to specific masculinities second one should check with you labs in clinic why collation tasks and a urinalysis and consider checking imaging studies and/or anti-nuclear feels like a plastic antibodies this will help to now down the differential diagnosis to one or two specific vascular disease to search for an Associated systemic on this that would make the basket by the secondary and to rule out vasculitis mimics all finally the diagnosis to the basket latest should be confirmed preferrably with biopsy of the clinically affected organ and/or and geography all in the final minute out very quickly review the basic treatment options for vasculitis although you should keep in mind that each individual about the latest may have specific treatment algorithms at this extremely brief overview may not capture all for limited 18 years past the latest for example mild I G a basket latest or hypersensitivity vasculitis sometimes observation and removal offending agents are all that is necessary the antihistamines and a short course a penicillin is often used all for mild systemic vasculitis prime his own on the order of 0.5 to 1 milligram per kilogram per day is common finally a rapidly progressive and life-threatening basket lettuce Prime is own plus they say the toxic aged such as cyclophosphamide is the most common regimen cyclophosphamide can be transition to a less toxic agent such as is a biplane or methotrexate after remission is achieved for FEMA there in
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Medical_Lectures
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How_to_Create_a_Differential_Diagnosis_Part_1_of_3.txt
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hello I'm Eric strong from Stanford University and the paloalto VA hospital this is a three-part video series entitled a guide to clinical reasoning or how to create an accurate differential diagnosis from a patient's presentation the learning objectives of the this video are first to demonstrate a standardized approach to generating a focused differential diagnosis from a patient's presentation second to create concise problem representations using semantic qualifiers and clinical syndromes next to understand the types of Frameworks to which the key features of a presentation should be applied and finally to know the categories of diagnosis which should be included in the differential unfortunately literature and textbooks that discuss and attempt to teach clinical reasoning often fail because of dense terminology and a focus on abstract Concepts that trainees find difficult to apply at the bedside with concrete examples in this video I'll be presenting an approach to clinical reasoning focusing on the differential diagnosis that is practical at the bedside accessible to students and other novice providers and minimizes unfamiliar terminology part one will introduce the clinical reasoning process and parts two and three will go through examples start to finish of how to apply that process to a real world patient part two will be an example at the level of a preclinical student and part three will be an example at the level of an intern I'm going to start by defining the term clinical reasoning clinical reasoning is the process by which a health care provider takes objective data acquired from an actual patient and interprets it using factual Knowledge from a textbook or the medical literature in order to either make a diagnosis or develop a treatment plan it isn't a single individual skill but rather a collection of related skills it involves interpretation of subjective data evaluation of the accuracy and validity of data synthesis of individual pieces of data into higher order groups determination of the relevance of scientific literature for a specific clinical situation critical evaluation of the arguments for and against diagnosis application of biostatistics and finally integration of different types of knowledge into a complete decision-making process although clinical reasoning isn't just about figuring out a diagnosis that will be the focus of this particular set of videos I consider there to be five steps to generating a differential diagnosis the first is to acquire data and to do this I recommend that you use all available sources that includes first and foremost the interview and examination of the patient but also diagnostic tests and chart review focusing on the medication list recent Primary Care notes and recent discharge summaries also depending on the circumstances you may consider talking to close family members and friends for collateral information provided of course that the patient cons sense as you gain experience with clinical reasoning you may begin to find yourself skipping the five-step process and actually start developing differential diagnosis as you are talking to and examining the patient that is as you are acquiring the data this skill dramatically helps with efficiency and helps choose the most appropriate diagnostic tests however it does not necessarily result in a more accurate diagnosis in the end as it sets one up for something called anchoring bias anchoring bias is a tendency to place too much weight on a single piece of information acquired early in the data acquisition process and a failure to update the differential diagnosis when conflicting information is later presented for example if a patient's uh HPI or history of present illness focuses on his shortness of breath and he happens to mention early on that it's worse when lying down a specific symptom called orthopnea that may lead a clinician to begin hypothesizing during the interview that the patient has heart failure additional investigation May then be appropriately directed towards confirming or refuting this hypothesis if the clinician then learns the patient has a fever has a focal decrease in breath sounds on exam and a consolidation on chest x-ray he or she should reconsider their original hypothesis if leici fails to reconsider the initial leading diagnosis of heart failure even after acquiring much data suggesting a different diagnosis in this case pneumonia he or she has committed anchoring bias so here's the bottom line for efficiency sake it's a good idea to consider the differential diagnosis in real time during the patient hmp but be aware of anchoring bias and once the entire data set is collected force yourself to return to the beginning of the clinical reasoning process so that each piece of data and element of the presentation is given its appropriate weight once you've acquired the data the next step is to identify the key features key features are the individual elements of the presentation which are likely to help differentiate one diagnosis from another for example in a patient with episodic chest pain whether or not the pain comes on with exercise will help to determine the likelihood it is from cardiac esia therefore this is a key feature in contrast the sever of chest pain on a 1 to 10 scale is surprisingly nonpredictive of the eventual diagnosis and thus I would generally not consider its severity to be a key feature key features include both positive and negative findings they may be from History exam Labs other tests or chart review the third step is to create a problem representation this should use semantic qualifiers and should synthesize with related findings into clinical syndromes some of these terms may be unfamiliar to you so let me explain what do I mean by the term problem representation this is a one to two sentence summary using precise medical terminology of the most highly relevant aspects of the patient's History exam and diagnostic tests sometimes problem representation is used synonymously with the terms summary statement as well as impression the latter term commonly used in written notes in the United States semantic qualifiers are qualitative abstractions of the symptom of a case in which an opposing abstraction is either explicit or implied they help to reframe a patient's symptom into terms more familiar to the clinician and easier to communicate to others common categories of qualifiers include the onset of symptom for example was it abrupt or Progressive and acute or chronic has the course of the symptom been continuous or episodic is the site unilateral or bilateral proximal or distal diffuse or localized what do the symptom trigger postprandial or exertional or ptic or positional and is the symptom associated with pain or is it painless the use of semantic qualifiers when reframing a patient symptom is thought to Aid in accessing chunks of information stored in the clinician's memory and is associated with a higher likelihood of arriving at the correct diagnosis let's look at some examples of how to use semantic qualifiers to reframe symptoms imagine a patient in the ER States for the last 30 minutes my chest is hurt whenever I take a deep breath we would reframe this as acute ptic chest pain or another patient states over the past several months both legs have been getting weaker and weaker this becomes chronic progressive bilateral lower extremity weakness a word of caution here some information is lost when the patient's presentation is translated into semantic qualifiers which can set one up for bias if the original history is never Revisited now what about that phrase synthesize into clinical syndromes a short elaboration on this is that a concept stellation of clearly related findings should be grouped together into a single clinical syndrome if possible for example if a patient has confusion a fever to 39° heart rate of 120 beats per minute blood pressure of 130 over 60 respiratory rate 24 white blood cell count 16,000 creatinine of 2.4 and positive blood cultures you can synthesize that as severe sepsis and if another patient has jaund sies confusion asterixis a total bity of 25 and an INR of two that can be summarized as hepatic failure another word of caution don't mistake the clinical syndrome for the diagnosis for example it's great if you recognize a patient has severe sepsis however severe sepsis is not a final diagnosis and your diagnostic reasoning should not end here you must also determine what has caused the severe sepsis is it community acquired pneumonia or urinary tract infection or appendicitis so now using semantic qualifiers and clinical syndromes how do we construct the problem representation there are at least two approaches to this the one I favor is to link four categories of information into a single sentence using a standardized order age and gender first then highly relevant past medical history followed by the primary symptom using semantic qualifiers and ending with the highly relevant diagnostic data using clinical syndromes when possible for example a 60-year-old woman with a history of poorly controlled diabetes presents with chronic progressive exertional dpia with exam and chest x-ray findings of volume overload and with unremarkable routine labs and EKG the other approach to the problem representation is to consider only this category so for this patient the problem representation becomes just chronic progressive exertional dpia I personally don't like this approach as much because I think the upside to being more concise is outweighed by the downside of eliminating the additional information but you certainly will come across the second approach from time to time all right so let's get back to our five steps that was a long one uh step four is to adopt a framework to better understand the patient problem this framework may be anatomic physiologic or some other type and it's commonly adapted from a reference source Frameworks typically take the form of a categorized General differential diagnosis where the strategy for categorization depends upon the specific problem what are some types of Frameworks a framework for acute renal failure May first divide diagnosis up into pre-renal intrarenal and post renol meaning is the problem before blood gets to the kidney inside the kidney or after urine leaves the kidney some of these categories can be further subdivided so pre-renal ideologies can either be from dehydration or from low cardiac output and intrarenal ideologies can either be glomular tubular or interstitial another example of a framework this time for anemia divides the ideologies into hypoproliferative and hyp proliferative hypoproliferative anemias can be from nutritional deficiencies bone marrow failure kidney disease or chronic disease Andor inflammation hyper proliferative anemia can be from acute blood loss or homolysis there is not just one acceptable framework for a specific patient problem for example let's consider the patient we just mentioned a minute ago a 60-year-old woman with a history of poorly controlled diabetes who presents with chronic progressive exertional dpia with examined chest x-ray findings of volume overload and with unremarkable routine labs and EKG what type of framework might we choose to adopt here in my experience the most likely framework a clinician would choose for this problem is an anatomic one also referred to as organ based so for this woman with dpia that means that it could be a cardiac problem a pulmonary problem or a heem problem as as we just saw briefly with the kidneys all of our organs can be further subdivided into functional components so a problem in the heart could be in The myocardium coronary vessels valves conduction system or paric cardium a problem in the lungs can be located in the Airways alvioli pulmonary vessels interstitium or plora and finally hematologic issues can involve any of the individual cell lines or coagulation problems or issues with paraproteins however with dnia as a chief complaint the major hematologic concern is of course anemia my thrownness with these Frameworks is limited by the minimum font size I want to use uh in in the uh diagrams here but if I was being more thorough I would also list the renal system here with subcategories for the renal arteries guli tubules interstitium and collecting system so that's the anatomic framework which is just one way of categorizing the differential diagnosis for this problem another completely acceptable framework that's based more on physiology might ask what are the pathophysiologic mechanisms that can trigger dpia there is hypoxia which can be from VQ mismatch impaired diffusion or shunt there is hypercapnea which can be from obstructive lung disease Central hypoventilation neuromuscular disease or decreased respiratory compliance less commonly is dpia from acidemia either from pathologic Acid production or from poor acid elimination finally input from the cerebral cortex from anxiety and pain can also lead to the subjective sensation of dnia this framework isn't necessarily better or worse than the anatomic one just different it's likely that some people will naturally gravitate towards one and some to the other one other type of framework that I feel both obligated and reluctant to mention is the pneumonic framework I feel obligated because many medical schools still teach and expect their students to use it I am reluctant because I think it is vastly inferior to other types of Frameworks but nevertheless here it is one such pneumonic that's taught is vindicate the v stands for vascular the I for inflammatory the N for neoplastic D is degenerative the second I is idiopathic C is congenital a is autoimmune t for traumatic and lastly e for endocrine okay one more step to go apply the key features to the framework which will generate the preliminary differential diagnosis when applying the key features the clinicians should use their presence or absence to estimate the likelihood of the diseases or pathophysiologic states that are suggested by the framework as a brief example imagine a 55-year-old man with a history of alcoholism and depression presents with chronic progressive bilateral lower extremity edema with an exam notable for anasara and a serum albumin of 1.5 G per deciliter one of the several Frameworks one might employ for this case might be a physiologic one where there are four categories for the four major mechanisms of hypo albuminemia there is impaired intake of protein seen in malnutrition impaired utilization and synthesis of protein seen in liver disease excessive glomular filtration of protein seen in the nephrotic syndrome and excessive GI loss of protein a syndrome known as protein losing enteropathy the key features for such a patient might include a history of depression how does Depression affect the probability of any of the categories in the framework it's not linked to liver disease independent of substance abuse and it's not associated with either Fric syndrome or protein losing enteropathy however it does increase the chance of malnutrition as a patient may not be eating properly in his depressed state if key features also included the presence of spider angom and splenomegaly on exam that would dramatically increase the likelihood of liver disease if the patient has no history of GI symptoms particularly diarrhea that would dramatically decrease the probability of protein losing uropathy as all pathologies that cause this General diagnosis also lead to diarrhea and a host of other symptoms and if the key features included a UA without any protura that would be definitive enough evidence as to completely Ru out nephrotic syndrome from the differential altogether estimating to what degree individual key features impact which components of the framework Andor differential is a skill that requires both textbook knowledge familiarity with scientific literature and experience in my opinion it is the single aspect of clinical reasoning that more than any other differentiates novice clinicians from the experts perhaps I should have started this video with a discussion of the next issue but although most of you are likely familiar with the term differential diagnosis let me Define it specifically so that we are all on the same page with how I'm using it a differential diagnosis often abbreviated as just the differential is a list of possible diagnoses which may explain the patient presentation it should include those diagnosis in which either its likelihood is high enough or the danger if it should be missed is high enough in order to Warrant additional testing to investigate that specific diagnosis it may or may not include additional diagnoses whose likelihood is low enough to not warrant immediate testing but which have not been completely ruled out the differential diagnosis should be prioritized in descending order of likelihood a solid Focus differential diagnosis should include the following the one diagnosis that you believe to be the most likely this is known as the working diagnosis or provisional diagnosis two to four diagnoses that are very common in general for which this patient's case could be either a typical or an atypical presentation any diagnoses which are rapidly fatal if untreated of which this patient's case could plausibly be the result this is often known as a quote don't miss diagnosis finally any diagnoses which are specifically suggested by standout features of the patient's history including unusual Hobbies or job and recent travel to an exotic location most of these unusual standout features will be what we refer to as red herrings a red herring is an unusual element of the presentation that falsely appears quite relevant But ultimately distracts the clinician away from the true diag nois it seems common for medical trainees to be instructed to keep their differential diagnosis broad what this is meant to mean is that the trainee should not prematurely jump to conclusions regarding the culprit organ system or determine that a single diagnosis is the only one worth considering unfortunately the recommendation to keep the differential broad is frequently misunderstood to mean that the differential should be very long and span every organ system in the body a long differential is more more problematic than a focused one even in training because it actually displays less thought and it can be difficult to formulate a diagnostic plan if there are 15 different conditions simultaneously under consideration for the typical Internal Medicine admission I would consider four to six diagnoses a good ballpark range to aim for When developing a practical differential my last words of caution first the framework and differential diagnosis are not the same thing the framework may be adopted directly from a reference source and is not specific for your patient the differential diagnosis on the otherand includes only those diagnoses relevant to the patient in question a differential diagnosis that has not been made specific to the patient is nearly worthless when prioritizing the differential and establishing the provisional diagnosis in general an atypical presentation of a common disease is more likely than a typical presentation of a rare disease finally the true typical presentation of a disease does not always match the textbook description of a disease or its historically taught presentation for example it's usually taught in medical schools and reinforced in suboptimally researched textbooks that spontaneous bacterial peritonitis or SBP usually presents with abdominal pain abdominal tenderness and so-called parital signs on physical exam in reality the most common presentation of SBP in a patient with known liver disease is altered mental status with or without a fever and without any abdominal signs at all as another example it's frequently believed that patients with pericardial tanod either usually or always present with hypotension in reality when studied it's been shown that the majority of patients with proven tanod are actually not hypotensive at presentation there are many many more examples of discrepancies between how an inaccurate classic presentation of a disease negatively impacts diagnostic reasoning most patients with heart attacks don't have crushing substernal chest pain most patients with migraine headaches don't experience a visual Aura before headache onset the list goes on and on so that's it for my five steps to a differential diagnosis once again they are first acquired data second identify the key features third create a problem representation using semantic qualifiers and clinical syndromes fourth adopt the framework and last apply the key features to the framework in order to generate the differential diagnosis that concludes part one of this guide on clinical reasoning focusing on how to create an accurate and focused differential diagnosis as I mentioned at the beginning in Parts 2 and three I'll go through examples from start to finish of how how to apply this approach to a real world patient as you listen to Parts two and three I'd consider pausing intermittently as you go in order to practice working through the case on your own la
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Medical_Lectures
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Immunology_Lecture_MiniCourse_3_of_14_Antigen_Recognition_by_T_lymphocytes.txt
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okay welcome back hope you had a good lunch and this is echo eggs it's okay okay so you know this is a you know it's very hard for people and listening because it's like drinking water from a fire hose you know getting all this stuff going very very high pressure and I know it's hard sometimes to keep track so I'm going to faster there's a concept you want me to repeat just raise your hand don't be ashamed and I'll go over it again and also I'm from New York City so people from New York tend to speak very quickly because if you notice B quickly then you know they're gone or were you speaking to so again if I'm speaking too quickly just tell me to slow down okay any questions or anything I'm going to change and you differently tell me don't be yes Christian you know you talked about how white cells can find their way out of the circulation at the site of inflammation and these adhesion molecules play a role in directing them the selections are they constitutionally expressed on epithelial are they up regulated as well could they appear to play an important role in sort of slowing down slowing down the cells and maintaining the cells in the periphery so they can access the they're expressed at low levels continuously but then what happens with a lot of these packages and I want is activation inflammation they can get raised and even higher levels of expression okay so now things will start getting interesting is now getting into T cells and B cells and also unfortunately you're more complicated so the third lecture is going to be discussing antigen recognition by T lymphocytes and the third and fourth lectures are really partner lectures because the third lecture conceptually is going to be looking at what goes on outside of the cell how t cells of pen are seeing antigen and the second lecture part which is lecture 4 will be discussing the structural motifs and how things go on inside the cell in terms of the peptides so against with their partner lecture so to start off with the questions want to consider are first of all how do b-cells and t-cells see antigen and how does the t-cell distinguish between whether a foreign antigen is being presented to recruit help or because the cells infected and obviously if it's to recruit help you don't want to kill the cell you want to help the cell and if it's because it cells terminally infected it can't be saved then you want to kill itself so how do t-cells know how to distinguish between those scenarios why do some t-cells kill another provide help now clearly there's a psychological explanation for that but as a monologist we have to come up with more of a structural understanding a local understanding of why that occurs and so again these are the things we're going to talk about in this particular lecture now this is actually one of my favorite slides because this is really what immunology was about like maybe 25 or 30 years ago in terms of its understanding of what was going on in the process but the other reason I find it fascinating is it really talks about how when you don't know something you truly lies it and say that it's not very important and then all of a sudden you start realizing that something that wasn't important turns out to be the critical link or most important aspect and this turned out to be the case for macrophages and monocytes right macrophages because about 25-30 years ago we basically thought that these cells were just phagocytic cells all they did there was a garbage people of the immune system they circulated around they saw something that needed to be taken up garbage picked up they just swallowed it they chewed it up and in essence that's what happened so you have this phagocytic cell it sees a bacteria a bacterium that's what they called it back then and then binds to it you have some membrane ruffling they were very proud that you've got membrane ruffling back then it ingests it it brings it inside you have fusion with the Fagor lysosomes and digestion and then this kind of nonspecific burping process that occurs when this is just kind of like shed out into the environment and so in essence macrophages were really looked upon is you know just non important cells and the really smart people and immunology didn't waste their time studying macrophages they were really more interested in other very more complex cells but now it turns out and as this lecture will show macro fighters are really clearly the critical cell in the process of t-cell and b-cell immunity and these cells now whatever you want to raise the importance of something you give it a title so instead of calling in a phagocytic cell now that you got a promotion and now it's called an antigen-presenting cell that sounds a lot more sophisticated and in new in new york what they did was we had people who are working taking out the garbage and they were called garbageman and they wanted to get a raise but the city was bankrupt so instead of giving them a raise they call them sanitation engineers and they were very happy with that they went back to work at the same salary so pretty much the same thing happened with these macrophages they really wanted to get something more important and therefore we renamed the antigen-presenting cells because that's teaching us that the role is not so much ingesting which also it does do but in terms of presenting antigen particularly to t-cells and the two major players I'll discuss initially are macrophages and dendritic cells and macrophages are critical phagocytic cells and in fact they have an important effect or function in terms of the their ability to phagocytose and kill bacteria and that plays a critical role in the immune system as a secondary process they also are involved in antigen presentation dendritic cells they have phagocytic capabilities but as opposed to macrophages where that phagocytic capacity is being used as an effector function but dendritic cells primarily their intake of their uptake of antigen plays a critical role in terms of enhancing its capacity to present antigen and again as a lecture goes all the distinctions between macrophages and dendritic cells in terms of their function now if we think in terms of infection with TB which is something several you are involved with we have two possible scenarios the first scenario is that you basically have the phagocytic cell that sees the micro bacteria to burgle a in circulation if phagocytosis it and that's the end of the story so it now has the MTB sitting in the vacuole but it apparently is not yet fully activated and if it's not yet fully activated it lacks the ability to destroy the TB inside the vacuole and this is what you would see now in this particular scenario people who are in the front could see large numbers of these classic acid-fast bacilli which represent MTB they haven't been fact they haven't been destroyed and therefore the infection continues very strongly and ultimately this will lead to significant disease in contrast if this macro Faiz is activated by virtue of interaction with the T helper cell this T helper cell secretes factors we'll discuss in a few minutes gamma interferon is a critical factor produced by T helper cells this now activates the macrophage makes it a much more effective killing cell and now these MTB that previously have been able to replicate inside these vacuoles and then ultimately escaped into the cytosol now because you up regulate the Fagor lysosomes they merge and fuse with the vacuoles and now kill the TB and now in this scenario the infection is being cleared so clearly the macrophages play a critical role in terms of its ability to control HIV infection but in order for that to occur it has to be activated macrophages arresting macro flies just does not have the digestive juices going enough to kill TB and what provides the stimulus for that is the T helper cell so this is this interplay that you need between macrophages and T helper cells and after a macrophage is activated again it has large numbers of these fabulous somes able to fuse with the vacuole also it up regulates a whole host of surface molecules so it will up regulate MHC class 1 MHC class to which all as I'll discuss later makes it much more efficient antigen-presenting cells it up regulates molecules such as b7 and cd4 T I'm going to discuss in a later lecture that played a critical role in co-stimulation T cell activation tumor necrosis factor as you know plays a critical role in terms of vascular permeability and cellular migration and nitrous oxide plays a critical role also in terms of killing this is an activated macrophage and again the immune system has the same motif and approach that I previously spoke about with the lymphocytes if there's no infection cells are not activated so if there's no active infection going on there's no need to waste all the energy to have this macrophage making all the proteins and all the factors that are required and therefore it's basically relatively quiet --scent but the trade-off of that is that in case infection occurs it takes a while to mobilize so it's kind of like you turn on a computer til the hard drive kicks in until everything kicks in takes a while to occur and during that window of time clearly the infection can go on uncontrolled now this is again you can see this slide a couple of times and just like this slide illustrates the secondary response so every time you're going to see the slide you can develop a rapid response to seeing this slide but just is just again to illustrate the first time you see antigen you take so there's a lag phase about a week before you get any response you get a slow response it's not very high however the second time you see antigen you have this very dramatic rapid response within a day or two and again the level of the response is dramatically better so what this allows us to really identify also is whether you've seen antigen and what the response has been now if we look in terms of the T cell macrophage interaction so here we have an infected macrophage it's been infected with TB present in the vacuole it's not been activated enough so it's not able to efficiently kill it now there T helper cell comes along it recognizes that the macrophage is infected because in this case this will discuss greater detail it's seeing a peptide derived from the TB in the MHC class 2 molecule and now it again showing co-stimulation with cd4 T city for T lie yet again discuss that great to tell later but just to focus now it's making large amounts of gamma interferon and the presence of gamete fer and activates the macrophage to make it more effective killing cells so therefore these are the major players now that are playing a role in T V immunity now this is another thought experiment for you to do but now you actually have graduated to using mice for your experiments so with that information that I gave you here's an innocent Mouse look in those eyes just staring at you saying please make the right decision to save my life and and the question would be transfer of which cells would protect this mouse from TB and again get ready to vote because I'm going to ask you to vote and and the choices basically are the following of either you could you could transfer activated monocytes from another infected Mouse so you infect another Mouse you activate the Manas I stand now revved up to kill and now you can give for example say a million cells transfer intravenously into this Mouse or you could transfer T cells to rise from an infected Mouse that has undergone the secondary that doubt is undergoing a secondary response to this mouse or you could transfer B cells from a mouse that also has been exposed to TV and undergoing a secondary response so you have one of three choices and you cannot choose all three okay so the first who votes you should use activated monocyte to protect this Mouse raise your hand now get if you're raising hand go for it okay you know the no halfway votes so who would go with activated monocytes raise your hand okay makes sense they're the ones that are the effector cells who would go with T cells raise your hand okay well T cells are always a safe answer because they're important cells that's a good one and again you know clearly the playing role who would go with these cells raise your hand let's that's a smart group because I haven't even talked about B cells so how possibly could that be answer and it's like any multiple-choice you always throw a third choice in just kind of out of the blue you know because you have to fill up you have two choices it would be pretty easy to question so now we have to ask the question between activated modeling the T cells it turns out that if you give the mouse activated monocytes it really doesn't help very much the mouse actually dies from TV infection however if you provide it with T cells from a mouse that is previously seen TV that actually saves the mouse's life and the question is why should that be and the answer basically is that that as I showed before an infected macrophage if it's not activated you won't be able to kill however once the T cell activates it now can efficiently kill when you're transferring over say a million activated macrophages in the greatest scheme of things it's not a lot of cells so therefore that's not enough to turn the tide against T V you may have to give who knows 100 million activated macrophages however when you transfer the T cells these T cells now recognize the TB peptide being presented to it so a single helper cell could go on to activate thousands and thousands of macrophages so you have a tremendous potential for amplifying the immune response and that turns out why T cells really are the cells that you'd want to transfer into that particularly cd4 T cells into that Mouse to save that Mouse is live so that's why that answer works but again doing that experiment allows us to really appreciate the capacity of T helper cells to rapidly mobilize and activate macrophages to make them be able to effectively kill it so in essence you know the T helper cell think about as a capacity building cell because what it's doing is it's coming into the environment and building the capacity of the macrophage to be able to illuminate T V and again the reason that's also important is if you'll be designing a vaccine the kind of vaccine you want to design is one that's going to activate T helper cells that can now turn on macrophages and make them able to eliminate TV okay any questions okay now well how does the t-cell know that the macrophage is infected with with T V again it's not like you know T cells are mind reader's they're able to just guess by the way that the macrophage is like is die a padi seeing that it looks like it's an effective molecule and it turns out it's going to have to do with the fact the way that antigens being presented to the T cell now this basically illustrates the how B cells and T cells the antigen as I had mentioned before they use different antigen specific receptors this is an antibody molecule again if you remember the heavy chain light chain the combination of the heavy and light chain is providing antigen binding site the immunoglobulin molecule in this case of for example IgG would have two binding sites one here and one here this is the T cell receptor same concept constant region and then a small hyper variable region that's recognizing antigen and again this structure and these are completely different however if you look at this picture what jumps out as a difference between the immunoglobulin molecule and a t-cell receptor what do you see that's different this is two of them two immunoglobulin and one t-cell receptor what about why are there two of them what's the difference between between this one and this one one's attached and one is not attached where the T cell receptor is only attached what is that teaching us that's teaching us a very important difference between the antibodies and T cell receptors because the antibody molecule has two functions one function is that is at the cellular antigen receptor so it's basically bound to the cell and it's telling the b-cell whether or not the antigen is pre-programmed to recognize is in the environment and hooked up to all the signal transduction pathways to activate that b-cell and make it big antibodies but in addition the effector molecule for the b-cell is the antibody molecules by that bezel so it basically secretes a eggs antibody molecule that matches the one expressed on the surface and this is the effector molecule of the bezel so the B cells don't have to be anywhere near the infection site in order to be effective because the antibodies are actually doing the clearing of the infection however the t-cell receptor only exists bound to the cell it only serves as the indicator to the T cell that the in the antigen it recognizes is that is out there it activates the T cell but the actual function is not carried out there's no soluble T C T cell receptors again to the circulation to look for peptide it now activates the T cell which means that the T cells themselves have to get to where the infection is they are like pant hand combat type affective cells in the immune system whereas you only think of the B cells as shooting missiles and the antibody molecules are the missiles that they shoot but they're very tightly guided by virtue of the antigen binding site that they recognize okay is that clear now in terms of the receptor gene I'm going to discuss the molecular rearrangement a little bit more detail when I discuss the you know globular molecule but again the concept is the same if we would have to have a single gene for each possible T cell receptor we would basically take up the entire genome with genes encoding T cell receptors and antibodies which is obviously very very inefficient but as important as immunologist think they are we're not important enough to take up the entire genome so it with the you mean a blonde woman and T cell receptor gene has done is use this incredibly efficient way of compart of basically picking and choosing and putting together different sequences to dramatically increase the capacity to generate diversity and again I'll discuss in more detail with even a blogroll lecture but all you need to know at this point is that this locus can have 70 to 80 different variable regions than these J regions or joining regions and each one of these is it has a completely different sequence each one of these has a completely different sequence and by virtue of picking one of these and one of these that gives you a completely unique sequence and think about it as like on those those luggage cases we have those combination locks we have those three wheels that you turn and three wheels don't take up a lot of space but by having zero through nine you can get like what about almost a thousand combinations and imagine if instead of having just just nine numbers or zero the ten numbers on each one you had 100 numbers on each one imagine how three wheels can give you that much high level of diversity that no one could ever randomly guess what it is so - by having all these multiple pieces you could how they combined together gives you tremendous diversity the beta locus in addition to having the variable region and the J region has an additional region called D region and again the way that I always remember it is again friends of mine are from Brooklyn and the way they were pronounced beta is beta you know beta cells so so think of beta with a D and that's how you remember that the beta locus has a D and the Alpha chain doesn't because there's no does sound in alpha so again you know these things look try to help you remember these things okay so now if you want to put it to context now you have all these variable reasons all these J regions and here you even have the additional variety of the D region and now you basically pick one V one J and this unique combination now gives you a unique gene that can see something so if you take this V and you bind to this J you get one sequence if you take the same V and bind it to a different J you get a different sequence which is going to recognize a different antigen and so forth and so on so by taking a single V and matching it with dozens of J's you get tremendous diversity and now that gives you an alpha chain the beta not only has the diversity of the J but also has all the diversities of the multiple DS and and therefore now in output pairs with the data and again the same way I'd mentioned with the heavy and light chain the same alpha paired with a different data will see different peptides so by now having two chains again you dramatically increase the amount of variability and again I'll discuss this again create a detail with the immunoglobulin rearrangement but again it's the same concept of this cassette approach of taking pieces of DNA bringing them together and then by having different pieces with different sequences it generates tremendous amount of diversity so now in order to understand how how t-cells and b-cells see antigen we take advantage of this secondary amplified response because the first time a b-cell and t-cell will see antigen or thicket seeing antigen it will amount a low level primary response however the second time it's seeing antigen it will amount the more rapid a higher level secondary response and that's a way of now using to find out how the b-cells and t-cells of singing antigen because it will change the antigen and see if the t-cell or B cell recognizes it as a new antigen or as an old antigen it's seen before so the experiment that we would utilize is by immunizing with in this case a native protein and then we would boost with a native protein well it's the same protein right so obviously both the B cell responds and the T cell response will provide you with a secondary response so if we you know if if we use this as a prop so this is up this is a obviously bag and if I blow it up this has this tertiary structure right that's and it looks a certain way but if I take the same bag and I basically flatten out like that it obviously has a completely different tertiary structure so in essence we can do the same thing with proteins we have a native protein that has the normal configuration but we denature it flattened out has a completely different appearance and now we ask the question do b-cells and t-cells see it as a different antigen from the native or as the same so we immunize with the DNA protein so immunize with a flat paper bag now will boost with the native protein and lo and behold the B cells do not develop a secondary immune response because to them it's a completely different antigen whereas the T cells mount the perfectly good secondary immune response if you immunize with the native protein come back with denature protein again these cells do not see it as a same protein whereas the T cells say yeah it looks the same to me and just to prove that this flatten denature protein is immuno genic if you've been immunized with the denature come back with denature b-cells were perfectly fine see it as a secondary response sees it as seen it before and again t-cells are always seeing at the same so clearly this teaching us an important lesson that b-cells and t-cells see antigen differently and it turns out as I alluded to initially that it depends upon what exactly that they're seeing and another important observation is that whereas b-cells can see antigen that's free-floating if you take free-floating antigen like this paper bag and add it to a t-cell the cd4 t-cell will have absolutely no reaction at all however if you take the same bag and you mix in some macrophages and now you mix it in with the cd4 T cells as I'll discuss what's now happening is that a peptide from the antigen presenting cell derived is being presented to the t-cell and now you get an immune response so in essence what's happening is the macrophage is taking this bag and it's basically ripping it apart into pieces of peptides and that's actually what the t-cell is seeing so in the absence of the macrophage it can't do anything with this bag only when this bag is digested into these little peptides now is it able to see antigen that clear but the other important thing is teaching us is why the beast' by a t-cell doesn't care what the structure of the protein is because it doesn't matter whether it's native form or whether it's the in Asian form at the end of the day it's seeing the exact same peptide so the structure of the proteins are relevant because at the end of day is only seeing a peptide and that's why it bounced the same secondary response whether or not it's see the antigen as a wild as a native form or indeed in nature because it ultimately gets digested up to the same peptide regardless of the structure tertiary structure of the starting protein set clear and this is just a kind of little more complicated experiment to basically show that the while the antigen must be processed to peptides still if you predigested it works just almost as well so in this case you basically take antigen Damac the integer presenting cell digests it into the peptides it places the peptide into the MHC molecule again I'll discuss this in a little bit a little more detail later in the lecture but now it presents it to the t-cell receptor you get a response if you take the same macrophage and ingest the antigen but you fix it with glutaraldehyde and therefore prevent the membrane from allowing any internal new peptides to be presented you get no immune response but if you fix the macrophage with glutaraldehyde so it has the MHC molecules but now you artificially digest the protein in the Constituent peptides these peptides can lodge in the MHC molecule and be presented to the T so the bottom line is is this you need two things you need to have peptide and you also need to have an MHC molecule to present it if you add three peptides to the T cells again that would be not be sufficient for the T cell to recognize okay is that clear and again now to come in how to be cells and T cells e antigen so what a V cell is seeing what an antibody molecule seeing is a three-dimensional structure of the protein so it's very superficial it just says well it looks like this that's one protein if it looks like this it's a completely different protein and therefore that's why when it sees natured or native it thinks of it as completely two different proteins however what the T soap is seeing is a peptide and the same peptide as may be derived whether or not the protein was wild-type or denature and that's why it doesn't matter to the T cell what the primary structure of the antigen is where it's the B cell that's incredibly important so it's because it's mesh is recognizing a three-dimensional structure provided by the antigen structure and the three-dimensional structure of native antigen is going to be different from the three-dimensional structure of a denatured antigen is that clear now this is again one of the critical questions well why don't you have self reactive T cells and our peptides that are being presented in the MHC molecule are their only foreign peptides and again this is a very common misconception when people first learn immunology they think that only foreign peptides are presented by MHC molecules and that's why we don't have any immune response targeted against itself it turns out to absolutely not be true in fact if you're not infected all your MHC molecules contain self peptides because it that's all there is out there so you just pretty much no such thing as an empty MHC molecule it always has to have a peptide in it if you're not infected the only other source our self peptides and now you have cell peptides and your MHC molecules why don't you now mount a immune response against self and the answer is because there are no T cells responsive to self peptides in circulation because they've been removed in the thymic selection process so even though you're expressing thousands and thousands of self peptides as long as there are no T cells to recognize them you won't get an immune response and therefore you won't target those cells you know the flip side also is early on in the infectious process you could be presenting viral peptides but if you haven't yet mounted a immune response again you won't be able to recognizing so the MHC molecule has absolutely no idea whether the peptide is a self peptide or whether it's a fired peptide it will display it and it's depending upon the presence of the T cells that recognizing in order to mount an immune response and fire peptides and self peptides are processed exactly the same way there's no that I know advantageous processing of foreign peptides over self peptides it all comes down to whether you have these T cells in circulation that recognize it is that clear it's a very important concept so there are two kinds of MHC molecules is MHC class 1 and MHC class 2 and dr. Janeway in his book showed it very clearly because MHC class 1 has one foot so this is I guess like that the person trying to learn all of Talman while standing on one foot about Judaism and MHC class 2 is 2 and again so whenever you see the picture is very obvious which one he's referring to but again clearly they have more differences to this structure than just whether they have one or two feet and they also turn out that different cells Express different groups of MHC molecules so in terms of MHC class 1 pretty much every nucleated cell in the body expresses MHC class 1 molecules you name a cell that's nucleated it will Express class 1 MHC molecules the only cells in the body that don't express it are red blood cells and again red blood cells are non nucleated cells however when you look at MHC class 2 expression it turns out to be very very specialized only to what we would call antigen presenting cells and these include T cells B cells again as I mentioned the important and the resenting cells and macrophages and dendritic cells epithelial cells we'll discuss when I talk about fine with maturation however all other cells in the body neutrophils you name them they do not express class 2 MHC molecules obviously a red blood cells don't either so what that's teaching us is the capacity to express MHC class 2 molecules is unique to immune cells and that means that this is playing a very different role in terms of an immune response that MHC class 1 MHC class 1 immune response is going to be focused in a way that every cell potentially can interact with a CDA cell which I'll discuss in more detail because immune surveillance to eliminate infected cells is applicable to all cells in the body whereas activating immune response recruiting help are only relevant to cells in the immune system that need to get that help in order to become better effector cells okay is that clear and in terms of well what cells recognize MHC class 1 versus class 2 as you all know T cells come in two types one group of T cells are cd4 cells one group or cd8 cells and they're defined by expression either of the cd4 molecule or the cd8 molecule this is just showing a ribbon structure of the molecules but actually is kind of interesting is that cd8 actually comes in alpha and beta heterodimer whereas the cd4 is just a single chain but now if we put the whole picture together it turns out that class 1 MHC molecules bind to CDA class 2 MHC molecules bind to cd4 so here's a target antigen presenting cell here is the class 1 MHC molecule here's the peptide here's a t-cell receptor but what gives the specificity of the cd8 t-cells the class 1 MHC is the fact that cd8 sequences are homologous to sequences present in the Alpha 3 domain of class 1 MHC and that tells the t-cell to be to only bind to class 1 MHC molecule in contrast there are different sequences in class 2 MHC molecule in this case the beta 2 region which binds to the cd4 molecule and this is what tells the cd4 T cell that is T cell receptor can only see MHC presented pipe by class 2 and again the way that I always teach students to remember is the product is always equal to 8 4 times 2 is 8 and 8 times 1 is 8 how you know class 1 MHC CDA class 2 is is cd4 ok and again this also explains the structural motif of the CDA and c4 molecules they're relatively tall in order to be able to bridge the across the t-cell receptor to see sequences in the MHC molecule that the t-cell receptor is being bound to okay any questions but what are the functional implication of this specificity of class 1 and class 2 MHC molecules for cd4 vs cd8 cells well as you all know a cd4 cell is a helper cell a CDA cell is a killer cell so if you are presenting antigen to a helper cell you're going to get helped if you're presenting antigen to a killer cell what's going to happen to you you get killed so in fact that's an important implication I would think your files a cell so so here you have a magnifies it's cruising along it sees an extracellular antigen it as a macro Frye's it just ingests it digests it it presents a peptide from it in this particular case is this macrophage infected if this is say a dead bacteria it's ingested digested or presented pipe is it infected no so when you want to kill it no or in this or another scenario is maybe it is alive but it's still inside the phagosome so it's eminently digestible so again we do want to kill this macrophage no you want to save it you want to recruit help so therefore now you have cd4 T cell for example it will make gamma interferon it'll stimulate lysosomes and now enable to to be able to destroy the phagocytose antigen and eliminate it or in the case of B cells the B cell binds to extracellular antigen by virtue of the antibody specificity ingest it presents peptide and class 2 MHC molecule now and recruit t-cell help as I'll discuss in greater detail to allow it to make antibody and class switch again it's not infected would you want to kill it no not at all however let's say that you have a cell listen and now as I'll discuss again later the peptides coming not from outside the body but the peptides coming from inside the body if you're making farm peptides inside your cell what does that mean you're infected and that means likely that you're not salvageable that helped help us won't from a cd4 cell won't save you and therefore in a ton means it's not as brutal to say but the cytotoxic t-cell says you know in order to save other cells from being infected we're going to have to sacrifice you to nip the infection in the bud and then any farm peptide seen in the context of class 1 MHC molecule by a cd8 cell again will basically stimulate lysis and killing of that target cell okay so clearly is very important distinctions and implications questions so then the airport so class 1 MHC molecules only present peptides derived from endogenous proteins proteins coming from inside the cell the cell gets infected by virus it encodes through the endoplasmic reticulum viral proteins being made now these proteins get digested into peptides and now B get incorporated into MHC class 1 molecules presented on the surface looking at a cd8 fahren t-cell see the sorry cd8 cell sees it recognizes a foreign antigen and kills the cell if these are self peptides they present self peptides to the outside environment is that a problem no because there are no circulating cd8 t-cells that recognize self peptides so therefore they're not going to kill the cell obviously autoimmune disease is so devastating because if you do have T cells cd8 T cells that recognize self peptides to disaster because then they'll start targeting all these cells expressing self Bo and the immune system is working it's not an issue and therefore cells expressing farm peptide class 1 MHC cd8 cell sees it and kills the cell but now for cd4 it's only Express oh it can only be a class 2 MHC II by antigen presenting cells macrophages dendritic sells and they only present peptides derived from exogenous peptides so either proteins from outside or peptides that are derived from a Eliza Somme because the class 2 MHC molecule comes into the Eliza home gets loaded by peptide now gets expressed on the surface these cells also have the ability to present in the context of class 2 MHC molecule but b-cells turn out to be rather selfish because in contrast to macrophages that will present any image and they see they just see it on Jessica presented B cells only internalize antigen if that antigen binds to its antibody molecule receptor so if if this is if this B cell is specific for example for tuberculosis and this is a Seibel tuberculosis antigen it recognizes it will internalize it digest the peptides and present it because that recruits help onto itself to make antibodies against this pathogen however if this is like a tetanus or an hour or an albumin or another hand engine that it doesn't recognize it'll let it go by and will not internalize it so it selectively only presents antigen derived from a protein that it itself recognizes whereas a macrophage will just ingest any antigen willy-nilly of whether or not it recognizes it ok is that clear and so therefore cd4 t-cells now calm in the case of the macrophage it have been infected with TB presents peptides from TB and class 2 MHC molecule it's still within the vacuole so it's definitely still salvageable t-cell sees the peptide stimulus count secretes gamma kefir on stem weighs it to be is more potent but in excel in case of a b-cell b-cell ingests it based on antigen receptor seeing it presents peptide and now activates this b cell to make antibody and again this is just a repeat from b-cell is presenting antigen from that itself so in so for example if this would be HIV for example this is an important concept so so let's say this b cell is gp120 specific so you all know gp120 antibodies are critical neutralizing infection so it sees the virus and now it internalizes the entire this is four examples of dead virus internalizing the entire virus chops it up and now it presents peptides it canonic could present not only peptides from gp120 but it could present peptides through any portion of the of the virus so it could be presenting peptides from b-24 from RT from integrates as long as it gets it inside the b cell it could enable peptides to be presented the t-cell has no idea what antigen the V cell recognizes all it knows is that it's presenting a peptide that has been pre-programmed to recognize and I'll discuss in a later lecture in terms of vaccine design how we use this approach to allow us to make vaccines that against carbohydrates it normally would not be very antigenic okay so now if we look in terms of where pathogens are located these are the where the MHC peptides will be presented basically have situs solid pathogens so for example TB once it leaves a vacuole to the cytosol now it will be presented in class one MHC molecule because now it's once it's in the cytosol the infection is not reversible presented the effector T cells and this cell is sentenced to death if the pathogen is still inside the vesicle so potentially you can still kill it by lysosome merging now present a class 2 MHC recruit cd4 T cell help and will kill the intravascular bacteria and for b-cells internalize it through antibody antigen binding and present the peptides and then stimulate the B cells to make antibodies through MHC class 2 cd4 helper T cell interaction ok any questions ok now how if you were a virus and you would want to evade the immune system will be a great way of evading the immune system throw it out any suggestions hmm well imitates itself but it's hard for virus to look like the cell now let's think how is how are we knowing that a cell is infected with virus what do we see what it without used to seeing what molecules well MHC 2 is not necessarily saying the cells infected what time the system immune system that the cell itself is infected right and how do those viral proteins being presented and what molecule MHC class 1 great now if you were a virus and you want to stop the CD AIDS from knowing that it's been infected what molecule would you down regulate the MHC class 1 so it's in a sense if if someone kidnaps you the first thing they do is they put a guard over your mouth so you can't scream for help so viruses have do exactly the same thing and it actually turns out this is a paper that that Bruce Walker was co-author on a number of years ago showing that HIV Neph actually down regulates class 1 MHC expression prevents it from being expressed in the surface and therefore prevents the cell from being killed by cytotoxic t-cells it's a brilliant approach that viruses have used that make sense there's really creative now of course ah we're not stupid we know that viruses would try to do something sneaky like that so therefore we've come up with a way of dealing with that and that way is by using another kind of cell called NK cells and then K cells are natural killer cells so are these helper cells now you get immunologist all your friends telling you that they're going to be doing some good killing and if that these it we're really actually originally described more for cancer immunology because cancer cells down regularly class 1 expression it turns out that NK cells have the ability to determine whether a cell is expressed in class 1 MHC or not if the cell is not expressing class 1 MHC then NK cells designed to kill it so the way it works is that the NK cells have two receptors on its surface one is called an inhibitory receptor that binds to MHC class 1 molecules if it sees that MHC class 1 it sends a negative signal to the NK cell that tells this other receptor called the activating receptor not to signal the cell to die these all cells turn out express molecules on their surface that can interact with this activating receptor however if there's no MHC molecule in this surface or it's been changed now this this this inhibitory receptor has nothing to bind to now this activating receptor has nothing stopping it there's like nothing turning on the brakes preventing it from killing and now what happens is that this activating receptor is unopposed it now sends a signal through this cellular ligand and tells the cell time P to undergo a batata death and then it kills the cell so therefore potentially a way that our immune system uses to circumvent immune evasion by down-regulating class one MHC molecules is by saying I know all cells should express class 1 MHC molecules if I don't see a cell expressing class 1 MHC molecule something fishy is going on therefore it's probably because it's been infected now I need to kill the cell is that makes sense and again this is actually very important in tumor immunity because very frequent that when tumor cells down regulate class 1 MHC to prevent cancer derived antigens were being presented and NK cells again play a critical role in killing tumor cells unfortunately and HIV is also relatively shrewd so rank so it's come up with another way of therefore avoiding NK cell killing and it turns out that net also down regulates that activating receptor this activating receptor that must be expressed by the cell in order to provide the death signal and down regulating this this also blunts the ability of NK cells to kill HIV infected cells so again this is another example of how very well developed way of circumventing pathogen evasion of the immune system but HIV is really amazing in terms of its ability to come up with new ways of evading the immune system okay is that okay so therefore really to end up with the questions I started talking about with how to b-cells and t-cells the antigen the answer is they see it differently these cells see three-dimensional structures whereas T cells basically see peptide so the three-dimensional tertiary structure isn't relevant to how a t-cell sees antigen has eventually seen only seen peptide how do t-cells distinguish between whether a foreign antigen is being presented to recruit help or because the cells infected and the way it distinguishes is based on what MHC molecule is presenting it if it's being presented by a class 2 MHC molecule that implies the protein came from outside so the cell itself is not infected or it came from a vacuole and the cell can still be saved however if it's presented by class 1 MHC that means it must have come from inside the cell either made by a virus or in the cytosol and therefore the cell needs to be killed and the class 2 MHC and why do some T cells kill and others provide help because cd8 t-cells C class 1 MHC which tells that the cells infected and can't be saved where a cd4 cell C class 2 MHC and means it still can be assisted and and therefore that kind of puts it in context and then we have a half hour break and then when the next lecture now I'll do the second part discussing exactly how MHC presents peptides and the structural elements that are involved in that process and thanks a lot for your attention
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Medical_Lectures
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Immunology_Lecture_MiniCourse_2_of_14_Innate_Immunity.txt
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and this lecture is first how can the immune system detect invasion by pathogens and alert the immune system rapidly you need to have like first responders how do you under house the immune system and do this and how does it have immunological triage to know this level of infection is extremely minimal we don't need to bother the t-cells and b-cells to mobilize or this is a really serious infection we really need to recruit b-cell and t-cell help how does that occur in the immune response and the question is well yeah b-cells and t-cells are really cool because they have these sophisticated t-cell receptors antibody receptors but is there another way of doing it that's low-tech and as we all know sometimes low-tech is is even more effective than high tech because it's simpler or less things to go wrong so can we develop a low-tech system a quick and dirty way to differentiate pathogens from self antigens and mount an incredibly rapid immune response and this is a cartoon just kind of illustrate the problem the immune system has and it going to apologize if it's not politically correct but first is that the immune system to do pathogen profiling has to have a certain level of sensitivity it has to be able to detect that something is wrong and in this cartoon it's someone who supposedly like a terrorist saying there it was driving bin Laden minding my own business with a couple of missiles in the trunk and they never even read me my Miranda rights and the point is is that this clearly they should have picked up on this something was wrong there was no sensitivity in this screening and the flipside is you need to have specificity you need to pick the right amount of response against the appropriate if there's an infection and this is a cartoon showing a the scene of the Magi where the three wise men come to the birth of Christ and well yeah they are of Arab extraction but clearly they're there for something very important very good and yet here you have the American attorney general back at that time saying they look Middle Eastern and detain them again you need to have specificity to really determine what is wrong and the immune system has exactly that same issue to deal with so how does the immune system contain pathogens you know once there's an infection you need to rapidly be able to contain the pathogens and how our immune system is directed to the sites of infection if you have an infection for example in your finger well how to be part of t-cells in your lymph node know to get into the circulation and get out at the right location where the infection is if the inflammatory cells don't go to the right location they're not going to be able to help you at all so clearly you have to have a mechanism to allow you to direct the effect itself to the appropriate site of infection so it all starts with the initiation of inflammatory response now does anybody here pick their nails anybody here pick their nails nobody you can admit it it's not like it's this horrible vise you know but you know bites their fingernails okay see you bite your fingernails have you ever got an infection from biting it exactly it happens all the time and in fact this is exactly what has to happen during the initiation of immune response and I think that's a great example of it so when you get infection in your fingernail with how do you know it's infected what are the signs it hurts painful what else it's swollen and what else and it's red this is you could give the lecture so this is probably where it looks like initially you may not be projecting well but you see it's nice and red and swollen so you basically it's painful why is it painful because the blood vessels that supply that area are now dilating they're getting larger well with things expand is not enough room it becomes painful why are they getting large to increase the blood flow and blood supply to that area it's read again increased blood flow makes it red and also it tends to be warm from the increased blood flow and why do you have that you also have Wehling and again why is it swollen because you have increased permeability of the blood vessels and now you have a fluid leaving the blood supply getting into the local tissue now why is this all happening this is all happening to enable inflammatory cells to specifically localize to that area the inflammatory cells are in the blood so you want to increase the blood supply make it more efficient getting there you also want them to be able to leave the capillaries into the tissue well if the capillaries are sealed they can't do that so you have to open these you have to increase permeability to now allow them to to migrate out and this is resulting into the symptoms that you have early on during infection but this is extremely mild infection so far and in fact usually this is all that happens it's like that for a day or two people have it and then it just goes away by itself we don't need to take antibiotics no need to do anything more because your innate immune system is being very effective at containing this response however and so what causes this inflammatory response to occur well the first major cause is a response to bacterial replication so you have a few bacteria that get into the wound now they start replicating and now with bacteria replicate they tend to damage tissue so in addition for the actual replication of bacteria to be a stimulus for immune response also tissue destruction is now the opposite of that is if if there are bacteria in the body and they are not damaging tissue the immune system has it they taunt with those bacteria we kind of make a deal you don't bother us we won't bother you now where does that happen in the gut our good is loaded with bacteria and yet do we mount a potent immune response against those bacteria that oh we call commensal bacteria we don't in fact are actually good for us because if it doesn't harm us we won't harm you however once bacteria start replicating and damaging tissue that's when the immune system says whoa now we have to mount an immune response because you've broken that treaty with us and if we don't do something you'd probably and of course significant damage does so again it's not just the bacteria being there it's also tissue damage tissue destruction that causes immune responses but in addition tissue damage in and of itself can cause an immune response because you need to mobilize inflammatory response to repair the tissue damage and the major effector molecule a cell in this is macrophages and macrophages are discussed in more detail release mediators to allow recruitment of these cells to the local area of infection now what is the goal of the immune response do I have to have a goal what does it want to accomplish first it wants to prevent the initial establishment of infection it basically wants to get the air eliminate the bacteria before it could establish a beachhead that's the most effective thing it could possibly do however failing that it wants to print prevent spread of infection from the invasion site so I always think of like oldies anyone ever seen one of these submarine movies you know so like let's say there's like a torpedo it blows a hole in the submarine what are the sailors do they seal the hatch they go you know crank it and close it because that will you contain the water so it doesn't spread throughout the submarine the immune system does exactly the same thing it gets those cells there and it wants to localize the bacteria so they can't spread into the body now if you are successful in doing that so if this is what your fingernail looks like now because it's gotten it's spread and you can see you don't have tight margins here anymore it's very first of all and the entire fingers swollen you have not contained the infection well now what do you do now is the time that you recruit effector cells for assistance because the local cells can't deal with it and at this stage you probably need to mobilize b-cells and t-cells the innate immune system the macrophages are not sufficient to contain the infection and that's when you call in B cells and T cells in order to mount the immune response so again conceptually innate immune response macrophages are the initial first responders invariably they're very effective in terms of containing infection and eliminating it but if they're unable to do a completely they must need to have the capacity to communicate with the b-cell and t-cell immune response and mobilize them when appropriate is that clear now the innate immune response as discussed previously is a layered response to have multiple layers and again as it's kind of like you have you have code yellow code Britta code red and whatever you have different ways of mounting the response so if you have let's say you pick your nail and you basically have scrubbed your finger down with alcohol before you've done that because you're an infectious disease expert you know that's the best thing to do you know that's when you know someone's paranoid when you see them like wiping down their finger with alcohol wipes before they pick their nails but anyway so but what that accomplishes is the bacterial load and the skin is not that high so therefore even though you guys are breaking the skin is not a lot of inoculation and therefore it is probably a best low level infection and you have preformed nonspecific effectors local located in the skin area they can remove the few bacteria that come in and you won't even it won't get red and won't get school and there were absolutely no signs of any kind of infections very very effective however let's say the bacterial load is a little bit higher and therefore now you have a larger inoculum now you need to still use your innate immune response because again you eat this is what's responding immediately after the exposure to the antigen but now it has to be a more complex innate immune response now is old go into a little bit more detail you recognize microbial associated patterns expressed by bacteria these now activate these macrophages make them more effective they secrete factors that also recruit more macro fires more innate responding cells and again if this this is effective then you remove the infectious agent maybe they'll be red and swollen but you've cut it off at that stage and it's really not that serious at all however sometimes as that dramatic picture showed it's not enough the innate immune system can contain infection completely and therefore now you have antigen gets transported to the lymphoid organs to the lymph nodes as you know that's where the b-cells and t-cells are sitting there waiting to see the antigen they're pre-programmed to recognize that antigen now stimulates the b-cells and t-cells to differentiate and now they need to migrate back to the finger to eliminate the infection and again we'll discuss later how do the t-cells know when they leave the lymph node where the infection is to go to the infection and clear but again this is just showing that when it's we could if it's a low-level knock ulam you could clear without even knowing you've been infected a higher level you still can use your innate immune system to clear it but if it's a significant path a specific variant bacteria large inoculum then you need to recruit the b-cell and t-cell acquired immune deficiency a t-cell acquired immunity in order to eliminate the infection any questions now this is just all to illustrate that here's your finger that's where the infection is going on where the b-cells and t-cells sitting there sitting all the way very distal from the site of infection so two things need to happen the first thing that needs to happen is that antigen present from the pathogen needs to be transported to the lymph nodes where it can interact with the antigen receptors on b-cells and t-cells that's phase one the second phase is once they've been activated particularly t-cells need to leave the comfort of the lymph node get into the circulation and then migrate back to where the site of the infection is and again that's that's a two-phase process again I'll discuss the details as the lecture progresses but that's what you should think about how does this occur well how does antigen get to the lymph node and how to let the t-cells get back to the site of infection as well as macrophages in the innate immune response now I personally like thought experiments and the reason is they're very cheap to do they don't really don't cost a lot of money you don't see use a lot of supplies and also you can do them anywhere so you're waiting in some queue somewhere you're going to do you can think that these thought these thoughts so if if if all sudden poof this angel appeared in front of you and the angel said that you've been not the best person in the world so I have to punish you and your punishment is that you have to give up one limb of your immune system and again this angel you know probably is you know has some immunological issues but but I mean who would ever think that I don't think any fable exists that this has ever happened in history but maybe we'll started you know Newton the fable so you have two choices you can give up the innate immune response macrophages complement or you can give up the adaptive immune response b-cells and t-cells so now the angel is making you make a choice so who here you get everyone has to vote you have to decide and you know get a guarantee there's no angel in the back that's going to go poof and you're going to lose that limb of the immune system but who here would say I'll give up my adaptive immune system these macrophages not too bright phagocytic cells because my b-cells and t-cells are so important raise your hand who's giving up innate immune system raise your hand just raise it you got a vote okay who here is giving up adaptive immune system raise your hand well you're not you know you got a foe but you're not voting okay come on what you keeping Pope that's not an option you know you know this is like you know angels are mean they'll say you know what I'm taking both you know you have to give one you know some time life you need to make choices so who here is going to give up adaptive raise your hand okay and who here would give up innate raise your hand okay that's that's fair so now we've done the third experiment let's do the real experiment and it turns out if you look at the y-axis is the level of inspection and the x-axis is how long the infection has gone on so if you look at a normal immune response this is in mice and humans initially you get infected you have a level of replication of the bacteria for example but then your immune system kicks in and now you have you eliminate the infectious agent and if you've cleared the infection if you knock out the adapter or acquired the b-cell and t-cell immune response so for example skin mouse or a rag knockout mouse what you see is is that you get level of replication it plateaus it obviously doesn't get cleared but it incrementally increases slowly over a period of time however if you knock out the innate immune system look what happens you don't even control it it just cramps up very very rapidly and in fact these mites die very rapidly this is an incredibly important experiment because it's very instructive because of what is teaching us is that what is the more important immune response the innate immune response because if you knock that out then you really are rapidly succumb to infection and so in fact now you've learned something very important when the if the Angel does appear to you you know you know then you know you're in trouble but but give up your adaptive immune system and keep the innate immune system but it's incredibly instructive because this is teaching us that this early response disability to ward off the infection to hold it in check until the adaptive immune response can we mobilize is that is so critical and even more important than in protecting a from infection now this is actually very interesting because this thought was used to tick in terms of how to treat bioterrorism in the United States we had these attacks with them they sent spores in the mail to actually centers if you don't want to get mobilization in terms of funding for research you know tax centers that seems to be the case but but very rapidly bioterrorism became an issue and the question was well if there's a bioterror attack how do you respond well one approach would be to all vaccinate that's what we do right well how long does it take for a vaccine to start working you know we two three four weeks well that by that time unfortunately you may succumb to infection so therefore the question was can we wrap up the innate immune system and in fact that's an approach that individuals are talking about doing namely to harness the innate immunity to potentially give people a shot to actually activate the macrophages very very rapidly and therefore to help contain infection and again this is still the experimental phase but at least this this paradigm of vaccinating to crank up the innate immune response is something that really is being applied based on the understanding of how important the innate immune system is now the innate immune system is not is actually fairly intelligent it may not have the sophisticated antibody molecules or t-cell receptors that b-cells and t-cells has but it still has receptors that are able to recognize the presence of a pathogen and the kind of receptors they utilize are very different from those used by b-cells and t-cells so in terms of the characteristics this is comparing the innate immune system to the adaptive immune system the specificity is inherent in the genome for innate immune systems absolutely yes the full gene encoding the receptor is there does not need to undergo rearrangement it's expressed by all macrophages so as opposed to adaptive where each and each cell recognizes a different antigen all the macrophages we reckon is the same pattern it triggers an immediate response so very rapid response it can recognize a very broad class of pathogens of range of bacteria or viruses whereas the adaptive amuses of an incredibly specific and it interacts with the range of molecular structures whereas the adaptive is highly focused on only the specific antigens pre-programmed to recognize in contrast the adaptive immune response encoded multiple gene sequences we'll go into that in more detail requires very complicated gene rearrangements to make diesel t-cell receptors you have to clone one cell one antigen specificity and is able to discriminate incredibly subtle differences in structure that may distinguish between a self reactive antigen versus a pathogen specific antigen whereas the in native community is more broad-based and not as fun and finely tuned to be able to make those subtle distinctions but at the end of the day it's this quick and dirty low-tech system that allows us to mount the rapid immune response that as you see from that experiment is so critical in our survival from infection now as I mentioned previously the critical aspect of the unique immune system is that it uses barriers that protect us from exposure to pathogen so skin as you know it when it's in tact it's actually very very protective these are fixed barriers they don't get better when it's been exposed antigen so you could wipe your skin with bacteria from fatal tomorrow it's not going to become better at protecting you however your adaptive immune system b-cells and t-cells at as I mentioned before does improve with antigenic exposure the more time to see the antigen the more rapid the response is the innate immune system is pretty much stays the same now in addition to the skin another critical component of the immune response of the of the innate immune response is the gut particularly your stomach so your stomach you have very high levels of acid in your stomach and this asset gives you low pH and that destroys for the most part almost all bacteria for example that you would ingest some actually that can be resistant but for the most part it's very effective and the experiment that that happened to me many many years ago was with my wife and my oldest son so anybody here have kids okay so if you know you have a two-year-old child right you go into your car and you kids like raisins eat raisins any sticky stuff they like to eat marshmallows great so you put them in the backseat of the car they see a marshmallow on the floor and the marshmallow is loaded with all sorts of dirt and gunk and what's what's your child's going to do he'll pick up no what's he going to do put it in his mouth and swallow it right I mean that's what kids do so I say in heaven me my kid went into the backseat and he holds it on this this disgusting raisin pops it into his mouth and swallows it and then my wife looks at me and he says says Harris look what he just did he swallowed it he's going to die now that was the backseat the car who knows what's growing on that raisin you know this is the end what are we going to do and we go to the bird room we should pump out of stomach oh you know what do you do right your parent and then and then she said you're a doctor right so you have to do something so you know I'm thinking what should I do and they realize wait a minute I mean immunologist so I said tomorrow you know honey um there is acid in the stomach and the pH is very very low and that will destroy any potential bacteria that may be in that raisin or marshmallow that I eat and there's really absolutely nothing to worry about and meanwhile I'm thinking I hope he doesn't get a fever I hope it doesn't get a fever oh he doesn't get sick and sure enough you know you find no problem at all and again all of us have survived our childhood because of our stomach pH the innate immune system is so critical in protecting us we have similarly in the lungs we have cilia that continuously be to help clear any inhaled pathogens that we come in contact with and our eyes and nose again our tears Sicilian saliva all have chemicals in that that breakdown bacteria lysozymes for example all which protect us and as I mentioned previously we have commensal flora that are present in our gut that also protect us from being overgrown by pathogenic bacteria so we all know for example if someone's been on antibiotics for a while and you knock out your commensal floor actually those individuals can now go ahead and get super infected and get diarrhea and other gi infections because you've limited this kind of protective barrier any questions now the first step in terms of the infectious process is that you have adherence of bacteria to the epithelium and they're just waiting there for an opportunity as long as the skin barrier is intact nothing's going to happen then it not going to be able to infect you and cause disease you're waiting behind the scenes we have tissue macrophages that are present we have dendritic cells we'll discuss in greater detail called Langerhans cells that are present waiting but this is kind of like the calm before the storm because if there's a break in the barrier so let's say you a person cuts themselves and now they've introduced bacteria into under the into the under the epithelium where the epidermis and now the bacteria gets introduced into localized environment well if the inoculum is very very small then preformed proteins of phagocytes that are present there will eliminate these bacteria before they're able to cause significant level of infection and in fact you probably won't even really like everyday you cut yourself you basically put on a band-aid you don't ever get infected because this is working very very well however if the particular bacteria that gets in there is a very very rapidly replicating Syria or the inoculum is very very high then now when it gets into the environment you have high level of replication high level of tissue destruction and now you need to recruit a much more rapid a larger number of macrophages and potentially dendritic cells in order to help contain the infection and again you trying to contain it and preventing it from from spreading you can see that there are some macrophages being recruited from the peripheral blood that Abadie seing through those leaky capillaries again to help provide reinforcements to the innate response to help control and potentially eliminate the infection however in addition also you can have induction of blood clotting to prevent spread of the infection into the peripheral tissue however if this is unsuccessful and the replication is so vast as they show before now you need to recruit adaptive immunity antigen goes into the lymph nodes recruit b-cells and t-cells they undergo proliferation t-cells now you see the physically being recruited into the area of predominantly to help ourselves for the most part and in addition antibodies that are secreted by B cells in the lymph node get into the circulation again because the capillary is leaky it allows them to more efficiently leave and now these antibody molecules will bind to the bacteria and now they'll illuminate it either through facilitating phagocytosis or directly with complemented mediation and T cells will stimulate the capacity of phagocytic cells to be effective now the invention before that phagocytic cells have to express receptors that have the capacity to recognize that a pathogen is there so instead of having again sophisticated antibody molecules of t-cell receptors it has receptors that recognize motifs that are known to be present in bacteria that they could now recognize that there's an infection going on so if you look at and each one of these factors has the commensurate receptor on the surface of the macro flies so for example bacteria have very high levels of glue cans in their cell walls our cells don't have cell walls so we don't have glue cans so therefore we have a glucan receptor if the macrophage sees glucan around it knows it's from a pathogen therefore it's able to bind to it bacteria have high levels of nanos present in the surface we don't therefore Manas receptor is specific for these moieties and lipopolysaccharide something almost all of you are familiar with LPS is present in gram-negative bacteria and again we have a receptor called cd14 that interacts with another receptor called told light receptor where T L R in this case for that now recognizes LPS and activates the cell now toll-like receptor is this actually incredibly cool discovery that was discovered from fruit flies and again for the most part who here would ever think that we would learn about immunity from a fruit fly from you know mice yeah makes a lot of sense but fruit flies well it turns out that fruit flies are very easy to mutate they have a very very rapid gestation so within a few hours or maybe less than a day you have a whole generation of fruit flies so geneticists love fruit flies so they basically knock genes out all the time in fruit flies and then ask the question what does that gene do so it turned out they knocked out this tall gene they had no clue what it was it just knocked it out and they said let's see what happens and it was discovered in 1996 and it was a developmental gene in Drosophila however what was very very dramatic was that when they knocked out the gene the Flies develop massive fungal infection now has anyone ever seen a fungally infected Mouse I mean a fly know it soon to mean I know probably the light the fly dies before the fungus can replicate enough to infect it yeah look at what the flies look like this is this is a skinny M of a fly and he recognized the the odd but what is this all this hairy stuff over here this is all fungus because these flies lacked all this hull and impound this was encoding what the flaw a critical fly immune response which turns out to be very very innate and lacking that they became infected so now individuals said well if total is playing a role in fruit flies let's look for the human homologue and see if we have toll and it turned out we do and it turned out that toll the family code an important immune molecule and that's why it's called toll like receptors that our receptors that we have very similar to the ones that softly uses a little bit more sophisticated but conceptually exactly the same so this discovery all evolved from studying fruit flies and so now the bacterial derived factors LPS binds the cd14 toll-like receptor mannose mantis receptor and glucan glucan receptor and sometimes immunologists are your friends so they name things in a way that makes it easy to remember so if someone asks you what binds to the mannose receptor that's pretty easy right what month what blinds to the glucan receptor that's pretty easy they didn't call CD 29 or they didn't call it like you know Joe's receptor they gave it a name that allows you to remember what it was and now toll-like receptor just think of that fruit fly that's infected with fungus and now if you look at at the cellular level the macrophage has these preforms receptors Mattos receptor glucan LPS receptor if there there's no infection they just basically sit there doing nothing and here's the toll-like receptor however if there's a bacterial infection now what happens is the bacteria expresses on its surface Manos for example it binds to the mannose receptor this triggers signal transduction of the macrophage causing it to release cytokines and chemokines to recruit other macrophages or in this case LPS binds to cd14 it now associates with toll-like receptor and again activates the macrophage make it more effective in terms of its ability to kill and now the macrophage is able to ingest the bacteria it's activated so it has a lot of fabulous tones which merge and kill it in addition we secrete cytokines and chemokines that serve to recruit other macrophages into the localized area to help in terms of clearing the infection again so therefore the binding of the bacteria itself to these specialized receptor is enough to trigger activation of macrophages in the complete absence of t-cells is that clear so macro price goes from being quiet not doing anything to being activated solely being activated by the presence of these bacteria because of these specific receptors and this process can turn on a whole host of genes and in this case LPS binds to cd14 and then it now associates with toll-like receptor and toll-like receptor actually uses a transcription factor NF kappa-b which many of you have heard of in terms of activating a whole host of genes to turn on the appropriate cytokines and chemokines to really initiate the immune response so this capacity to specifically recognize these motifs that a president pathogen allows that the innate immune system to activate itself in the complete absence of t-cells and this a whole range of toll-like receptors each one of them has specificity for different compounds or molecules that are derived from different pathogens so for example gram positive bacteria have tri acetylated lipoprotein dice literally like plant protein and they're recognized by toll receptor 1 2 & 6 of flagellin is president some bacteria is recognized by 12 receptor 5 in addition some of these tall receptors can be activated by viral infections so the innate immune response is not only extracellular for bacteria there are also toll-like receptors that are present inside of cells in the nucleus that can be activate the cells after infection with a virus and make the cell make interferon alpha that allows the cell to basically protect itself from viral infection so it turns out that PG&E which is which is present Hylian viruses can bind TLR nine double-stranded RNA which again is present in retroviruses for example can activate tlr3 and again allowing the cell to protect itself from infection in the complete absence potentially of a t-cell response the innate first-line immune response okay so it's a lot more complicated than people initially thought but again the initial insight was by seeing fungally infected fruit flies okay any questions on that so people yeah so the question is in the absence of tolai receptor is there other mechanisms and yes because the mannose receptor the glucan receptor are not toll-like receptors and then that's another way of recognizing it but clearly toll-like receptors are critical for a wide range of pathogens so if you knock out toll-like receptor but still have nanos and glucan you're compromised and you again you you'll get sick but not as bad as completely knocking out your innate immune system now does this have relevance to HIV you know you know well if you think about it you probably know why would you know t-cells are basically your cd4 t-cells are infected with HIV that should not have an impact on the immune system in fact Marcus Advil who actually I know visited here a while ago had a paper in streamer ology two years ago we basically reported that HIV actually encodes toll-like receptor ligands that bind to macrophages and activate macrophages well does that make sense why should HIV want to activate macrophages right you think that a pathogen we want to turn off immune response not activated why would anyone have any suggestions why would we want to activate it yeah tries to get into the body so it just uses as a taxi back into the macrophage R&B okay and let's say let's say you have the buyers that's already infected the macrophage but the macrophage is quite a scent is there going to be efficient replication of the virus in a quiescent macrophage probably not because it's not a lot of proteins being made so the virus would want me to activate the macrophage to rev it up to how to make a lot of proteins and now can make a lot of virus so even though it's counterintuitive initially it actually makes sense why HIV would want to activate in this case macropods to make them a better Factory through making virus and in fact it can be the same thing for t-cells as well but as interesting that HIV is actually harnessed that these innate immune response toll-like receptor to activate macrophages to make more virus again one of the many examples of how HIV is hijacked our normal immune response to allow to become more efficient and more effective pathogen now again to ship back to the innate immune system clearly a clip a critical issue is how does antigen get from the periphery to the lymph node where's the bridge between the innate immune system and the adaptive the salty cell immune system and it turns out that Langerhans cells which are present in the skin and in the mucosa provide an important link between the innate and adaptive immune system so basically immature dendritic cells are present in the skin after activation they migrate from the skin to lymph nodes and transport skin the rive antigens that are coming from pathogens from the infection inside the skin and in the lymph nodes now they become activated dendritic cells become very efficient antigen presenting cells again I'll be discussing this in greater detail in subsequent lectures and now activating antigen specific lymphocytes and then again these cells are the way that the innate immune system activates the adaptive immune response how they need immune system talks to the adaptive immune response and pictorially what you can see for example is that here's a Langerhans cell that's in the epithelium or epidermis and in this case it's been infected with the bacteria coming in through a break in the skin it binds in this case in this case LPS cd14 recruits toll-like receptor for this now activates the Langerhans cell when their Langerhans cell is activated and now knows as to migrate from the skin into the lymph nodes so that goes into the draining lymphatics it gets into the lymph node and here in the lymph node it differentiates into a very highly effective antigen presenting cell and as discussed great in the next lecture expresses MHC molecules and now it's able to present antigen from this bacteria that is digested both two T cells as well as to be sultans mobilize them to then it now to go back and fight infection so again this is the critical bridge between the peripheral tissue and the lymph node that enables recruitment of the adaptive immune response is that clear any questions now in addition to having these receptors that are patterned motif recognizers the innate immune system also has soluble factors that have the same function so the same way that these cells make antibodies that are very highly specifically recognized pathogenic antigens and an immune system has soluble factors that also recognize the same kind of pattern MO to you again a quick and dirty approach to recognizing pathogens but one that allows you to have very rapid response because you have these present all the time in the body and again you have the same way of a Manas receptor you also have a soluble nanos binding binding lectin you have a c-reactive protein which bonds to motifs present and again this is actually interesting for people who are infectious disease at people because our physicians because one of the earliest signs of infection that you could use to monitor infection is c-reactive protein y-you get infected you trigger your innate immune response it makes it soluble factors to protect itself one of them to see reactive protein it's actually a clinical test you actually test patients for c-reactive protein to determine if they have a bacterial infection and in this case for example if the c-reactive protein binds to possible colon which is a compound present on bacterial surfaces and allows it as an obstinate and basically more effectively engulfed by macrophages or mannose binding lectin by Tomatoes again also linking the soluble factor making the phagocytic cells that much more effective and ingesting it okay is that clear so again the innate immune system really has mirrored what the adaptive immune system has with antigen specificity receptors on the surface of cells as well as making soluble factors that also recognize these patterns now once monocytes have been activated by pathogens it makes a whole host of cytokines and chemokines designed to both recruit to recruit new T cells B cells as well as other macrophages activate macrophages and also trigger a range of batteries died but to range up if you have a lined up one that probably works better on but to again to compartmentalize the infection and localize it so it doesn't spread so to focus for example on tumor necrosis factor that's a compound that were very familiar with clinically and at the appropriate doses tumor necrosis factor is very very important because what it does is local infection with gram-negative bacteria you may so now you've infected the skin over here you want to localize it so you have release of plasma proteins TNF for example makes leaky allows the matter of the phagocytic cells to leave the capillaries to the local area of infection in addition you have a small clot that you would reduced to prevent the bacteria from getting into the bloodstream leaving and giving you infection in your bloodstream about the distal size that's the appropriate job for two minutes factors doing good however what happens if you make too much to minik rosing factor that is potentially fatal and causes will be known as septic shock so too much to necrosis factor well now you have open capillaries throughout the body if you open capillary throughout the body you've basically depleted the blood volume because you get swelling all of the body if we don't have enough blood to pump and you go into shock your blood pressure dramatically get slower you get inflammatory cells leaking out throughout the body causing damage throughout the body in addition you have development of clots so you have disseminated you quite a lot with ease you have clots that preventing the perfusion of critical tissues for example the spleen more importantly the kidney going to kidney failure and in fact too much to necrosis factor is a terminal event at least septic shock has anyone here ever seen septic shock is very dramatic now how would you use this information to come up with a treatment for septic shock I'm sure you want to antagonize T&A okay she's OneNote agonized Tina and think of how could you harness the immune system to attack antagonize Tina what would you make in a device is he enough in fact you can inject you know your method of choice make monoclonal antibodies against TNF person goes into septic shock you give them antibodies against to necrosis factor it sequester's it it stops the whole process in fact this has been used as a treatment for septic shock giving antibodies Matunuck Ross this factory skin understanding the underlying immune methodology behind the septic shock allows us to develop therapies to treat those individuals again this is an important point is that new system is good but sometimes too much of immune response can be bad for the patient and that's why this delicate regulation is so critical in having the immune system work so well now how can affect yourself be recruited to the side of infection well first you have a local site that's infected it needs to express signals that recruit specific cells that are needed to contain infection and the flipside the effector cells need to have the capacity to recognize those signals stick in vibrating from the circulation into the inflammatory site so this morning I had a great example of that because you know I was looking I was being driven by the person who runs the bed records I'm staying at to come to this location we have never been here so as you know if you've ever been looking for an address or something you can drive up and down the street for a long time looking for something and you won't be able to find it you need that familiar thing to hang on to to pull you in so we're driving around and then we see Victoria waiting out there for us and we say oh that's our recognition molecule out there we know that the car should screeched to a halt stop is this in the right place for us to be and is a combination of both you basically the we were looking the she was providing the signal this is the right place to come and we were the effector cells looking for the signal tells that's where we should be coming and this same thing happens for infection now in order to recruit cells you need to have a whole host of chemokines and this used to be a very simple field of ten years ago but now you can see there's a whole broad range of chemokines that are being produced by macrophages and other cells this gives you a list of what cells made them this gives you a list of what cells are specifically attracted by these of chemokines and what effect they have and again this allows the specific and focal recruitment of different populations of cells to mount the appropriate immune response and again we'll discuss in the HIV lecture that is almost all you know HIV has utilized this to you as a co-receptor for HIV so HIV can utilize ccr5 as well as cxcr4 as its co-receptor so to get another example of how HIV is co-opted our normal immune response in order to enable more efficiently effective but again this is incredibly complex this is the innate immune system yet by having different chemokines made by different cells you could fold this recruitment of a very very tight specific population of cells to give you the appropriate immune response but whatever pathogen is initiating it now how two cells get mobilized that area they have to stick and in order to stick they have to have adhesion molecules and this is actually incredibly complicated field but in a simplistic point just appreciate that you have basically the one molecule is expressed and that molecule can bind specifically to another molecule that serves as its lying-in so in this case this is showing you P selected for example binds to see a Lewis which we'll discuss in a few minutes LFA one the very very important adhesion molecule specifically wanted to several molecules but predominately the one we'll focus on is item and again this is another example of how we mean ologists are your friends because I cam stands for interest cellular adhesion molecule so whenever you see am at the end of our immunological work the chances are very high that it represents an adhesion molecule so again if you see I Pam inside of the heater molecule LFA one you earn is lucky because sometimes you have to memorize that but LFA one I camp is a very very important interaction between two adhesion molecules that's utilized through multiple motifs in the immune response but you need to network have a way of getting the cell to stick so in fact Victoria served as out adhesion molecule because we recognize her she recognized us and then go down and then we came into the Nelson Mandela University to the right lecture hall so it actually very well this is just showing you item one and its expressive at high levels of activated endothelium when endothelium is not activated there's no I can water very low levels are expressed only after it's been activated and I can one finds very efficiently to LFA one well how can we put this together in order to allow us understand how infection focuses inflammatory cells to the site of infection well the first point to appreciate is that inflammatory cells macrophages neutrophils even b-cells and t-cells they express adhesion molecules so in this case they suppress high levels of LFA 1 an LFA 1 is very efficiently going to bind both - I am - as well as item 1 and therefore now if there's a site that's expressing high levels of I cam 1 and the macro visor is expressing high levels of LFA 1 has been activated sticking will occur and then I'm allowed specific targeting to that area so now if we think in terms of what happens on normally in the bloodstream so normally think of the bloodstream as a highway now if you needed to let's say you were picking somebody up on this on the sidewalk and you didn't know where they were you knew they were within like a 10 block area where the highway would you be will you be in this in the old way in the fast lane in the center of the highway you know because if it's a six-lane highway whatever would you be adjacent to the side walking is slowly where would you be anybody in this lower leg because if you see the person and you're in the fast lane it's across over three or four lanes of traffic you want to be digging up that person that's for sure on so that so that's what you would do we'll think about the bloodstream the blood is basically circulating in a very large blood vessel the possibility exists that you're circulating macrophages may need to diabetes into the local tissue because infection if those white blood cells were in the center of the bloodstream how efficiently would be able to like quickly veer off into a side of infection not very efficiently so there has to be a mechanism to allowed macrophages to maintain their proximity to the vascular epithelium to be ready to dive into a site of infection the way this is done is by having the expression of these low-level affinity adhesion molecules between the phagocytic cell and the an affiliate in this case you have CL Lewis expressed on the surface of the back acidic cells and I selected expressed on the surface of the endothelial cell this is another example of why immunologists are your friend because e stands for endothelium so II selected is the kind of selectiveness expressed in endothelial cells the interaction between these adhesion molecules is not very strong so think about kind of like velcro or like you know stick of notes you put it against the wall take it all around not it's enough to stick but it's not as permanent and therefore what happens is is that the guidance itself rattles along the surface of the endothelium it doesn't get stuck in one spot but it's enough to more to the sign of the endothelium so therefore if there's any signs of infection it could rapidly dive in so now let's see what happens when there's an infection so now again you've had a break in here in your in this case reptilian infectious process going on you have a phagocytic cells now have been activated by binding of the pathogen to their motif recognition receptors they're secreting chemokines those chemokines now are getting into the circulation in addition they're making cytokines that have an effect on the endothelium making these endothelium cells now to express high levels of IM one molecules in the absence of not being expressed now that is infected they are expressed and in addition to a necrosis factor also being made by macrophages oh is making the epithelium leaky so therefore the cell to cell junctions aren't as tight and then where you can get the appetit source of the cells so now as we put everything together in this case interleukin 1 diffuses into the circulation the macro file is rolling along the side Lewisohn deselected however the ILA activates the Fitness phagocytic cell it may up regulate LFA 1 expression now this magnifies comes down on an I cam 1 which is being expressed because of the infection the inflammation it binds tightly to the I cam 1 once it binds to the I can work it stops it grinds to a halt it's like you know you hit the brake line and stops and now once it stops it now sees hope is a leaky membrane I can deputies through it it's being drawn by the grade of the chemokine into the tissue here and now comes in and now we could be a phagocytic cell and this is the effective way how all these Dino sitting cells are rapidly localized to the area of the infection the ICAM ones expressed it's flags it down chemokines are expressed and it draws it in through gradient and now this is how you rapidly recruit inflammation to the appropriate site of infection if there's no infection no I cam expression no chemokines the magnifies keep going on does that make sense is that clear very very elegant system ok now again HIV is amazing instability to take advantage of the immune system so HIV wants to stick to cells in order to infect cells so when HIV buds from the membrane of t-cells for example it can actually take I cam 1 with it so in addition to expressing gp120 which is you know the binds of cd4 and ccr5 HIV said I know I'm not proud I'll take another stick molecule I'll take a 10-1 so now HIV is I can one so now HIV comes to a cell wants to infect in addition to being able to bind to cd4 and ccr5 now it can bind to LFA run so in fact you actually amplify the ability for HIV to infect a cell by virtue of its co-opting these adhesion molecule and allowing a different to be well in this case showing that expression of elephant one sorry expression to my cam one by viruses it amplifies the ability to infect so again HIV is co-opted does that heejun molecule approach that we utilized by the innate immune system to stimulate and increase its ability to be infectious so basically the question is that I pull in at the beginning of the lecture was how can they use this to detect invasion by pathogens and alert the immune system we discussed the innate immune system how it's there it's present the front lines where infection can occur and that plays a critical role in the first step it uses pathogen profiling by expressing receptors that recognize motifs that are uniquely expressed by bacteria also secretes factors that are unique with a recognized motifs that are uniquely expressed by by pathogens and that in and of itself binding to it can activate the cell and it will allow the immune system to contain the pathogens by recruiting the innate immune phagocytic cells to allow to engulf the pathogens before they have a chance to spread plus it makes factors like two minutes factor that see you left the local air blood supply to help also contain the pathogen and prevent spread but and the way that the immune cells on the right to size them in fact is by selected expression and up gradation of adhesion molecules at the size inspection as well as Lighting's for those molecules by the inflammatory cells themselves and if all else fails using Langerhans cells allow them now to bring in the adaptive immune response to really all hope clear the infection so again I think the if the angel comes to you don't forget keep the innate immune system it's a very important part okay thank you very much for your attention
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Medical_Lectures
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Biochemistry_Review_Session_1_for_Kevin_Aherns_BB_450550.txt
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Kevin Ahern: Good evening everyone! How's everybody doing? You're wasting valuable studying time by coming and listening to this. Student: [inaudible] Kevin Ahern: What's that? Student: [inaudible] So hopefully this is providing you more than you would get if you were studying. So if it's not then obviously you're wasting your time here, aren't you? I never thought about that before but it seems like the pressure's on me now to be productive so if it's not then you have someone to blame for why this isn't working. So how's studying going? [murmured responses] So the good news is that with that flood day we actually have about one less day of material on this exam than we usually have. So in that sense it is a lighter exam. My exams as you know always have a lot of material on them so that's, I guess, a bit of good news. But it means that it'll catch up with us later and it'll probably catch up with us on the second exam. So the second exam will probably have a lot of material on it is my gut feeling. This will be like I've done before so I'll be here answering questions for as long as you would like and we'll go from there. I will say if you haven't looked at the practice exam you should. The practice exam is the format of the exam is different this year, this term than it was last term. In fact it's significantly different. So I would encourage you to look at the practice exam and make sure you read the instructions carefully. One of the things that students mess up with on this exam is not reading instructions carefully, particularly for the first section. So make sure you understand what's in that first section. I meant to mention in class today and I didn't, and I'll try to put this out to everybody in an email tonight, but the exam has as I recall twenty points in the first section, fifty points in the second section, and thirty points in the third section. And there is an extra credit question on there for you. For singing loudly. I liked the singing today. That was kinda good. Alright, so where are we at? What's up? Giovanno? Student: [inaudible] Kevin Ahern: These are all academic questions. We're all academics. [Kevin laughs] Student: I was looking at the pyruvate dehydrogenase. In one of the figures you described it as being E1, E2, E3. And I have it here, I can show you. But actually when you're going through the process it's not that. Because the E3 that you're describing is actually... Is that just three forms? Kevin Ahern: Yeah a lot of people have concerns about what I said about E1, E2 and E3. Student: I don't quite understand it. Kevin Ahern: No that's fine. And I have no problem with that. I'm happy to. I'm glad you asked the question because that gives me a chance to just point out some stuff. One of the things that happens here is you have three subunits and you have things happening between them. So we don't have this thing where we have three enzymes. Enzyme A passes off to separate enzyme B passes off to a separate enzyme C, which is what we're used to thinking about with enzymes. This is all happening within the confines of an enzyme. So where we say exactly this happens and where we say exactly this happens and where we say exactly this, words are going to fail us. And that's part of what you're seeing the difference between what I'm telling you and what the book tells you. I will say, and I pointed out to some of you who've seen this, or who've sent me this, that this schematic that the book gives actually is completely consistent with what I said in words. And so my word descriptions actually come from this figure and I think this figure is a good, is a very accurate description of what's happening in that overall process. So as I said in class I'm not really obsessed with E1, E2, E3. I don't honestly think that they have that much relevance to us. I think the important thing is what's happening in this process. And if we call what happens, you know, a decarboxylation in E1, okay. But that isn't the important thing. The important thing is that a decarboxylation happens and then an activated intermediate is formed, and that activated intermediate is all in TPP. Whether we say that's E1, E2, E3, that's really not important for our purposes. And I really want to emphasize that. But I do think the steps in that mechanism that I talked about are important. So you have a decarboxylation. That decarboxylation if we're talking about yeast or bacteria can be a stopping point or actually a branch point to making ethanol if they don't have oxygen, that is they have no NAD. In our cells we don't have that option, and in bacteria and yeast who have oxygen, they don't stop there. They continue along the process. And that decarboxylation creates a reactive intermediate called activated acid aldehyde. And that simply gets passed to a lipoamide residue. So that lipoamide residue is a carrier of that reactive acid aldehyde and it simply takes that reactive acid aldehyde and passes it off to lipoamide. So it's just a pass, pass, pass that's happening. That pass happens to lipoamide and it's at the transfer to lipoamide where that oxidation occurs. And somebody came and asked me the other day, "Exactly where is that oxidation?" Is it as it's being passed or is it after it's on here, or whatever? The oxidation is happening as it's transferred. As it's transferred we see that this is going now to a thioester where before it was an aldehyde. Actually it was a reduced aldehyde up here. But this is definitely an oxidation that's happened and that's happened on the attachment of this guy to the lipoamide. And you can also see by the fact that the single bond which was the sulfhydryl, I'm sorry, the disulfide bond that was there has been reduced which is also evidence that oxidation of this guy has happened because it's given up electrons to these two sulfurs. The two sulfurs now are reduced. This guy gets passed off to CoA. That's not an oxidation. This is simply a transfer of this guy from one sulfur thioester to another sulfur thioester bond. And then everything else is just recycling back. This guy is reduced. It passes its electrons off to FAD. FAD passes its electrons off to NAD and we're back where we started. So those steps are what matter. Where we say E1, E2, and E3 fit in really doesn't matter. Does that hopefully clarify for you? And I sort of apologize for bringing up E1, E2, E3. In previous years I haven't done that and it's one of those things you learn in teaching is that when you give people more information than what they need they do get worried about that information, and that was my fault. My phone is telling me something. Sorry, somebody wants, I like this message. It says, "Hi, I'm in town and I have a pie for you. "Can I bring it by?" [laughing] Damn! Why did I schedule this? Yes, back here? Student: Are you expecting us to know the structures of the molecules in the citric acid cycle? Kevin Ahern: So I have addressed that in the lecture so if you look back on the... Student: I was just wondering cause it was on the practice exam. Kevin Ahern: Well again, don't study practice exams. Learn what I've talked about. Yeah? Student: Can you go over how the net synthesis of glucose is made with the glyoxalate cycle? Kevin Ahern: Yeah certainly. Student: Where does the extra carbon come from? Kevin Ahern: Yeah sure, that's a good question. Let me get you that. So let's go to the glyoxylate cycle. Where is the glyoxalate cycle? Student: Bottom of the page. Kevin Ahern: Right there, the very bottom. Thank you. You stare at it and your mind doesn't tell you. So the glyoxylate cycle as I said overlaps with the citric acid cycle in plants, yeast, and bacteria. And by overlap, all I mean is some of the enzymes are common between the two. If we look at citrate synthase it's common for the two. We look at aconitase it's common for the two. And if we look at the enzymes that convert succinate over to malate, they're not shown on here, but those same enzymes that are in the citric acid cycle convert succinate to succinyl CoA. I'm sorry, succinate to fumarate, fumarate to malate, malate to oxaloacetate. Those enzymes are common as well. The difference for the glyoxylate cycle is it has these two enzymes here that short circuit the citric acid cycle. So by short circuiting it isocitrate lyase and malate synthase are avoiding the decarboxylations that happen in the citric acid cycle. So in the citric acid cycle, when we get to isocitrate the next step is to make alpha-ketoglutarate with isocitrate dehydrogenase, okay? Well this is bypassing that. It's taking this isocitrate and making something else out of it. What's it making? It's making a glyoxalate, which is what gives the cycle its name, and it's making a succinate. Now what this means is that we started with six carbons here. We had six carbons here, six carbons here. Now we have four carbons here plus two carbons here. We haven't lost anything. In the citric acid cycle we would have lost one carbon and then we would have lost another carbon to get to this guy. So those two carbons are essentially saved right here. Well these two carbons actually become useful to us because now the cycle can take this second enzyme and add a second acetyl CoA, and in doing so make malate and then oxaloacetate. Now your question had to do with how do we have net synthesis of glucose? And the answer is as follows. If we go around the circle, we start with one oxaloacetate and we end up with one oxaloacetate. That's just like the citric acid cycle. However, we haven't accounted for this guy. This guy can be made into fumarate, malate, oxaloacetate, and now we've got an extra one. So it's that extra oxaloacetate that's used to make glucose. We don't have the luxury of that extra oxaloacetate in the citric acid cycle because we start with one, we add two carbons, we get two decarboxylations, and we end up with one oxaloacetate. Does that answer your question? Good. Yes, Liz? Student: I feel like this is kind of a stupid question, but why is that... Kevin Ahern: You know, about a third of the questions I get are start with that very thing right there. Student: Why is that advantageous? Like why do you want to make it into an extra? Because doesn't that just go through the citric acid cycle [inaudible]? Kevin Ahern: Well that's a good question. It depends on cellular needs. So I was talking to somebody in my office the other day about this, and this gives the bacteria, plant, and yeast cells flexibility that we don't have. Now for yeast and bacteria that's probably not a big deal, but for plants it could be a real big deal. Why is that? Well plants need a lot of glucose, not just for storage, but they use it to make cellulose. So now you've got an extra means of making glucose that you didn't have before. So you could, for example, live off of the oil that you made and make glucose out of it very readily. We can't make glucose out of fat. Fat is our storage medium and the only thing we can do with glucose is burn it in the citric acid cycle or make ketone bodies. We can't make glucose out of it. So this gives these guys much more flexibility in that process. Yes sir? Student: So my question has to do with the sodium/potassium channels. It's really just out of curiosity. So since in nerve transmission the sodium channels open first, they are the smaller of the two channels. Would I be correct in saying that? Kevin Ahern: That's correct. Student: So that's convenient because they don't even have to deal with... Kevin Ahern: Don't even have to deal with it. Excellent, yep. Student: So theoretically, or I guess hypothetically, if the two channels were switched in order would the potential energy of the electrochemical gradient be sufficient to overcome the unfavorable energy of desolvation and resolvation of the sodium going through the potassium channel? Kevin Ahern: You know that's a very good question. I probably won't have a very satisfying answer for you but I'll wing it. I'm a professor. I get paid to wing it so I'll wing it for you. I will wager that if you had it reversed, let's say that it were the potassium channels that were opening first, you would have not quite as clean of a signal. And part of that is, and I didn't really emphasize this in class, but more sodium ions leak in through the potassium channel than potassium ions leak in through the sodium channel. That sodium channel, because it is size exclusion almost purely, makes, I don't know, it's like a million to one selectivity for sodium. If you look at the selectivity of the potassium channel against sodium it's about a hundred to one to a thousand to one. So as you can see you get a lot more sodium in through that potassium channel. And what that would do would mean that that would really confuse and counter that gradient going the other direction. So I think it would have at the very least a small effect, but it could add up and make a larger effect. So that's what I would say based on what I know about the channels. Yes, Connie? Student: Going along the same topic, I guess I'm kind of confused how opening up sodium and potassium gradients will allow the information to be transmitted along the cell. Kevin Ahern: So her question is how does opening of these channels allow information to be transmitted along the cell. I guess I'll just draw a little thing here on the board. Let's imagine I have this nerve cell here. And here's the end where all the action starts. So I see, the very first thing that happens is I open the sodium channels and the sodium comes rushing in. [chalk scraping on board] And that changes the electrochemical potential and causes now the potassium, which is in here, those channels to open up and the potassium to rush out. What that creates at this end of this guy right here, what it creates is a high concentration of sodium and a lower concentration of potassium, where up here they're already low sodium and high potassium. Well what's going to happen when we have a difference between the two? Diffusion's going to happen. So we're going to see potassium moving this way, we're going to see sodium moving this way, and that's going to change the electrochemical environment over here, for example, as it moves down. So we see a wave of voltage change moving down that nerve. Student: Okay, so the fact that potassium is diffusing that within the cell is causing the channel to open? Kevin Ahern: Hold on. What's happening is as the diffusion is happening is you're seeing a voltage change. Do you agree with that? Student: Yes. Kevin Ahern: So we see a voltage change moving across here. The voltage change is what opens the gates downstream. So the thing that opens the gates down here, let's say we had a sense, we had a touch thing. So that fired the end of that nerve cell but everything else along here, all these gates along the side of the thing are sensitive to voltage changes. They're exquisitely sensitive to voltage changes. So the slightest voltage change occurring as a result of this wave causes a new wave to start, and it goes du, du, du, du, du, du, all the way back. That's how it happens. Does that make sense? Liz? Student: Does the sodium-potassium-ATPase keep running during that? Kevin Ahern: Yeah good question. Does the sodium- potassium-ATPase keep running during that? And it does. The gates opening up allows things to move faster than that potassium- sodium-ATPase keeps up. And these will close fairly quickly. This doesn't stay open very long because it would be counterproductive for the cell to do that because you're going to have one putting it out, it's kind of like the futile cycle, right? We've got the-not futile cycle, the uncoupled mitochondria where we had the protons coming in and we had the electron transport pumping them out. So exactly the same thing would happen if this were open for any period of time. Yes sir? Student: Do both the sodium and potassium channels have that free phase gated ball and channel protein? Kevin Ahern: They don't use the ball and channel thing, no. There are some things that do that. But they do have a pretty specific gate that's there, yes. Yes. Yes? Student: Not with the nerve transmission but with the P type ATPase with the sodium- potassium-ATPase, it says that the movement of sodium and potassium is central for the cell to be able to maintain osmatic balance. Can you go over how that is exactly? Kevin Ahern: Yeah, that's a good question also. So when we look at a cell, I talked a little bit in class about how you do the experiment in a basic biology class where you take a dialysis tube and you fill it with let's say a protein of some sort, and that protein is big enough that it can't pass out through the dialysis tubing. You close off the ends, you put it in water, and what you'll see is that the tube will swell, swell, swell because what's happening is water is diffusing in to dilute out the protein, but the protein can't come out. So if you let that go long enough what will happen is you'll actually burst the membrane because until you burst the membrane there's no way of equalizing that pressure. That pressure in that examples very analogous to what's happening inside of cells. If we think about what's happening inside of cells, cells are in fact membranous, or like that dialysis tubing. Water can move across their lipid bilayer very readily. It doesn't have problems like ions, like sodium or potassium or protons. Those have trouble moving across the lipid bilayer. But in fact water moves across as readily as anything. There's four things that move across the lipid bilayer easily. Water, carbon dioxide, carbon monoxide, and what's the fourth one? Student: Nitrous oxide. Kevin Ahern: I'm sorry? Student: Nitrous oxide. Kevin Ahern: Nitrous oxide will as well but that's not the one I'm thinking of. Water, carbon monoxide, carbon dioxide, and oxygen! Oxygen. If oxygen didn't we're in trouble. So all four of those move across the membrane very readily. Well water's the problematic one because water is the liquid one of that mess. So if we don't have some sort of trick to balance the pressure so that water doesn't want to diffuse in, water will and the cell will burst. And so the sodium/potassium imbalance that's created by those gates is one of the ways of dealing with that osmotic balance. And the mechanism of that is way beyond what we'd talk about in this class but it's just an example of a way to balance an osmotic pressure. Does that help? Student: It's only that it tricks it? Kevin Ahern: It is a way, if you want to say tricks it, that's fine yeah. Yes, Jodie? Student: So with tetrodotoxin affecting sodium gates, if there was enough of that in the system would that eventually lead to osmolysis of the cell? Kevin Ahern: Osmolysis of the cell. No, I don't think so because tetrodotoxin is a neurotoxin and the sodium gates are for letting sodium back in. That's not the pump. So don't confuse the pump with the gates. They're different things. So tetrodotoxin kills you because it kills your nerves. Yes? Student: Can you talk a little about liposomes and how they work? Kevin Ahern: Yeah, sure. Liposomes, let me give you a figure. I like that figure that your book shows of liposomes. There's "Lipids and Membranes." Where are we at? Anybody see it? Here it is. There we go. Preparation. So liposomes are essentially artificial membranous enclosures. If I were to ask you a definition that's what I would call them. They're artificial membranous enclosures and they're rather easy to make. To make them you have to get a group of molecules that would normally comprise a lipid bilayer. These would include the glycerophospholipids. These would include the sphingolipids that we've talked about. And generally in making liposomes those are the two things that people put together. They don't mess with cholesterol or anything like that, at least as far as I know. And the idea being left alone they will sit and associate with themselves, but if you force them to interact with water they will spontaneously form the lipid bilayer that we see. And that's really based on their chemistry. When you force them to interact with water those hydrophobic regions of the glycerophospholipids and the sphingolipids don't want to associate with water. So they actually orient themselves and that lipid bilayer arises as a result of that trying to avoid water. So those hydrophobic things, their geometry is perfect for being able to get on the inside and then leave the hydrophilic phosphates and things like that on the outside. So they arrange in that way. So if we shake it up or in this case sonicate it which is another way of doing that, we get them out of their comfort zone and they go out here and they start making these structures. Well the structures are random in how they form, so some of those structures will contain, if we have molecules in the solution, some of those structures will contain the thing that we had in the solution to start with. Well the beauty of that is that it's, just as these guys are relatively easy to make, so too are they relatively easy to fuse with natural membranes. And so by sort of forcing them together with natural membranes they will in fact fuse and when they fuse they dump inside of the cells the thing that we had inside of them. Student: [inaudible] Kevin Ahern: Is it a way for cancer treatment? It is a way to get cancer drugs inside of cells. If we have time later in the term, I'm going to tell you about a novel way of treating any kind of cell that you want to kill. I'll briefly tell you here just since you asked the question. So, has anybody ever heard of AVI BioPharma here in Corvallis? Nobody's heard of it? Okay, it's the biggest biotech company in Corvallis. It was actually created by a person who was a postdoc in this department back in the 1970s and he had a very simple idea that he said was going to cure every disease in the world. And his idea is simple enough that it will do that. So he spent, I don't know, probably ten years, literally, doing research in his garage, I'm not making this up, making these compounds that would literally target anything that you wanted to kill or stop. I'll tell you how they work in a second. So after a long period of time he got some investors and they saw the wisdom in this. They got a bunch of patents and they actually have some things that are currently in testing. They're perfect on paper. The problem is they will only work inside of a cell and getting them across the cell membrane is a challenge. So they've got some strategies for doing that. How do they work? Well everybody knows what messenger RNA is, right? So messenger RNA contains the coding for protein. His idea was, "Well messenger RNA is only one strand "and if I know the sequence of that strand "I can make a complementary sequence that'll pair with it." It's called antisense. Well rather than make it out of DNA or RNA which would get dissolved in the cell by the cell's enzymes he made an artificial one that was like DNA and RNA but then it wouldn't be destroyed by the enzymes. So it forms basepairs. It forms a duplex with the RNA just like a regular RNA or DNA would but it can't be destroyed by the cell. In the laboratory it works absolutely perfectly, because what happens is you put this in the place where the ribosome would go along and make protein and it can't make protein because there's a duplex there instead of a single strand. So if he has a bacterium that he wants to kill, all he has to know is a critical messenger RNA sequence. He chemically synthesizes this artificial sequence and he can kill them! I mean it's remarkable. If he has a cancer cell that has a single mutation and that single mutation is in a critical gene, he can stop that gene from being made. It's perfect. I mean it's brilliant, it's absolutely perfect, and he would be a Nobel Prize winner today if this were easily deliverable across cells. So the getting things across cells is a very, very important consideration. You can design a perfect drug but if you can't get it across cells you don't have a perfect drug. [Kevin laughs] Duh. So anyway, I find his ideas neat. He's a genius. He has absolutely brilliant ideas. He sold out when AVI went public and then started his own company in Philomath now called Gene Tools where he makes these synthetic drugs and sells them back to AVI to test. [laughing] It's a pretty good deal, you know? [Kevin laughing] Anyways, so. He knows more about making money than I do. So Jodie? Student: I had a question relating to the Tm point of membranes [inaudible] and that whole thing. So membranes can adjust their fluidity, sort of over time based upon lengthening, shortening, and saturated versus unsaturated. So why integrate cholesterol at all if it all it does is expand the Tm point [inaudible] fast enough response time? Kevin Ahern: Yeah that's a good question. I don't know fully the answer to that question. I wondered that myself. But it appears that expanding that Tm range does seem to have some important role in the cell, and I'm guessing it gives more flexibility and there's something in that transition range that is actually advantageous for cell flexibility, but I don't know the answer to your question. Student: Are there very many anchored proteins or any at all that actually use cholesterol as an anchor? Kevin Ahern: Are there any proteins that use cholesterol as an anchor? To my knowledge I don't know of any, no. Things that use anchors tend to use fatty acids an anchors, so you'll see a lot of them will use myristic acid which has fourteen carbons. It's a nonpolar thing and that fourteen-carbon thing just buries itself in the nonpolar region of the lipid bilayer and the other portion sticks out and holds on to the protein. And that's what's commonly used for anchors. I don't know of cholesterol being used as an anchor, no. Liz, you had a question also? Student: Could you do the same thing with I don't know if- Kevin Ahern: With what? Student: I don't know if I'm pronouncing this correctly, but micelles? Kevin Ahern: So micelles, no, you really couldn't because micelles don't form that enclosure in the same way. Do I have micelles on here or not? Okay there's a micelle. You'll notice that with this there's really nothing to put in here and it's not the same as the lipid bilayer. So we don't have an aqueous environment in here, essentially. We have, these tails are all just associated with each other. Would this be a way to carry in something nonpolar? Perhaps. But I'm trying to think. I'm not sure how well this would fuse with the membrane. That would be the one consideration because, for example, this would be like what a detergent does. And when we wash our hands we don't integrate that detergent into our membranes to any significant extent that I'm aware of. So I don't think that would go. Yes? Student: Back to the pumps, so there's the sodium/potassium pump and then calcium is pumped out due to the sodium being pumped in- Kevin Ahern: Oh, the sodium/calcium pump. Student: So are those pumps just completely unrelated to the amount of, are they in different cells or how does the potassium/sodium pump not affect calcium/sodium- Kevin Ahern: Oh okay. That's a good question. Let's start with the sodium/calcium pump. The sodium/calcium pump uses a sodium gradient to drive out the calcium that's inside of the cell. So it's an antiport. Sodium in, calcium out. And your question is if I have a pump over here that's pumping out sodium out and I have one over here that's letting sodium in are they at cross purposes? And the answer is no. What the sodium/calcium pump is doing is taking advantage of what the other pump is doing, just like what a nerve cell is doing in taking advantage of that sodium concentration gradient. So cells are extremely opportunistic and this is a prime example of that. Another place where we see this opportunistic thing is if we look at bacterial cells with respect to the lactose permease. You saw that protons on the outside were carrying in that lactose? How do we get protons on the outside? Well bacteria don't have mitochondria. They kick protons out of their cell and so those protons are carrying lactose back in. So yeah, cells are really advantageous at that and that's not counter, they're not going to cross purposes at all. Jodie? Student: Is the sodium/potassium pump and the sodium/calcium antiport, is the sodium/potassium pump able to sort of outrun the other one since it's a three-to-two whereas the other one is in a one-to-one sodium/calcium? [Kevin laughing] Student: I guess I'm just asking kind of relative rates. Kevin Ahern: So he's asking about relative rates. So he points out that the sodium, I'm repeating this for the people who didn't hear what you said on the TV. So the sodium/calcium, I'm sorry the sodium/potassium pump pumps out three sodiums and the sodium/calcium is letting in one sodium for each one. But we haven't said anything about how fast either one of these work. So I would be reluctant to say one outruns the other just based on how many that are there. I would agree that if the rate of pumping of one was the rate of movement of the other one that yes, the sodium/potassium would win. But you have to keep in mind that cells are striking a balance here. They're not doing a race. And that balance is rooted in that osmotic pressure. So once they get to that point where they want that osmotic pressure they're going to slow down. They're not going to keep pumping it out. That would be completely counter to their purposes in doing that just like in the mitochondria where we see the proton gradient gets very high, the mitochondrion can't keep pumping protons out even if electrons want to flow because the gradient's too high for it to do that. So that's really the determinant that's there. So it's not really, as I say, a race with any of those. Does that make sense? Student: So at that point would they just scale back on the transcription of what created those particular pumps and allow membrane levels to fall a little bit? Kevin Ahern: So would they slow down the transcription of those pumps and let them fall? I suppose ultimately that would happen, yes, but that would be a longer term proposition. When you affect transcription and translation usually, especially in eukaryotic cells those are longer processes. Yeah? Student: You said there was the osmotic pressure that controlled the speed of the [inaudible] pump. Do they have a way of sensing that pressure or-? Kevin Ahern: Do they have a way of sensing that pressure? And that's a good question also. The way they have of sensing that pressure is actually with the electrochemical potential that's there and so that does, as we've seen, that's why I was sort of drawing the analogy of the mitochondria, we see that that potential gets very high and they don't keep pumping protons out because they just simply can't do it. There's a similar thing that's happening with that sodium/potassium potential as well. That gradient gets high enough it's not going to fight that, yeah. Yeah, Liz? Student: Is the word "pump" synonymous with active transport? Kevin Ahern: Yeah, is the word "pump" synonymous with active transport, and the way I use it yes it is, absolutely. Whenever I say pump I'm thinking of an active transport process. And when I'm thinking of an active transport process, what am I thinking of? [class responding] At least one thing is always moving against the gradient. That's what active transport is. Yeah? Student: Could you talk about the mechanism of the iron copper peroxide bridge? Kevin Ahern: Yeah, did I talk about the mechanism of the iron copper peroxide bridge? Student: Like what it's for? Kevin Ahern: So let me show you. That's Complex 4 so let me pull that up and show it to you. I haven't said anything about relative to mechanism, no. And part of that reason for that as I understand, I'm not an expert in this field, but as I understand it, this peroxide bridge itself is a bit controversial. It used to be accepted as pretty much fact and now I've heard that there are other models that are explaining that. So I haven't really talked about the mechanism there. Let's see, Complex 4 is "Mechanism of Action" right here. So here's what you're talking about. I don't really have much to say about it other than the fact that this is a way of grabbing and holding on to oxygen and providing a place for the electrons to come and act. So because it's held in this configuration in the confines of this complex, the two protons and two electrons can come here, and then further two more protons can come over here and release that. So we had two electrons here and we had two electrons here for a total of four electrons. But I haven't talked about how this goes to this, goes to this. And this actually is a good and important question because it's actually in the mechanism by which this occurs that we see that the reactive oxygen species being produced as a byproduct. All I've said in class is that they can be produced as a byproduct but I haven't said how. And that's actually beyond the scope of this class to be honest with you. Yes? Student: So electrons come in one at a time through Complex 4? Kevin Ahern: Electrons come in one at a time through, they actually come from cytochrome C but they're being dropped at the Complex 4, that's correct. Student: So for all the other complexes do they come through in pairs? Kevin Ahern: Okay, so electrons come into the electron transport system in pairs. So NADH and FADH2 drop off electrons in pairs. So Complex 1 and Complex 2 both accept pairs of electrons. They dump off pairs of electrons to coenzyme Q but coenzyme Q passes them off one at a time in the Q cycle through Complex 3. So everything after coenzyme Q is one electron at a time. Student: Can we take another look at the two stages of the Q cycle? Kevin Ahern: Sure. Q cycle, yeah. Q cycle I think is pretty cool actually. I can't make that go over. Can I shrink it? No I can't do that. For some reason that's running off the side over there and I don't know why. Actually hold on. I'll bring it up. Does that bring it on? Oh, maybe not. Let's try this then. That didn't. Oh here we go. I think we have it on there now. So there is our Q cycle. Let's orient ourselves. This big guy here is Complex 3. So all the action of the Q cycle is happening in Complex 3 and this Q pool simply means the mixture of Q's with two electrons and Q's with no electrons. Both are necessary for this cycle. So we start this process with an empty Complex 3. So this is after we've loaded it. We're starting with an empty Complex 3. In the very first step we can imagine that an empty Q, that is with no electrons, binds down here. And then we have QH2 back up here. We have this empty. We start with a Q down here and we have a cytochrome C up here. We get a QH2, that is a Q that has two electrons that comes in. At this point we're ready to start this process. So to get started we have to have a cytochrome C, we have to have a Q with two, and we have to have a Q with none. So that split happens, as I've talked about here, one electron going to cytochrome C, one electron going to this other Q to make a Q with one electron. And that leaves this Q that started out with two with no electrons and no protons. The protons are kicked out. This guy goes back to the Q pool. In the next step, notice this guy stays in here. It hasn't gone anywhere. It stayed in the confines of Complex 3 and a Q with two electrons comes in. The Q with two electrons and a new cytochrome C, don't forget we need a new cytochrome C because this guy can only handle them one at a time, so we brought another one in that had no electrons, and this guy does the same thing that this guy did. One electron here, one electron here. And when that happens two electrons have been transferred, two more protons got pumped, this Q goes out, and this Q goes back out. So now we're back with the QH2. We've moved two electrons which essentially means we've converted one QH2 to one Q. Connie? Student: It almost feels like the coenzyme Q doesn't even give its electrons to Complex 3. It just kind of uses Complex 3... Kevin Ahern: Exactly. You're quite accurate in what you just said. It doesn't give its electrons to Complex 3. Complex 3 is basically a dock for, it's like a network center for these things to come together and do their thing. That's exactly what it is. Yes? Student: So protons are pumped out of Complex 1, 3 and 4? Kevin Ahern: Protons pumped out of 1, 3, and 4. That's correct. Student: How come they're not pumped out of Complex 2? Kevin Ahern: I'm Sorry? Student: How come they're not pumped out of Complex 2? Kevin Ahern: Well there's not enough energy there to do that and that hasn't, certainly hasn't evolved that way. So if there were the energy to do it then we'd get that. It turns out that getting electrons off of FADH2 actually takes some energy and so that energy isn't available as a consequence for the pumping of protons. Student: And that's why FADH2 doesn't produce as many ATPs? Kevin Ahern: And that's why FADH2 does not produce as many ATPs because not as many protons get pumped. Exactly right. Student: So is cytochrome C pretty much just like an electron acceptor? Kevin Ahern: Well cytochrome C is an electron transporter because it's accepting electrons here but it's donating them to Complex 4. So I like to think of it as a shuttle. It picks it up at one, it drops it at the other. It picks it up at one, it drops it at the other. So it's back and forth, kind of like the Ping-Pong mechanism of the enzyme, right? On the one side it's got electrons. On the other side, no electrons. Electrons, no electrons. Yes sir? Student: So this is all happening in the lipid complex? Kevin Ahern: This is all happening in the inner mitochondrial membrane which is a lipid bilayer, yes, but this is happening in the inner mitochondrial membrane. Student: So where does cytochrome C act? Is that in the matrix or is it in the membrane as well? Kevin Ahern: So cytochrome C, that's a very good question, is a peripheral membrane protein. So it doesn't extend all the way through but it's linked with that one layer of the lipid bilayer and it's because of that that it's very mobile. If you think about this this guy is acting as a shuttle. It has to move quite a bit between Complex 3 and Complex 4 and so the more mobile it is the faster those electrons can get moved. That's also a consideration for coenzyme Q, and coenzyme Q turns out to be a very small molecule. So small molecules can move very rapidly in the lipid bilayer and that works out well. So coenzyme Q is bringing things between Complexes 1 and 3 or 2 and 3, and cytochrome C is taking things between 3 and 4. The big complexes don't move very fast and so it's really nice to have those little shuttle things moving things quickly between them. Yes sir? Student: I may be getting my terminology messed up here but is coenzyme Q also referred to as ubiquitin? Kevin Ahern: Coenzyme Q is also referred to as ubiquitin, yeah. Actually, let me back up. It's not called ubiquitin, it's called ubiquinone. There's a difference. Ubiquinone. I make that mistake myself sometimes. Yes? Student: You said that FADH2 doesn't have as much energy as an NAD, but back in, it's like E2 or E3 we're you're trying to recharge, what was it... Kevin Ahern: Yeah I know exactly what you're going to say. That's a very, very good observation. So Connie, let me show you what you're getting ready to ask. It's rare I have somebody notice this but it's a very, very good observation, so let me get my- Ah, come on. Oh, that's not what I wanted. Green is what I wanted. So what Connie's asking is she's noted that when I talked about this reaction scheme here, look what happens. This guy is donating electrons from FADH2 to NAD. And I just told you something contrary to that. I said that it was harder to get these electrons and protons off than it was off of this guy, so this guy shouldn't be able to donate electrons there. Anybody know why or how it does it? Student: Is it because it's part of the [inaudible] complex [inaudible]? Kevin Ahern: Exactly. This guy is covalently linked to the protein and the electronic environment that this is in is different than that of a free FADH2. So we call it FADH2 because it has that ring but it's actually covalently linked to the protein and that's not the same as an FADH2 that's floating around. Very rare I have somebody notice that. That's a very good question. Yes sir? Student: Speaking of FADH2, well I guess the shuttles of the electron transport chain, is there movement between the complexes more or less random from diffusion? Because it seems like they're not specifically linked to a protein that is moving them directly to another complex. Or maybe I'm mistaken. Kevin Ahern: I misunderstood the first part of your question. Student: Cytochrome C, for instance, moving the electrons from Complex 3 to Complex 4. Student: Is that random? Kevin Ahern: Oh, is the movement of the complex random? Yeah, it is, yea it's random. Now keep in mind that there's a lot of protein in this membrane. Remember I said the inner mitochondrial membrane is loaded with protein. So that random movement has to have advantages of proximity if it's going to be effective. And that's one of the reasons it has a lot of protein in there, so it's much more likely it's going to bump into the right thing and donate that electron, but that's exactly it. I need to make a phone call, folks. I hate to do this but my pie is going to disappear if I don't and I just can't let that happen. I would never do this in class, but let me just do this really quickly. They'll like the fact that I'm doing this in front of the review session. I could have you guys say hi to them, how about that? Student: Give them the link, they can watch it online. [Kevin laughing] Student: I'm really curious what sort of pie this is. Kevin Ahern: Yeah me too! [Kevin laughing] Bring some up here for the whole class! [Kevin laughing] Jessica, how you doing kid? Well I'm not avoiding you, but you're not going to believe where I'm at right now. No, I'm standing up in front of my BB 451 class giving a review session and I couldn't respond to your thing and I said, "You know there's a pie at stake here," so the class said, "Hell yeah, get on there!" [class laughing] Twist my arm! I'm in ALS. Oh, well okay. Come on up. We're in 4001. Maybe people will help me eat it. [laughing] Or if the exam doesn't go well they may throw it at me. [laughing] So I'll see you in a bit. Take care. Bye. I've never done that before. That's weird. But it's for pie, folks, I mean come on. Student: I think this is probably a stupid question but it has to do with the inner mitochondrial membrane cells. So we have the DAHP, we have the cytoplasmic, cytoplasm 3, G3P-dehydrogenase. Why is it going, because usually when you have a dehydrogenase you always have a reduction of NADH and here instead you're having a production of NAD. Kevin Ahern: Okay so you're talking about the shuttle system, right? So let me go to that shuttle system. Student: But then it's the same terminology in the other [inaudible] it goes in the other direction. Kevin Ahern: Right. So let's see. The shuttle right here. This is the one you're talking about? Student: No, it's the glycerol. Kevin Ahern: The glycerol, okay. So here's your shuttle. And your question was why are we making NAD? Well remember that what we're trying to do here is we're trying to move electrons from NADH into here. Student: No I mean I understand why that's happening. It's the name because every time... Kevin Ahern: Oh, oh, oh... Student:...[inaudible] in that situation there may be an NAD while being [inaudible] Kevin Ahern: I see your question. Very good question. I hadn't even thought about this. It's a very good question. It's named for the reverse reaction. So if we go that way we make an NADH. I hadn't even thought about that before but that's the reason why. But it is a reversible reaction and that's what's happening right there. Yeah? Student: So if you get less energy from the electrons because you're transferring it to FAD, why wouldn't it just transfer into NAD? Kevin Ahern: There's no NAD to transfer it into. So the NAD is in here. In other words this guy can't cross the membrane and neither can the electrons. Student: So NAD can't be attached to an enzyme [inaudible] Kevin Ahern: NAD is not attached to anything out in the cytoplasm. You've got two choices. Student: [inaudible] Kevin Ahern: Right, but I'm saying that there's nothing that attaches to NAD like that. NAD is free in the cytoplasm. There's no attachment. There's no shuttle, I mean there's no mechanism to get it in but there's only this shuttle that's there. Yeah? Student: So is FAD giving its electrons to coenzyme Q or is that [inaudible]? Kevin Ahern: Yes, so what's happening here is this is accepting the electrons here and then you're seeing the other reaction where this is dumping its electrons to coenzyme Q, exactly. So what's happened is this guy has gone from electrons at NADH to electrons at coenzyme Q but we bypassed Complex 1. So we didn't have any pumping happening from Complex 1. Liz? Student: In the citric acid cycle when you take out a GTP, does the cell use that GTP [inaudible] ATP? Kevin Ahern: Yeah, so the question is when the cell makes GTP in the citric acid cycle does it use it in the same way as it uses ATP and the answer is yes. Exactly the same energy and GTP is an energy source for protein synthesis. It's a very important energy source. Absolutely. And there's other ways of making GTP, so this isn't the only way cells have of making GTP. But it's one. Yes? Student: [inaudible] Kevin Ahern: Is there a use difference meaning some reactions use it, some others don't? Yeah there's definitely a use difference. So in protein synthesis we don't see anything except GTP that's used. You can't substitute ATP. And if look at, say, hexokinase. So hexokinase you remember used ATP to put a phosphate onto glucose. GTP won't work. So in essence those aren't barriers that get crossed. Enzymes have evolved one or the other. Yeah? Student: Did you say this one was involve in insect muscle? Kevin Ahern: This is involved in insect muscle. Student: Okay, and then the malate/aspartate one is in us? Kevin Ahern: In us, mmhmm. Good questions tonight. I told Indira, I said, "Oh they'll probably "have a couple questions. "There's no problems to solve." So there's been really good questions tonight. Connie, a question? Student: Just totally random but in the sodium/glucose port, do they, since they come in together, do they just go through one channel or are there two separate channels? Kevin Ahern: Yeah somebody asked me that after class the other day. With the sodium/glucose channel, are there two different channels that are used. I don't know the answer to the question. It wouldn't surprise me if you didn't have a separate one for each one but I don't know the answer to that. They are very different. One's charged, one's not. One's a ring, the other is not. So I would think that there would be separate ways of doing it but there also may be an association that's made between them that brings that through. So I just don't know the answer to that. Yes sir? Student: According to the conversion between pyruvate and acetyl CoA, I believe you mentioned that the independence of the decarboxylation is important for the oxidation for fermentation? Kevin Ahern: Yes. Student: If there's implications can you go over that? Kevin Ahern: Yeah sure. So the question had to do with the ability of the decarboxylation of pyruvate to be independent of the oxidation and is that important, and the answer is yes it is. So if we look at that mechanism shown right here, we see as I noted a decarboxylation first followed by an oxidation followed by transfer to get acetyl CoA. This is where I called E1, E2, E3 and people didn't like that. So that was my mistake. Now, your question has to do with is it important that, hey there she is! There's my famous pie [Kevin laughing] This is cool. I just want to make sure they don't throw it at me. Thank you Jessica. Jessica: You're welcome. Kevin Ahern: This is one of my former students and she is currently at OHSU in her second year, third year of medical school. Jessica: Yeah... Kevin Ahern: And she didn't throw it at me. Jessica: This matters. Kevin Ahern: Thank you Jessica. I'm sorry I can't get together with you guys tonight. If you try to call Indira she's giving an exam right now so both of us are a little tied up. Sorry. Thanks again. Take care. So the significance of the independence here is that because this can happen independently there's no NAD that's needed. Now the significance of that is since no NAD was needed we can use the product of this guy and then take electrons off of it, or I'm sorry, take the, use the様et me back up. We can take the product of that, which is basically acid aldehyde and we can reduce it with electrons from NADH to make NAD. If we were using NAD then it'd be an even trait. So the significance is that this now provides a substrate that will accept electrons and allow yeast to make NAD and, of course, when they're doing that they're making alcohol. Student: [inaudible] Kevin Ahern: That's when oxygen is not present. That's right. So like in any cell when oxygen is not present NAD is limiting and so you have to have other ways of making NAD and this provides a way for them to do that. Student: [inaudible] Kevin Ahern: Yeah, so can I repeat that? So to repeat that, what I said was the significance of the decarboxylation being independent of the oxidation is that this making of this molecule does not require NAD. If they weren't that way then this would require NAD and it would produce NADH. Then if the cell reduced that you would use an NADH. You would use an NAD and you would make an NAD. There would be no net gain. But if it sidetracks here it sidetracks before the oxidation happens and it's able to reduce it and so now you can have a net synthesis of NAD if you go down here and do the fork in the road. Does that make sense? No? Is that a yes or a no? I see some people saying no, so let me... If I go through here. If I go all the way over here. If I don't subvert that, this requires an NAD. To go right here at this step something's got to take those two electrons. Somebody's got to take those two electrons. NAD would take those two electrons and make an NADH. So if we go this route which is what we go through there is no way of making NAD. We can't do it. But if you stop it here before the oxidation has happened, which is what his question was, is there significance that this is separate from this, yes it is because if we stop it here we can run down and add electrons from NAD to this to make ethanol. That's what yeast are doing. And when they're making ethanol they're also, the reason they're making ethanol is that generates NAD which keeps their glycolysis going. Yes? Student: I thought that the [inaudible] Kevin Ahern: No, no, no. So I'm not sure what you're saying. Oh you're talking about here in the mechanism? Yeah well ultimately they do go to NA-, ultimately you make an NADH, right. So those are two more steps after this but the product is NADH, not NAD. Yes sir? Student: Can we look at how the ABC transporters are different from the P-type transporters? Kevin Ahern: Well I haven't said anything about how they're different other than the fact that his question has to do with the difference between the ABC transporters and the P-type transporters. But to address your question what I did say about the P-type was that they involved the phosphoaspartate. And that's the only thing I said about them. The ABC transporters simply don't involve a phosphoaspartate. So that phosphoaspartate is a covalent intermediate that's transient. It's like the serine protease where we have the reactive alkoxide that does the attack and does its thing. In the case of the P-type ATP transporters they have, to show you the mechanism, they get that phosphoaspartate sidechain in the middle of their mechanism and that helps to drive the overall process. And that is shown here, I think. So here is the chamber. The transfer of the phosphate from ATP to the transporter puts a phosphoaspartate there. That causes a charge change in the protein. It's a driving force for the shape changes that happen and the shape changes that happen result in the movement of the calcium out, the release of the ADP. Then in the next round in the cycle the removal of that phosphate causes the system, ultimately, to get back to its original state. So we have two states that arise from that phosphoaspartate. Student: So in step five it releases that other phosphate but where does that water that comes in end up at? Kevin Ahern: Where does it end up at? So when you release a phosphate, one of the protons goes here and the OH goes here. That's what a hydrolysis reaction will always do. What's that? Student: [inaudible] Kevin Ahern: It's Monday, yeah. Yeah? Student: Did you explain how the ABC transporters work? Kevin Ahern: I haven't explained how they work. I just simply said they were another transport system. I used to go through all the transporters and I would show the steps in all of them, and to be honest with you, A I'm not a mechanism person, and B, it's really hard to ask mechanistic questions like that without simply regurgitating the whole thing. And I just don't see that that's really very useful. You saw as much mechanism when I talked about serine proteases as I will talk about. And I talk about it there because I think it's important to know a catalytic mechanism and how it occurs. But mechanisms in general like this, I mean what can I say? There's a shape change, you know? I mean that's really what's happening in all of these things and it's really hard for us to conceptualize that and sort of do something with it on the exams. So I don't really talk much about them anymore for that reason. Yes Connie? Student: How does succinate dehydrogenase differ from other dehydrogenases? Kevin Ahern: How is succinate dehydrogenase different from other dehydrogenases? Well it's different in the sense, only in the type of reaction that it's catalyzing. So succinate dehydrogenase is an enzyme that is taking electrons directly off of a carbon-hydrogen system. The other dehydrogenases that you've seen in class up until this point have involved oxidation of an OH to make a ketone or an aldehyde. You've also seen decarboxylations that have happened of alpha-keto acids like alpha-ketoglutarate and so forth, but you haven't seen something take protons and electrons from a single bond like that. And that takes something with greater pulling power to get those electrons out and that's why FAD is used. So it's the FAD I would say that's more interesting and more unusual than is the enzyme itself because the FAD provides that pulling power to get those guys off of there. Student: Can you repeat that part? Kevin Ahern: I'm not sure what to repeat. Student: You were just talking about dehydrogenases we have seen. Kevin Ahern: Oh the type of dehydrogenases you've seen in the past have been things that converted an alcohol to an aldehyde or a ketone, or they did a decarboxylation, but you have not seen dehydrogenases that are acting on a bond system like this. Now this one is not unique. We'll see when we talk about fatty acid oxidation that the very first step in fatty acid oxidation involves an oxidation just like this and it uses FAD for exactly the same reason. These are hard to get protons and electrons off and it takes the pulling power of that FAD to make that happen. The second step of fatty acid oxidation is exactly like the malate悠'm sorry, is exactly like going from fumarate to malate. You add water across the double bond. The third step is just like malate going to oxaloacetate. You're converting an alcohol to a ketone. So these reactions that you see here in the citric acid cycle you will see again when we talk about fatty acid oxidation. Yeah? Student: Just going off what you said about the succinate dehydrogenase was that in the membrane [inaudible]. Kevin Ahern: It is found in the membrane and others are generally soluble. That is a difference of the enzyme. That's correct. But there are other enzymes. You've seen, for example, adenylate cyclase last term. That was embedded in the membrane. So there are enzymes that are in fact embedded in membranes. That's not that unusual. Good questions. Connie? Student: Can you go over pyruvate dehydrogenase regulation? [inaudible] kinase and phosphatase? Kevin Ahern: Pyruvate dehydrogenase regulation is, let me think about this for a second. So I talked about that relevant to... Student: [inaudible] Kevin Ahern: There it is. I knew it was there somewhere. So with this enzyme we have a couple of reaction mechanisms. One is that when we phosphorylate the enzyme, the enzyme is inactive. When we dephosphorylate the enzyme it is made active. That's just simply a kinase/phosphatase type reaction. That's not to my knowledge protein kinase A. That's a different kinase that's doing that, I believe. I'm pretty sure that's right. The other consideration is the enzyme, because it is using NAD, it is sensitive to NAD concentrations, and these factors negatively affect it. ATP, NADH, and acetyl CoA. And acetyl CoA being a product not surprisingly is going to have a negative effect on the enzyme. But again what I said about regulation, I really want to emphasize it. When we think about the regulation of not only this enzyme but also the citric acid cycle, the biggest consideration is the amount of NAD. That's what it comes down to. NAD and FAD. Yeah phosphorylation is a factor. We haven't talked much about it this term, and we won't, but the biggest factors for us to consider are the oxidation state of the cell. If the cell has plenty of NADH this ain't gonna go. Yes sir? Student: We talked a little bit in class and then we read in the book about the different levels that NADH and FADH2 have relatively for pumping protons across the membrane to contribute to that gradient. Which did you want us to use? Kevin Ahern: I'm not sure I understand the... Student: Because NADH and FADH2 contribute the electrons that result in protons being pumped across the inner mitochondrial membrane. Kevin Ahern: Right. Student: But they contribute in disparate amounts. One of them contributes more. I believe the book said it was 2.5 to 1.5. Kevin Ahern: I said 3 and 2. That's what I would use. And again this is not about getting hung up on numbers there but you can see that there's more protons pumped if it's coming from NADH than if it's coming through FADH2. Student: And the accepted range overall for ATP generated was between 30 and 38? Kevin Ahern: That's right. So 30 and 38 encompasses pretty much what people will all agree on, yes. Yes, Connie? Student: You made an anecdote that carbons aren't really lost after the second round of the citric acid cycle? Or something like that. I don't know if I mis- Kevin Ahern: Well carbons are lost in every round. I think what you're thinking about is where I as an aside said the carbons from acetyl CoA that come in aren't lost in the first round. So let me show you the cycle. So where's the cycle? Here. No that's not it. Student: [inaudible] Kevin Ahern: Which one? Student: [inaudible] Kevin Ahern: This one? This is color-coded. Here's an acetyl CoA coming in. They're in green. There's the green, there's the green, there's the decarboxylation. You haven't lost any green. There's the green after the second decarboxylation. So in the first round you didn't lose any of the carbons of acetyl CoA. Only on the next time around will you lose them. Student: Okay so it just chews off the opposite ends each time? Kevin Ahern: Well they're just not in a position to get decarboxylated in the first time. It's not a major point. As I said in class, and that's the reason I made it an aside was some people get really, their panties in a wad about this. But I mean to be honest with you it's okay, that's nice, but so what? Student: Can you talk about neurotransmitter recycling? Kevin Ahern: Neurotransmitter recycling I think is actually kind of interesting, although I just mentioned it with respect to cocaine addiction. But I think it's interesting to think about. So let's see. "Membrane Transport" here. And when we see... So this is how nerve cells talk to each other and that communication between nerve cells is obviously important for that information to get transmitted. That happens when this voltage wave that's moving down here that happens from the movement of these sodium and potassium ions comes in, that voltage wave causes these synaptic vesicles which contain the neurotransmitters to fuse with the membrane and release the neurotransmitters. So the neurotransmitters are this guy right here. The binding of the neurotransmitters, the exit of the neurotransmitters here cause them to bind to this nerve cell and stimulate another wave of sodium and potassium moving through here. And normally these guys would have to be recycled and normally they are recycled just fine. But in, cocaine is one example of a drug that in some case, with some neurotransmitters interferes with their ability to be reabsorbed. And it tends to be associated with pleasure centers and so when that happens then the cocaine enhances that pleasurable feeling by leaving those neurotransmitters stimulating this pleasure neuron right here longer. Student: [inaudible] Kevin Ahern: It's actually not going into the green. It's binding to the green. It's not entering the green. So you see it's being released here and so it's exiting here and it's going to bind to receptors on here. And then when this thing's done with them it lets them go. Well if it lets them go and they're not getting recycled they can come back and bind and bind and bind and so it prolongs that good feeling. That's why it gives people that high. Make sense? Student: Isn't that one of the unique targets of neurotropic medication like SSRIs do? Kevin Ahern: Don't know. Student: Reuptake prohibitors to prolong... Kevin Ahern: There are reuptake inhibitors that are used in treatment but I don't know, again. It depends on the neurotransmitter and the neuron itself that's being affected, Jodie. I'm not a medical doctor so I can't tell you that. I don't know. Student: I actually have a yes/no question. Kevin Ahern: A yes/no question. The answer is no. Student: Well the question is does arsenite prevent lipoamide from reforming? Kevin Ahern: Does arsenite prevent lipoamide from reforming? What does that mean? Student: Once lopoamide becomes sulfhydryls. It needs to reform before it can get used- Kevin Ahern: Oh good question. So I'll ask everybody else that question. So the product of the... ...treatment of arsenate, and where was that? That was my phone. It's going nuts here. Where was that arsenite? That must have been here, right? Student: [inaudible] Kevin Ahern: Right, right. Second from the bottom. So your question is that we have this guy right here but for it to be active it's got to be in the disulfide state, right? Well what happens to these things in cells? We learned that last term. They're readily oxidized. That is it doesn't take much to oxidize them. You put them in close proximity they will readily oxidize, and if they do that you've made a disulfide again. So the answer is nothing extra would be needed to do that, no. Student: I guess i just got confused. Clear as mud? Alright guys, that's a long one. So I will get the video posted as quickly as I can and thank you for a bunch of good questions and thank you for letting me talk to the pie lady. This is my treat. [END]
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Medical_Lectures
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Hyperglycemic_Crises_DKA_and_HHS_Part_2_of_2.txt
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a word about effective plasma osmolality this is a frequently overlooked value in dka and HHS but it's critical to be aware of its significance it's calculated as 2 * the sodium plus the glucose divided by 18 the reason for its importance is that patients have a significant risk of neurologic decompensation if their effective PL plasma osmolality is above 320 mosmos per kilogram I can't actually imagine a scenario which this value increases further with appropriate treatment so as with the ketones it's not needed to follow serial levels there are a whole slew of additional lab abnormalities in dka and HHS hyponatremia at presentation is nearly Universal as a consequence of salt loss during osmotic diuresis poor po intake and the trans cellular shifts due to hyperosmolarity patients typically have total body deficits of other electrolytes as well as listed here this is probably an appropriate place to first point out that while these patients may have a total body deficit of potassium patients in dka often will have normal or even elevated serum potassium levels due to trans cellular shifts of potassium as part of the complex mechanisms which regulate and attempt to normalize extracellular pH at this point I'm going to shift the focus of the talk to discussing the management of these conditions the four General domains of management of both dka and HHS are IV fluids insulin electrolyte abnormalities and the precipitating causes dehydration is a major component of both dka and HHS in IV fluids Remain the most important intervention for both conveniently a simple algorithm for the use of IV fluids can be applied to either start by bolusing one to two liters of either normal saline or lactate ringers if the patient is hyperemic you probably should go with normal saline however in most cases the choice of fluid doesn't actually matter you should consider a series of smaller bises in patients with a history of cardiac dysfunction once one to two laders are in reassess the patient's volume status if they still appear severely volumed completed that is hypotensive tartic with Poe urine output and with nonvisible neck veins continue the fluid bishes of NS or LR until the volume status is more improved once the patient is in the realm of mild volume depletion look at the serum sodium if it is low or normal switch to a continuous infusion of normal saline at a rate of 250 to 500 CC per hour if the serum sodium is high use half normal saline at 250 to 500 CC per hour in patients with a history of heart failure or in those presented with any degree of hypoxia you may want to consider lower infusion rates in all cases remember that serum sodium shouldn't change more than 0.5 mil equivalents per liter per hour increasing the sodium faster than this can trigger Central Ponting myelinolysis while decreasing the sodium faster than this can trigger cerebral edema fear over excessively fast correction should not prevent addressing severe volume depletion and shock very early in management since the risk from shock is greater than that from these two other complications generally an overly rapid sodium correction becomes more of a concern after you switch over to a continuous infusion rate as a very general guide typical total body fluid deficits in dka range from about 3 to six liters in most patients while in HHS it is closer to 8 to 10 L so now what about insulin myal dka that is dka with an arterial pH greater than 7.25 and serum bicarbonate level greater than 15 mil equivalents per liter and with an alert patient subq insulin is a reasonable choice however in any dka more severe than this or in any form of HHS at all a continuous infusion of intravenous regular insulin is the preferred route of delivery there are several different protocols for selection of IV insulin regimens but there has been no difference seen when compared head-to head therefore for Simplicity and ease of remembering I prefer the following option A 0.1 unit per kilogram bolus followed by a 0.1 units per kilogram per hour continuous rate we are not aiming to immediately or rapidly bring down the glucose back to normal rather we are targeting a rate of improvement of no more than 50 to 70 milligrams per deciliter each hour remember that glucose is osmotically active therefore dropping the serum concentration too quickly has the same risk of precipitating cereal edema as decreasing the serum sodium too quickly once the serum sodium is below 250 dextr should be added to whatever IV fluid you're using this prevents hypoglycemia by overshooting and also helps to prevent the aformentioned cerebral edema as we've mentioned before hyponatremia is very common to both dka and HHS due to a transcellular shift of water from the intracellular to extracellular compartments that is triggered by the serum hyperosmolarity this situation is given the very obvious name of hyperosmolar hyponatremia as a consequence these patients who are hyponic by numbers may not actually have a separate path logic condition that directly causes hypona to correct for this effect you can take the measured serum sodium and add 2.4 time the measured glucose minus 100 over 100 for some reason which I have never been able to determine a lot of pocketbooks that house staff carry around on the wards cite a correction factor of 1.6 but the literature supports a value of 2.4 now this idea of the corrected sodium can be very confusing the initial point to make is that the measured serum sodium is the actual true serum sodium it's not like the glucose somehow interferes with the Sodium assay rather the adjusted sodium is essentially an estimate of what the body's sodium level would be if the hypoglycemia wasn't present so what now how does this affect our assessment first when determining effect of plasma osity according to the previously mentioned equation the measured serum sodium should be used not the adjusted serum sodium also when determining the appropriate IV fluid between normal saline and half normal saline the measured serum sodium is the appropriate one to use as well so why do we care about the adjusted sodium at all the reason it's important is in the situation where the adjusted serum sodium is still below the normal range if that's the case the patient must have another pathologic process contributing to hyponatremia Beyond just the hypoglycemia and hyperosmolality for example sadh or curosis or nephrotic syndrome or something like that that's the only reason you should ever bother calculating this value so what about potassium in both dka and HHS potassium levels almost always drop during treatment in both conditions insulin directly stimulates potassium uptake by cells and the IV fluids will improve GFR and its renal excretion in addition in dka the correction of acidosis will promote exchange of intracellular hydrogen ions for extracellular potassium ions because of that additional mechanism abrupt and a dangerous hypokalemia is more or less limited to dka and only rarely seen in HHS in both conditions however once the potassium is below 5.3 mil equivalents per liter the IV fluids should contain at least 20 m equival per liter of potassium and if potassium is below 3.3 mil equivalents per liter the insulin should be held entirely until hypo glycemia is addressed keep in mind that these cut offs are based on guidelines which are themselves a little arbitrary also remember that pottassium is osmotically active in itself so adding it to normal saline will make a modestly hypertonic fluid even more so phosphate is another electrolyte to monitor hypophosphatemia can develop a worse enduring treatment as insulin stimulates intracellular phosphate utilization such as in the formation of ATP however routine replacement in dka and HHS has been shown to have no outcome benefit therefore aggressive repletion is usually reserved for those patients with symptoms or complications of hypophosphatemia or when serum phosphate drops below 1 milligram per deciliter which is quite uncommon to remind you complications of hypophosphatemia include muscle weakness or even Frank rabdom mysis cardiac arrhythmias seizures hemolytic anemia and altered mental status another major component of treatment is the identification and management of the precipitating cause of the hypoglycemic crisis the list of potential causes is quite long but statistically the three most common triggers are gastroenteritis UTI and pneumonia these lead to at least 50% of dka events because monoc cardiio esmia is another important precipitant every patient with dka or HHS requires an EKG and at least a basic assessment for ACS one last intervention to consider in dka in particular is IV bicarbonate historically this has been used with some frequency with the thought that it would help correct the patient's acidemia more rapidly unfortunately there are multiple theoretical reasons why this may actually be harmful and literature's evidence agrees that it is not beneficial therefore guidelines recommend against routine use unless the pH reaches truly extreme levels with a cut off suggested by some authors of less than 6.90 how frequently should one reassess these patients with hypoglycemic crisis the short answer is very frequently to be a little more specific to monitor dehydration their vitals and urine output should be checked at least q1 to two hours their blood sugar should be checked Q 1 hour until it is below 250 and stable for several sequential readings their electrolyte should be checked every 2 to 4 hours until the crisis is resolved that brings us to this last slide how do we determine when the episode of dka or HHS is resolved things to look for include normalization of volume status normalization of mental status normalization of the an Gap a serum bicarb above 18 mil equivalents per liter and a pH above 7.30 perhaps counterintuitively a normalization of serum glucose is not on this list and should not necessarily be expected or even particularly desired during the treatment of hypoglycemic crisis once the dka or HHS is is felt to be resolved the insulin drip if the patient was on one should be converted over to subq insulin remembering to overlap the two by at least one to two hours I hope you have found this lecture on the identification and management of hypoglycemic crises both informative and useful once again this has been Eric strong of the paloa veterans hospital and Stanford [Music] University [Music]
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Medical_Lectures
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Immunology_Lecture_MiniCourse_14_of_14_EvasionImmune_System_by_Pathogens.txt
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okay I think we get it started again so this is it this is a Harlem stretch this is the last lecture I think it's amazing to me how quickly it went and I think next time we're gonna do 14 lectures in one day you know this is like no big deal like actually it is a canoe marathon going on I was I heard what yeah so we could actually maybe talk to Victoria next time we'll do a combination canoe marathon and the ology thing so what we'll do is we'll all kind of canoe at the same time guys can have been creative Oh Crocs hold on a second you want a croc I'll show you a frog how's that yeah actually this is like a macrophage this idiom is an e/m of a macrophage okay okay so so this is the final lecture that will be appropriate and and and the way I was thinking of titling it was revenge of the pathogen and the idea is that the immune system is so complicated and does so much to try to eliminate pathogens but as you know for example with HIV the pathogens don't just sit there twiddling their thumbs and saying okay I'll I'll just take whatever you give and if I lose I lose they've generated ways of trying to evade the immune system and what's also interesting is principle it allows us to design ways of countering that but again it really helps us understand how different limbs of the immune system are actually relatively effective but again the ability of pathogens to come up with creative ways of the BEI immune response is really incredibly impressive so the questions to think about first of all are how do pathogens how do pathogens subvert the immune system to prevent their elimination and again they're multiple livers of the immune response pathogens come up with multiple ways or circles of getting around that and how could the immune responses contribute to pathogenesis that we assume that the immune response is the good guys but sometimes an over-exuberant immune response actually can cause a lot of damage and the simplistic example I think about which I mentioned to some of you is that think about having an army going out to battle and the army is passing through your town well sometimes that those soldiers do bad things while they're passing through even though you're not the enemy they just can't help help themselves and so to having a lot know if they're stomping through your house with their dirty boots and they don't you know rub their feet on the mat before they come in and they're swinging their guns around this and they've wrinkled your china and crystal that's going to cause a lot of damage even though theoretically they're supposed to be helping you so again the same thing could happen in the immune system as well that an overage movement through immune response may actually cause more damage than it then the underlying infection would have done so in order to kind of give a global perspective this is to kind of say what makes a pathogen pathogenic so for the most part what defines a pathogenic microbe as opposed to a non-pathogenic microbe is the its ability to replicate in the host if it can't replicate in the host it's not going to be very efficient in terms of generating sufficient numbers to survive and again so it was an exception so tapeworm is one exception it basically directly gets outside of the post but again it probably makes so many so many eggs when it's outside and the inoculum is so large it doesn't need to be a problem but also one have the ability to spread to new hosts here if you want a dead end in terms of transmission that's also not going to result in significant infections because the classic pathogen basically infects an individual generates a lot of organisms and then generally has a way wherein that individual can infect other people so positively it would be through respiratory transmission through potentially stool transmission through some way of being able to infect or blood-borne transmission that was something like malaria you have you have a mosquito able to help out in terms of that blood drip or in transmission if you have mosquito netting and you get rid of the mosquitoes then malaria will dry out because you're not gonna get transmissions so and clearly with HIV we're very familiar with that if we take precautions and use barrier and other techniques to prevent the transmission again you're not going to you're going to stop that a lot of little infection from going on so getting the ability to spread is really critical for pathogenic microbes they're stimulating strong responses and again the people will tell you that a pathogen that's a failure is one that attention to itself because it makes a lot of sense if you might have a pathogen Minds its own business and doesn't run quite a lot of tissue destruction then your immune system is not going to be mounted in a strong way is on discuss in a minute certain pathogens can become latent where they basically don't replicate very much and then they use it to basically just ignore them we all know commensal bacteria are going to spoil them they don't then ultimately strong responses because they don't damage tissue so a good rule of human being half engine is in the same way when you're a guest if you ever want to be invited back to a house or is a good idea not to break the furniture not to break the crockery and and so always you know it's clean up after yourself if you leave a mess I strongly suspect you won't be invited back a second time the same way with the pathogen if you cause destruction of tissues of self tissues then union system is going to get upset or map to miss normal response against you this is an awesome word don't kill the host very quickly if you kill the most very rapidly you're not going to give the host an opportunity to spread the infection what you rather do is infect the host have a large inoculum and then let the host run around you know breathe anymore fall on people and then spread the infection in general pathogens that kill Merriwether so Ebola virus for example kills people very very rapidly it's incredibly traumatic but first of all because of the fact that the infection is so dramatic you only people know who is infected and you stay away or in that person and in addition this is that person basically dies so rapidly these kind of epidemics tend to burn themselves out so this is another and again hopefully no pathogens are listening to this flat shirt so get tips you know you know these microscopic pathogens are like writing a little microscopic notes you know we know there were billions of pathogens that came to this course hopefully and maybe they'll fill out evaluations and so in general many pathogenic microbes can persist because either they do not elicit an effective immune response and or they are able to evade the response once it occurs and that that's what this lecture is going to focus on so in terms of how can Pantages evade or subvert posed defenses there are four specific mechanisms that I just want to mention this lecture is from dr. arturo positive all that he gives in einstein and that the mucosal lecture that I gave yesterday is one that modifies warm that dr. Betsy Harold game and Einstein so thank them and of course they are very happy to share their slides with this group they're very you know proud that they're helping and teaching students here in Durban South Africa so these are four different strategies that pathogens can use in order to evade your musical antigenic variation changing antigens again people doing HIV research and appreciate that very much you may mount a very very strong immune response so what does HIV do it mutates so those antibodies are completely worthless other pathogens have similar approaches that they utilize latency if they stop replicating they kind of live a hibernating hi again the immune system is pretty much going to ignore it they somehow evade killing and they kind of dodged the bullet and that's how they evade the immune system and finally some pathogens and suppress the underlying immune response HIV is the classic example of that but it's so doing therefore they also protect themselves so now to start with theme one antigenic variation simplistically we mounted an antibody response and that antibody response recognizes surface antigens in either a virus or bacteria and therefore pathogens that are cleared by antibody and particularly are susceptible to alter to two antigenic variation think about it when a t-cell recognizes peptides it's recognizing a very wide range of peptides coming from a wide range of proteins in contrast antibodies are already recognizing surface antigens so you only seen the outside of the pathogen so they're important there's a much more limited repertoire of antigens that antibodies recognise and the other one makes a lot easier for antibodies to to defer for pathogens to mutate those few it'll show for example flu is a great example of the relevance of that and the mechanism of antigen variation as I'll show you in a few minutes can be fixed such as pneumococcus and noodle caucus there are almost a hundred different strains of pneumococcus each water would has a different antigen gen pattern on their surface so therefore even though so think of it like pneumococcus as human beings so in this room we have where there are 50 different people each one of them is completely unique so I meet one of you I recognize one of you and in developing relationship know who you are but then if a new person wants you to room with anything that I've learned about that person and recognize them is basically starting all over again because in completely different that's the technique that the pneumococcus utilizes there's also random mutagenesis which is influenza and as I'll show you again influence just randomly mutated if reshuffles or picks up genes or gets genes from all the strains of influenza and therefore changes itself so therefore instead of having a room full of different people that pneumococcus is maybe one person and this one person just randomly changes their appearance so they'll grow here the role mustache till like frizz they're here they'll die to here so that they look differently no they are the same individual and Japan Japan's ohms have as a stratum in a minute they basically have cassettes of different genes that they can bring that has new antigens that they could express so Japan is ohms you could think of like someone who's a master of the sky and they have this giant suitcase full of wigs and all sort of fake nose in this fake you know ever fake lips big eyes different contact lens colors you know the whole thing the only spies do and then rapidly change their appearance okay so now if we focus on start with pneumococcus as you've made nose it encapsulated bacteria the antibody response is absolutely essential for clearance of infection and in fact the the seminal studies on pneumococcal vaccines were actually done here in South Africa in gold miners because apparently gold miners had a high incidence of pneumococcal disease and in fact they demonstrated in that population that you could fascinate them and prevent them from getting infection but it's due to antibody responses there are at least 84 different serotypes of pneumococcus that have been identified and eat the structural difference in the capsular polysaccharide which is what and this is a scanning and pneumococcus which what the antibody recognizes translates into different and antigenic differences and different types and the important point is that neither T is type specific the immune system deals with each pneumococcus like it's a new infectious agent because the antigens are completely different so therefore and this is a slide showing what pneumococcal pneumonia was like this is a people long and again you don't have to be a pathologist to realize this doesn't look very healthy you know along these light pink and very like a sponge and this is this red fog a very thick lung which is basically due to both destruction of tissue by pneumococcus it as well as a robust immune response and again this is what normal lung tissue looks like is pretty much a lot of empty space because that's where the gets exchanged in the alveoli but this is John where the new-moniest you can see you've completely obliterated a normal lung tissue and clearly that's why individuals with pneumonia have trouble reading because they don't have they've lost a significant amount of their lung capacity for oxygen exchange and that this keeps on going that ultimately these individuals may die so how does the this work this is just a cartoon showing different types of pneumococcus and what you see is that each type has different color and different shape on the outside so as I had mentioned previously antibodies are very superficial they only look at outside appearances and therefore even though the inside stays exactly the same that you just change what the outside looks like if you made an antibody for example so infected with one type of strapped ammonia now you have antibodies against this particular yellow triangle Apatow and what this is actually showing is that make antibodies and these antibodies what-what molecules is antibody binding to on a polymorphonuclear side of our macro box FC receptor and this is hype flirting with Monica's because now it's going to ingest pneumococcus to destroy well now a few explained individuals exposed to another type of pneumococcus this particular type instead of having general triangles on service now has these blue circles well there were wonderful antibodies against yellow triangles but is this going to be of any help against this strain absolutely not and that person is going to get sick again and so forth and so on so every one of these strains is viewed the types of is basically it is a different organism so that's a major problem with pneumococcal infection so therefore if you want to design a vaccine against pneumococcus how would you design a vaccine yeah exactly you basically make a vaccine that's a cocktail of some yellows and greens some blue some purple whatever it is and now has several different antigens now human eyes with that and then that individual will make a blue base and about a response and now that they're infected with one of these strains they will have pre-existing antibody responses I mean it's problematic as you know because you probably have all different levels of antibodies against different strains and that's why you need to have like an abdomen in the animal in the vaccine to make sure that you get very potent immune responses and what we generally do is you basically determine what the most common strains are in that population or types and then that's how you design your your vaccine to focus on the ones that are in your population some harmless and others that have ways I mean what you try to focus on the look at these different types of pneumococcus well is just wanted remote a bad pneumococcus and this one not as bad and therefore that's the one you want to focus on I really I can't give you a definitive answer for that my god feeling would be that they're probably very similar because it's the only thing that they're changing is the external code unless this plays a critical role in adherence or some other variance factor since since the rest of the pneumococcus david relatively constant that i would think that three people leave very little but again i can't give a definitive answer i'm not a bacteriologist but it's interesting you always want to focus on the bad stuff this is this is influenza virus and again this is something that i think everyone now is aware of because of the whole h1n1 epidemic it was this in South Africa affected as well yeah and then who here has gotten vaccinated against h1n1 really I should have brought some with me in this case but but what you need to know about influenza is that influenza has two major proteins and it's embedded in the lipid bilayer one protein is hemagglutinin and one protein is nurettin neuraminidase and again these are outside the virus so that's what an antibody response is going to be focused against and in influenza antigenic variation is caused by two pathways one pathway antigenic drift and the second pathway is called antigenic shift now being a linguist right which do you think is one of them change drift or shift shift absolutely drift is kind of like the kind of thing now you fall asleep and you can you canoe drifts a little bit and shifts is when you're you know you basically move into a very very different location and influenza has segmented RNA genome which allows it to undergo antigenic changes and antigenic drift predominantly results from point mutations to eat a servicing of lutein in Turner neuraminidase so you can appreciate that's not going to be a very major change however antigenic shift results from restoring the RNA genome so you get a piece of RNA from another strain of influenza that gets popped into this or maybe even influence that infects another another species so for example maybe some influence that infects pigs or Birds jumps into human influenza and is something completely different that no one has ever seen before that's when we have it gonna have a disaster so the 19 the 1918 flu epidemic which I'll show you in a second was one of those devastating epidemics that we know of was an example of shift completely change the antigen nobody had fries community and therefore it was incredibly dramatic in terms of this because both its morbidity and mortality the 2008 flu epidemic was an example of drift so it was only a minor change and therefore not as serious the swine flu 2009 is an example of shift because now you put a whole new RNA genome in and that's why there was so much concern in terms of what the potential impact that that would be on health and this is just showing pictorially this is a e/m of the influenza virus and again you can see that the spikes the hemagglutinin spikes and ineuron neuraminidase spikes that are on the capsule and this is what the antibody is going to bind to in case of antigenic drift neutralizing antibodies against hemagglutinin block binding to cells and you see blogging and therefore it can attach itself and causes and replicate and then ultimately a little bit clearer in this case the italians orphan hemagglutinin that neutralizing antibody is no longer fine but this may be only hemagglutinin but you probably still have great advise against neural mayonnaise and therefore it can protect you however antigenic shift means that you basically not only change a little small areas where you may have other antibodies that can also pick up the slack but now you basically have a completely new new RNA so this strain of virus for example uses this RNA that gives it this you know gluten and now this is exchange of RNA segments and now this strain of flu has exchanged as RNA and now instead of expressing this hemagglutinin and now as expression is completely new one and now you absolutely have no antibodies that recognize it and then clearly you can have a much more devastating disease this starting from scratch and in terms of numbers is very dramatic the great pandemics and and enemies are great word because it really does sound scary the great pendant wrote Russian flu in 1889 in 18min 1890 killed a million people Spanish flu devastating 40 to a hundred million deaths it's mind-boggling and if you read history about what was happening that is just incomprehensible and in a very relatively narrow window of time Asian flu people probably here don't remember it anyone remember agency Benton yeah good I mean we're all young it basically killed a million people Hong Kong flu I barely remember Hong Kong flu I think I was a little over ten at that time it's you know it's a great name because you know they named it after like a country or whatever so it's so you can relate to and swine flu this card epidemic as about as September 12 2009 and resulted in a little over 3,000 deaths so clearly the major concern was that we would see similar kind of numbers over there and that's why everyone was getting very excited in terms of trying to develop a vaccine and again this age h1n1 was a resaurant of four strains of previously circulating influenza A virus and luckily it just seems that it's just not as pathogenic and as a Spanish flu okay so we're sure from the flu to panas ohms and this is a I don't think this really does not affect this region of Africa but it causes sleeping sickness just to the point of amia logically the main service area is it's called very specific lipoprotein VSG and there are over a thousand VSG genes in this organism they're kind of like sitting on the side and what happens is and this is especially what it looks like what basically happens is that any given time you have you have a gene that's being expressed with the appropriate promoter but kind of like us is on the side our thousand genes that are not being expressed and what your panels can do is just randomly pop out the one that's being expressed and put a new one in so like I said it's like someone that is a master of disguise and they have like a hundred different fake noses that every time they just stick a new one on so as soon as you recognize them they could change their appearance and this is what happens you basically get infected you basically have a large level of amount of parasite make a really good antibody response to that particular sequence and now you control the infection so you basically think that your cure and what the trypanosome does is okay fine get rid of this antigen put a new one on you don't have antibodies against dad and now the level of infection goes back up make antibodies against that particular VSD gene control it and now you put on a new one so in essence it does the same thing conception that HIV does except instead of relying upon mutations it almost has like prefabricated mutations that are ready to be popped into changes antigenic appearance is that clear now the only thing I would say is is that the immune system should feel complimented that these pathogens have gone to so much trouble to develop ways of evading it what this is actually teaching us is how good the immune system really is the second theme is latency some viruses can persist by seasoning replication until immunity wanes great example of that is herpes simplex as well as EBV and because the viruses aren't replicating they're not making viral hepatitis and if it's not making viral peptides what can be recruited what kind of cells cd8 cells can't be recruited to to to kill the cell because the cd8 comes by checks out the MHC class 1 molecules for us it couldn't see any foreign peptides no because they're not being made and again HIV does the same thing when Lately infects memory t-cells it's not replicating not making HIV peptides that are being put into MHC molecules and that's why those cells are able to Ebay the museum and for many many many years so that if you don't make a Penta that's the Achilles heel of cytotoxic t-cells Norma peptide it's like like leaving no footprints if something doesn't leave footprints you can't detect that it's there and and therefore that's how we evade and based on the bottom line is that they basically you'll I love herby simplex you may know causes cold sores and herpes encephalitis and persistent sensory neurons and the things like stress bacterial infections can can activate the virus the immune system can control local manifestations like cold sores but it can eradicate it herpes zoster and who here has had chickenpox yeah so you've all had chickenpox a very dramatic infection when you're a kid but after you clear the infection you haven't completely eliminated those viruses those viruses also can can continue to persist in nerve nerve tissues and they get reactivated as shingles when when one gets older and this is just a picture of herpes his occult sorties of herpes virus and again the idea is that the virus persists in the trigeminal ganglion basically but not replicating and also the immune system doesn't eliminate it but now you have a stressful event in your life like let's say you're giving a major lecture at a major meeting to present the work that you've spent four years doing in your masters or PhD thesis and you're all stressed out because you're going to be speaking in front of a thousand people showing your data and that's stress now because all the sudden going to activate the herpes replication and the next morning you wake up you look in the mirror and it's like unfortunately boom you have a cold sore so they tend to emerge at like the least opportune time in your life is stressed absolute and stimulated replication horribly it won't happen to anybody here the other thing is for as far as a zoster as as far as as as far as herpes ox is concerned what shamers occur they tend to occur in the distribution of the nerve that is an so if anyone's ever seen shingles or got shingles you know it's very dramatic because it basically has a third of narrow range which is the distribution of the nerve where and is hiding and because that's where it's coming out of epstein-barr virus causes infectious mononucleosis it's also herpes it affects B cells and causes them to proliferate and this is controlled by stalked by cd8 positive T cells which kills the proliferating B cells but again a fraction of the B cells can survive with EBV infection and the mechanism by which it causes lengthen C is basically it is replicating but it makes a viral protein that interferes with the degradation of viral proteins and in viral proteins can't be digested into peptides then they can't be presented in class one MHC molecules so this is a way it gets around that and some and latent infection may be responsible some for some lymphomas such as burgers and Hodgkins because again these BBB can cause hyperbole operation of these cells through the theme is much more dramatic basically escaping killing your immune system is all poised to kill the pathogen and the path had just dodges the bullet has a way of avoiding the killing mechanism and certain pathogens are very strong immune response but they escape killing and the persistence is due to survival and the bottom-line strategy is basically stay away from the volts now Mycobacterium tuberculosis is one that all of you familiar with and the way that micro bacterium one way that micro bacterial tuberculosis is able to survive inside of macropods after it's ingested is it's inside the phagocytic that vesicle you're very familiar with but as you know in order for the TB to get killed it has to have fusion of lysosomes by the lysosomes that contain all of the enzymes and free radicals that when they fuse with the vesicle to destroy Michael bacterial tuberculosis however apparently TB secretes factors that prevent these lysosomes from fusing with the phagocytic vesicle and then TB is able to live there no problem at all in addition some TB does also as I'll show you in the next slide have the ability to escape the phagocytic vesicle once you're outside the vessel in the cytoplasm the normal order can get killed because the lysosome has to fuse and dump the enzyme into some container that's gonna not harm the cell once you're in the cytoplasm that mechanism is no longer viable for killing and is just showing Mycobacterium macrophages Listeria is another inch of cellular bacteria and the way to evade the immune system is it's in the phagocytic vesicle the vacuole but now it basically gives out of there really quickly normally it's hard to get out of the phagocytic vacuole I mean that's how it's designed but somehow the Listeria is like Houdini is able to escape from this that the vacuole before any of the fabuloso is confused with it once it's in the cytoplasm has the ability to replicate in the cytoplasm and can't be killed anymore by the cell and third approach is used by Toxoplasma and in in the United States we have parties where where the host doesn't want to spend a lot of money so he tells people BYOB I know if you have that here you know bring your own beer or bring your own whatever so so the Toxoplasma gondii has a has basically a system that it utilizes it's called V yov bring your own vacuole and basically what it does is it basically makes its own vacuole inside of the cell and the vacuole that it makes is one that lysosomes don't recognize and therefore won't fuse with so now it's kind of pitching its own tent inside of the cell and it basically is impervious to any of the cellular machinery for eliminating it and can live inside the self perfectly fine okay any questions and Treponema is which which causes simplest is a spirochete a kind of bacteria and the way that it protects itself from the immune response is by camouflaging itself so for example when soldiers do jungle warfare they put on these they could put on these actual netting that have leaves all over them and they could lie on the ground and it looks to you like they're just a pile of leaves and you basically ignore them and then of course they get out from under them and they you know like Rambo they start shooting but but that's how they hide well trementina does exactly the same thing it basically has has codes itself with host proteins and those host proteins code its own fire proteins and therefore antibodies don't see them and therefore they they avoid being seen Borrelia burgdorferi is another spirochete cause lyme disease and again it could avoid killing by complement by coating itself with host proteins that the Hauptmann won't recognize now another way is the viruses subvert the immune system so one of the ways of doing that is by inhibiting humoral immunity inhibiting inflammatory responses blocking of antigen for processing and you know suppression of the host itself and these again as many ways as we have of fighting infection viruses come up with ways of avoiding the immune system so for example heard these simplest and seed cytomegalovirus they encode an FC receptor really like it's amazing they make their own FC receptor but this FC receptor is a dead was a dead end it doesn't do anything it's not wired into any of the cellular machinery so now antibody binds to the FC receptor thinking okay now I've docked in the FC receptor now macro files you know do you know do whatever you supposed to get activated or something and nothing happens because it's basically a decoy FC receptor herpes that the same thing with complement receptor it makes a decoy complement receptor common binds to it nothing happens and and vaccinia actually makes a complement control protein again I didn't discuss compliment maybe next year a new compliment but but you we have certain proteins and enzymes that turn off our complement pathway which is not we need to do it or not to harm ourselves well vaccinia business is no problem I'll make my own and therefore turn off the complement system appropriately and therefore protect itself from complement the set of mega buyers basically makes a cytokine homologue we don't really know what that why that works that cydia makes soluble cytokine receptors these soluble cytokine receptors bind to the cytokine chelate them so they can activate the immune system so it's basically neutralizing them EBV basically reduces expression of adhesion molecules and now you know if you don't express adhesion molecules the cells won't stick to ensure that you won't get answers your presentation efficiently and therefore this protects the virus from from for example CTL killing and vaccinia actually prevents NF kappa-b activation so that cells infected it can't get activated and therefore won't mount an immune response so these are like all these different amazing things that viruses can do now sometimes though the pathogens cause inappropriate immune responses and this is something I was discussing with someone earlier that the immune response sometimes can do mates to be over destructive and one that I'll show you again that I've previously shown you is stamp operating as a super antigen to coerce very dramatic disease and HIV as you know to please see for cells and again this is a review with leprosy showing you how it's suppressed or causing inappropriate immune response toxin says super antigens basically get dysregulated cytokine T cell proliferation depletion of T cells and this is caused by staph strapped super antigens and this is what I mentioned previously to you this is a this is a picture of this brand of tampon called rely tampon and and and that about 20 years ago it was dynamic of toxic shock syndrome and menstruating women who are using this particular tampon 40 individuals woman died from this the staph aureus strain produced Thompson as it was a super antigen that activated t-cells globally and these particular tampons were designed to absorb a lot of fluid but that provided a unique environment wherein the step could replicate and release its Thompson again we do see it occasionally but now that these tampons have been taken off the market we don't see it as much this is again reach showing again the difference between lepromatous leprosy and tuberculosis that very severe lepromatous and much more less severe tuberculoid and this is just showing pathologically that tuberculoid leprosy has few few bacteria and the parameters leprosy has a lot of bacteria again discusses in detail previously so I'm just kind of going through it quickly and the cause is making the wrong side of mine response it's an intra cellular pathogen instead of macrophages what you want to do is drive that macrophage to make to kill and make of lysosomes to fuse and kill it and what satifying would you want to stimulate to do that so what side of mine do you want to stimulate to regulate the capacity of macrophages to kill intracellularly fax a toast path interfering gamma exactly and in fact the individuals that that may interfere in gamma CH 1 cytokines they have a tubercular leprosy because they're able to control it however individuals who whatever reason for example they may be infected with a parasite and they have a th one follow a th to polarized response they don't make significant gamma interferon and therefore they don't control it and then they develop a little more severe lepromatous leprosy and we discussed that potentially you could treat these pastes with the gamma interferon and shift their immune response to the appropriate one and then potentially shift them from more severe lepromatous leprosy to a model there's a burglar love to see in addition the immune responses can to pathogenesis and two examples of this are respiratory syncytial virus and mouse mammary tumor virus respiratory syncytial virus RSV is made of pulmonary pathogenic and children too we see a lot of our SP in South Africa and the class Lee is going to infect young children less than a year to AB age they get very significant pneumonia that can get a bronchial IIST which actually resembles asthma because got a lot of bronchospasm it's a major cause of morbidity and mortality in the states during winter months the emergency room and wars are full of kids are Frankie Elias we don't have a treatment for it we put him in a mist tent we give them oxygen that's really all we can do you is your epidemic like in june/july and probably Durbin doesn't see any because it's never winter here you know again you know when the when the Chamber of Commerce you know tries to encourage people to move to Durbin or when when University of kwazulu-natal wants to recruit students we just have some kind of tagline you know it's never winter in Durbin something like that as a way you know leave your coats at home or something like that and maybe you know we don't see RSV and if that's the case here in Durban I don't know but ours bene is responsible for almost forty forty five hundred deaths and with devastating zits kids that's what's really so glad about it of course bronchial is now this was a major disaster in the States because they developed a vaccine that and again you know one hand you think vaccine development is very very easy because look at what Jenner did Jenner took Cal pop scraped it up made a vaccine next thing you know smallpox is cured in the entire world that on the other hand you know clearly people knowing who are making HIV vaccines also know how difficult it can be so for RSV people said okay we know how to make vaccines let's grow up RSV and then let's kill it and kill it by fixing it in formalin so they did that and then they immunize children with his a vaccine and lo and behold not only did our speed not protect children but in fact more kids that the kids that got vaccinated had more serious and more significant disease so actually the vaccine was worst and for people who are familiar with with one of the HIV vaccine trials where the patients that got vaccinated actually did worse than ones that aren't vaccinated appreciate what a devastating effect that is when you make a vaccine it actually does more harm that doesn't mean then it doesn't even phenomena to protect but it actually harms the vaccinated patient and it's unclear exactly what the mechanism by which they actually got worse it has been suggested that there was something about the vaccine that polarized the response to with th2 response and as you know bronchiolitis causes this kind of wheezing process which is asthma is a similar to asthma which is th2 mediated disease and potentially by polarizing if the patients got infected instead of mounting a th1 response they mounted a tease to response and therefore had more of the manifestations of of wheezing of bronchospasm okay so basically that's that's the lecture we asked how do pathogen evader subvert the immune system again they do that basically targeting different mechanisms that the immune system uses to eliminate them and by targeting them now that's how they subvert the immune response but it's amazing this broader array of methods they're used by pathogens it's not like they just use one or two they pick apart the muses and at different stages and then attack it it that way how come you immune response contribute to the pathogenesis and again if you may either the rural community spawns such as in leprosy in lepromatous leprosy or you basically have an over exuberant immune response like toxic shock syndrome or again make the inappropriate one in the case of the RSV vaccination again that's how the immune system actually is bad for you so this is the last lecture so I really want to give acknowledge to people that really made this course possible and first of all I want to thank the leadership of catered the kwazulu-natal Research Institute for tuberculosis and HIV Phillip Jacobs is a close colleague of mine at Albert Einstein College of Medicine and one thing about bill Jacobs is he got like a triple dose of the enthusiasm gene because anyone who's ever dealt with him knows that he's like this bundle of enthusiasm and again he's visited here I know previously and some of you may have met him and he's just incredible in terms of his enthusiasm for science as well as enthusiasm for curing tuberculosis submitting a vaccine Bruce Walker old you know again is a tremendous role model in terms of what he does and his vision and what and not only does he have a vision but he also has to have ability to get things done and surmount unbelievable odds to get things accomplished and again he got me to come here you all know dr. Vince Stern and slim Kareem here at kwazulu-natal and again thank them very much for organizing but of course all the real work was not done by these people these are the vision people they say let's do XY and Z but at the end of the day that's not what's going to get the job done was gonna get the job done is that people who actually do it and I want to thank Victoria who really has been spectacular in organizing this and again this is you know she's very very busy doing all the other stuff that she does and on top of that she had organized this and did a fantastic job doing it and Zoe Rogers who's he the administrator came here from Boston to really get everything set up and make sure everything works smoothly and the fact that this works so smoothly I'm out I don't think it was one glitch that I know of really is a testament to their amount of detail and effort that they put into it putting these booklets out making sure everyone has them again this is all done didn't happen by itself I want to take Carolyn Wright who is the proprietor of the bed-and-breakfast I stayed in the st. Anne's I'm giving a plug to it and again you know on DVD all you out there if you ever come to the urban State and wonderful hope you don't edit that out because of a problem with that and she's great not only is she great in terms of running the place but the first day I was here the cab didn't show up and it was like this whole panic like how am I gonna get here in time because if you come here nine o'clock and I wouldn't be here you just leave and never come back again it would have been disaster you probably just wait five minutes for me you probably I he's not coming that's it I have a free week when Carolyn just threw me into her car and drove me here to make sure that I got here Sylvia Miller actually was my caterer she provided all the food for me and did a fantastic job and again you know without guests the car doesn't go and without food would give me four fourteen lectures in one seminar it would have been a puddle of protoplasm and Luisa Gonzales is my secretary and she's really spectacular making sure all the details were taken care of and it was important to have a sentence my wife Elia types Goldstein because you know when when I talked about possibly doing this I wasn't sure because again and Louisa actually cleared my schedule so I could be here for a week that wasn't a trivial task either and my wife said is really important for you to do it the students are really gonna appreciate it it's really something as to have a tremendous impact and just do it don't worry about our four kids don't worry about running the house don't worry about the fact that our kids are having finals right now and normally you have study with them don't worry about any of that kind of stuff you know just go and do it so again I really want to thank my wife the most and finally the most important people I want to thank is you because because again a rainy day out there but the attendees because again is a tremendous commitment on your part to sit in a room for four four lecturers a day for four four days it's really a tremendous amount of dedication and really showing the enthusiasm you have for learning and again I have to say them so impressed that your lag at all you know you really have that showed up the same amount of show the same enthusiasm and even as time went on I think people really got even more interested in more through by learning the material and whenever you teach and many of you who have been have been teachers as well you know that you're only as good as your students and if these students don't have that level of interest and excitement and enthusiasm then the lecturer kind of sucks into that negative energy and kind of starts wilting and here was exactly the opposite I felt that positive energy from you that enthusiasm for learning that curiosity and it made me work even harder to try to convey the information so I really want to thank you and I really want to applaud you thank you very much and I also I want to thank our cameraman now I have no idea if you did a good job and not because I haven't seen the video you know hopefully you know from through one of my son's Bar Mitzvah is actually the film didn't come out so we have absolutely no record of that but I'm sure everything here is fine and Victoria said that yesterday was a rainy day and it was very depressing to stand outside for a group shot so we wanted to maybe try one more time out in the Sun and get a picture of everyone's smiling and and and showing it had a really good time again thank you very much to all of you you
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Medical_Lectures
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Immunology_Lecture_MiniCourse_1_of_14_Components_of_the_Immune_System.txt
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oh first of all it's such a privilege to be here and uh really am very happy and uh also I'm grateful to all of you who are attending because I know that everybody is busy and has a lot of things that they have to do and have taken time out from the busy schedule to come here and in addition I just want to uh thank first of all uh uh both Howard 's Medical Institute and University of quas Z Einstein which together our partners excuse me it's not working this is like a placebo microphone uh okay I'll just is it you know they always bang on it and this is for this is for the for that okay well can you hear me in the back okay so yeah just a question of if I'll last 14 lectures doing this but I want to thank the uh kth as you know which is a combination of University of quaz zul and atal uh Dr slim Kim and whim Stern whim Stern uh uh Albert Einstein Bill Jacobs and Howard use which is uh Bruce Walker and I want to also thank the sentence for Aid research which is a network of research institutes funded by the NIH in the United States and that's really where Bruce and I first got to know each other and again Bruce has really been an inspiration to me in terms of coming here to South Africa and I also want to thank uh Victoria caspit and Zoe Brown who've done a fantastic job organizing it and one thing about organizing these things is that as you can see all these little things have to add up and be put together and they've done a really fantastic job doing that okay uh now when um we're uh uh uh learning Immunology in very short time and uh this reminds me of a a talmudic story that happened over 2,000 years ago and this uh person came up to a famous Jewish sage and he said I want to learn all about Judaism but I don't have a lot of time so I want to learn while I'm standing on one leg and the sage was like probably a little crotchy and said you're get out of here you're just making fun of me you know threw him out of the of the school and then this person really really was very sincere and curious so he went to another Sage who had a reputation of being a lot more uh flexible and he said the same thing Rabbi I don't have a lot of time I want to learn all about Judaism while I'm standing on one leg so the rabbi looked at him for a second and said okay stand on one leg and he stood on one one leg and while he was doing that he said all you need to know about Judaism is don't do on to your neighbor what you wouldn't want to have done to you and everything else is commentary so now you can spend the rest of your life learning the commentary but now you know all you need to know about Judaism so the same thing about Immunology in a sense you're all here learning Immunology while standing on one leg you know in such a short period of time so therefore oh this is now I'm live okay whoa he's like the voice of God so is this too loud that's okay so therefore uh you could really summarize and you can all stand on one leg if you want you can summarize all you need to about know about Immunology basically kill the pathogen and don't harm the host that's it everything else the rest of the 14 lectures are commentary on how the body goes about doing that how the body kills the pathogen but it does it in a manner that doesn't harm the host and things go wrong sometimes we don't kill the pathogen and have infections and sometimes we harm the host and have autoimmune diseases and maintaining that balance is really all you need to know about Immunology everything else is commentary okay is that clear so again some people who are busy you could leave now because you know all you know about Immunology okay so I like framing lectures with questions and then use the lecture to answer the questions and then at the end we'll summarize and make sure that we've answered the questions that we've set out to do so a first question I would like to pose is why do you get some infections like chickenpox for example only once like who here has had chickenpox you okay who here has had it twice twice in the [Music] back okay so he had um a weak case of it so so but the point is that you only get it once so why is that second of all how can we generate a system the immune system that could recognize a broad array of pathogens that are out there and with incredibly High sensitivity and specificity and yet only use a small amount of DNA to encode the immune system when you think about encoding millions and millions of te- cell receptors and millions and millions of imunoglobulin geneses how can you do that with a limited amount of DNA that we have in our cell you can't devote the entire genome to the immune system so how does that happen and why the first time you get an infection you get really really sick but the second time either you don't get it for example like chickenpox or for other infections you get them but you don't get as sick so why do you get a more potent and more rapid immune response the second time you're exposed to an infection how does the immune system provide a high degree of sensitivity and specificity there are thousands of pathogens viruses bacteria parasites your immune system has to be able to Target those diverse types of pathogens some of which are inside the cell some outside the cell some large worms how does your immune system be able to have this broad diversity to handle it and yet be very specific for each specific pathogen and finally we have B cells and te- cells and why is it that B cells are important for some pathogens whereas t- cells are very critical for other pathogens what is it that they do together and what makes one more important than another and how do they interact with each other okay any questions on the questions and feel free to ask questions now this is a picture of a boy that um some of you may be familiar with and this is uh David are people familiar with the boy in the bubble and what's amazing about David is that as you know he was born without functioning immune system in terms of B cells and t- cells and and and K cells and his two older siblings uh died within the first year of life from infections the infections they died from were ones that were unusual so they pretty much figured out that they had an imuno deficiency when David's mother was pregnant with him they said well the likelihood is that he may have the same disease if we let him come out the same way as brothers the chances were he would die in the first year of life so they had this brilliant idea and they said well if you don't have a functioning immune system you can't protect yourself from pathogens but wol protect you from pathogens so he was born through cesarian Section and he basically lived his entire life in this plastic bubble all his air was filtered all his food was autoclaved and sterilized because since he couldn't fight infection therefore he had to be protected from pathogens and in a sense the congenital immuno deficiency that David had is very similar to what people with HIV infection have except that's acquired whereas this happened congenitally and David Lon grew up in this bubble um but when he was about 16 years of age they presented him with an option of getting a therapy of buar transplant from a unrelated donor in order to reconstitute his immune system it was an experimental Approach at that time and unfortunately he succumbed to the disease but at that time if David looked at the doctor looked at you in the eye and said doc why don't I have an immune system what's wrong with me no one could answer that question if they if he asked what did my B cells do what did my t- cells do at that point we really didn't know very much and I think a lot of that ignorance that we had Dave was an inspiration and really led to a lot of what we know about Immunology so I think this is again it's amazing how patients that have diseases can really Inspire both Physicians and researchers to really redouble their efforts and come up with Solutions and I think that's what a lot of people in this room are hoping to do with the rest of their lives so again this is an example of how a patient can really have tremendous impact now if we think in terms of uh infections one of the most dramatic infections that mankind has ever seen is small poox and has anyone here ever seen small poox infection absolutely not but within the history of the human race small poox has killed probably billions of people and small poox is incredibly dramatic so this is a picture of an individual that's infected with small pox skin the skin lesions are very very dramatic and even more dramatic is when a child is infected so there's no way that you're going to miss this diagnosis there's no way that if someone's infected you're not going to see it so this really has a tremendous impact because this really led to the development of Immunology because immunity uh comes from the Latin word immunitas and in the United States what doctors do is they take these Latin words for different diseases and use them in their practice because that tends to impress the patients so if instead of saying you know you're sick you just come up with some Latin word to describe sick and it sounds a lot more impressive so so immunitas was a concept they had back in Rome which basically meant people who had certain jobs didn't have to uh do any kind of Civic duties but his historically this came about that people noticed that if someone was infected with chickenpox with small poox and small pox had a basically a lethality rate of 30% so an average of 30% of people that were infected with small poox died which is dramatic on the other hand 70% of people survived and yet what was noticed was people infected with small poox only got it once and again it's such a dramatic infection that that's something that people appreciate and therefore the concept was once you got infected you had immunitas you got Exempted from reinfection there must be something protecting you and that factor of protection was therefore called immunity and what I think is really important to also appreciate is that people were just as smart then as we are now H and they lacked a lot of the technological tools but they were just as intelligent and they also made observations and that's something that all of us really should be trained to do make observations and act upon them and in fact uh in China uh the observation was made once you get it you don't get it again and they came up with this approach where they actually took small pox lesions they scraped them they dried them they made a powder out of it and then they had children inhale powder from these lesions and in essence what they were doing is no different from attenuating the virus and now infecting individuals with the attenuated virus this actually worked the downside was it was about uh5 or 1% incidents of infecting uh through this approach but the people that uh didn't get infected actually ended up being protected so again this is a very very crud vaccine but again they didn't know any science they didn't know Immunology they made an observation they acted on that observation now the major as you know advance in small pox was Edward Jenner and again Edward Jenner knew no Immunology at all but he was smart he made observations and he acted on those observations and he observed that uh women who were functioning as milkmaids would get this infection called cowpox that was similar to small poox but instead of getting hundreds of lesions they only got a handful of lesions but these individuals never got smallpox now millions of people may have made that observation but Edward Jenner was the one who asked the question why why don't they get small poox and the ability to ask that question why is what makes scientific research what it's all about that's the what should drive us is asking the why but in addition to asking the why Edward Jenner basically did the experiments and basically he took uh crusted lesions from the cowpox injected actually a young boy with it and it turned out the boy was protected from small poox and thus the small poox vaccine was generated and this this is the reason why no one in this room has ever seen small poox because the vaccine was so effective and the reason it was so effective was the only host for small pox are human beings and once you eradicate it in the human population there's no place for the virus to live and by 1976 after a worldwide effort to immunize the entire world the last case of small pox was seen and in 1979 small pox was officially eradicated and again Edward Jenner did all this initiate all this not knowing any Immunology at all he didn't know about B cells he didn't know about antibodies he didn't know about cytotoxic te cells he had no technology he just had a brain and he used his brain in order to do this but again a lot of what this lecture is going to be discussing is exactly how this worked how do vaccines work and how could we potentially uh utilize them and make better vaccines okay any questions so far so if now we start thinking in terms of Designing the immune system how would you design an immune system that would protect you from being infected well one way of doing that is by developing a multiple layer type system and again people Everyone goes to airports now and you have to pass through all this kind of security and one of the Hallmarks of security is you have more than one layer because if you just have one guard or One Security agent you're only as good as that individual if they take a break if they're not that good that could be a major vulnerability so what you tend to have is more than one checkpoint to allow you basically to protect yourself the immune system uses the same approach and the way it does it actually is by having a more lowlevel but always ready immune resp uh system called the innate immune system and then more sophisticated portion of the immune system called the acquired immune system and the innate immune system and the acquired basically work at multiple different layers using multiple different infector approaches but there they're different in terms of the specific spefic method that they utilize so for example the innate immune system a barrier as you know is the most important part to protect you anyone has ever T anyone here has ever taken care of a burn patient okay why why do burn patients die from infections and that's what we need to protect them so as soon as they get burned we cover their burn area with antibiotic cream with gauze pads to protect them from being infected what that really underlines is how critical the skin is in terms are protecting you from infection it's very rare for intact skin to get infected however you know if you get a cut and there's a lot of dirt or exr around that's very likely to get infected because you've broken that barrier the acquired immune system basically has mucosal immunity is the dominant barrier that's utilized there and we'll discuss that in more detail in a subsequent lecture also sble proteins are present both in the bloodstream as well as other secretions that protect individuals inate immun system uses complement and the acquired immune system uses antibodies there are cells that are infect your molecules innate immune system uses fosic cells such as macrophages polymorphonuclear lucaites they don't really know what a bacteria is they can't tell a pacus from a strepto cacus from a brucelosis or from tuberculosis but they just look at it they know somehow we'll discuss in in the second lecture how they know that it's foreign and they just swallow it up and digest it t- cells and cells are much more sophisticated and again we'll be discussing that in the afternoon lecture exactly how they function and how they're so specific and they all use mediators so the inate immune system secretes Incan one which provide which generates fever tumin necrosis factor and the acquired immune system utilizes gamma to feron as among other Sol mediators to activate and stimulate other cells now if we now want to break down and think about like sitting down and designing immune system so imagine this would be a question on an exam you get a blank piece of paper and they say how do you design an immune system and it's actually a fairly complicated thing to do from the ground up but to try to break it down into what the immune system needs to do the first thing it needs to be able to do is distinguish between foreign and self between a an invading pathogen and your own tissues if you can't do that job you're not going to be able to function as an immune system so again as I said you know kill the pathogen don't harm the host that's the first thing it needs to do once it's determined that something is foreign it really wants to amplify the foreign nature of that Invader so therefore now other limbs of the immune system can more readily recognize it and know that it's actually uh something that needs to be responded to after that's occurred you need to recruit helpers if you're a maage and you see something bacteria you know you need help because if you don't do it effectively there may be other bacteria coming you need to recruit help so you need to and we'll discuss how you mobilize Defector cells both innate as well as uh B cells and t- cells after you've uh uh identified it mobilized defectors they have to clear the pathogen and finally you want to prevent recurrence the Hallmark of the acquired immune response is that you basically the first time you get infected you may get a bad infection but subsequent infections either never occur or much Milder because the immune system's philosophy is uh fool me once shame on you fool me twice shame on me so the first time I'll give it to you hopefully I'll survive the infection but then the next time I'm going to be prepared and won't get infected okay any questions okay so and again you know for a lot of you this is stuff you know very very well and it's a lot of its review but I'm just trying to develop a whole kind of the uh philosophical basis of the immune system then then we could put the detail into place so if you want to kind of toer in order for the immune system to work it needs to have a very high level of sensitivity so you need to know what's infectious agent what's not an infectious agent but you also have to have that high level of specificity you need to distinguish between cellular proteins and foreign proteins between a bacteria between a virus between a fungus different bacteria and that determines the effectiveness of immune surveillance and I apologize um so okay so so if now you again we're still designing the immune system so it has to have specificity be able to distinguish very subtle changes in protein structure because one change of protein may be yourself protein a slightly different one may be faren different carbohydrates different glucoses one could be self one could be forign you need to be able to distinguish between just a few amino acids or a few just slight stru structural differences you need to have a large diversity because you need to recognize a host of hundreds maybe even more different potential pathogens that can infect you you need to have that memory to be able once you've been infected the first time remember you've been infected to mount a much more rapid response what's also critical is the immune system needs to be able to demobilize itself once you've cleared the infection you need to be able to get rid of those infector cells because you don't need them anymore the infection is gone because as we all know from after a war if you have a lot of soldiers running around with their guns and the battle's over they can get into trouble uh you don't want to have a lot of people with guns for no reason running around and you de demobilize the Armed Forces after a war is one so to in the immune system otherwise they have a high potential of attacking self and again this is so critical in the immune response you need to distinguish self between nonself you give tremendous power to immune cells you want to make sure that they use that power wisely and only target it against infectious agent not against the body itself and how do we mobilize the immune system well it turns out you could break it down into different phases so the first phase is called the cognitive phase and cognitive from cognition thinking it basically is the first step where the antigen binds to the specific cell that recognizes it that's makes a lot of sense that's step number one number two is and again getting into a lot greater detail this is just an overview once the antigen binds to the specific cell and we'll discuss how the cell recognizes antigen Etc but it now changes the cell activates the cell and makes the cell undergo either proliferation or differentiation into a uh an inent specific cell in the case of B cells and t- cells a more potent F acidic cell in the case of macrofagos that now could better fight infection and then now it goes into the infector phase where now the cell is fully armed and activated to allow to mobilize a response to eliminate infection and now this is just to uh illustrate the diversity of pathogens that are out there but it does that in a way to break it down into ways you can start understanding how the immune system could Target pathogens in a very organized fashion so if we think simplistically one group of pathogens are extracellular they're outside of the cell they float around in the bloodstream and these are bacteria parasites and and fungi so these are just some examples strep ponia claustrum and they cause a range of diseases but one could appreciate that you need to have an immune response that could Target pathogens outside of the cells and what do you think is going to be very effective against that things like antibody and compliment because those are soble factors outside of a cell however antibodies really can't get inside of a cell so once a bacteria gets inside of a cell antibodies probably are not very effective so therefore there are intracellular bacteria and one that almost all of you are familiar with is U microbacteria tubercula tuberculosis it gets inside of macrophages once it's there antibodies can't do anything so you need to have another method of dealing with those infectious Asians viruses something all of you are familiar with they're also inside of cells so antibodies may be able to be useful before they get inside of the cell to prevent it from entering the cell but once it's inside of the cell again antibodies probably don't do anything need another way of dealing with those virally infected cells and finally parasitic worms I mean who here has seen a parasitic worm I'm sure a wide range of you have how how big are they they're like you know they're very very depressive just imagine if you depended upon cytotoxic te- cells to kill a you know you'd have this little microscopic t- cell you know throwing these little couple of molecules of perin into the worm and the room will be laughing like you know give me a break you know is that the best you can do or imagine like IGG binding to a worm and then mobilizing compliment making microscopic holes and in the worm it be a joke you need to have a completely different immune response and again that's why you have the whole IG mediated immune system which just giv is incredibly vigorous response again a completely different approach to dealing with parasites than for viruses and bacteria okay any questions now one thing about immunology and I have to apologize is it can be incredibly confusing because Immunology is very nomenclature rich and if you don't know the nomenclature these words is Blow by you and frequently you're kind of too embarrassed to stop and say what is that what are you talking about just kind of Nod your head sagely like yeah I know what that is and me while inside go whoa I'm just lost so the the the first so again you know have to apologize for that but I mean it wasn't me who invented this stuff but the first thing you always hear about is CD and CD this cdnet different numbers and it's very very difficult to realize what they are and what CD basically stands for is cluster of differentiation and the idea is is that cells Express and again all of you are familiar with it cells Express unique membrane proteins identified by monocl antibodies and the cells can be defined by what pattern of proteins they express on their surface so all of you know CD4 typifies a helper T cell cd8 a suppressor cell cd19 a cell these are things you can kind of memorize there's no way around it but the CD proteins are identified by sequential numbers and I remember when I first started learning Immunology I thought the lower the number the more important the protein right doesn't that make sense like cd1 must be the most important protein you know like you're driving around and you see a license plate and it says one on it right you feel that must be a pretty important person you see a license plate like 2,000 9,000 whatever you say oh that's probably some schleper but it turns out out that this was all numbered based on when they were discovered it turns out that a lot of the not very important proteins were discovered first and a lot of the extremely important proteins were discovered much later so you can't just look at the number and try to guess what's important so cd1 does anyone know what cd1 is no it's like a thite marker but like you know pretty much is not that critical cd2 no CD3 now oh yeah te- cells that's pretty important CD4 they kind of got in the groove after a while they got started getting then they CD4 they want to run then cd5 cd6 cd7 kind of like low all cd8 big one so again unfortunately you just have to memorize which are the important ones but at least you get a sense of that the number really can't help you in in terms of what's important and what's not but the good news is it's not like they named it after people can you imagine if everybody that discovered one of these proteins had it named after them you know so you had the caspero protein prot or the brown protein you know whatever and then it's terrible it's like HIV kills all those CD you know caspit expressing cells you know it wouldn't be very good and then forget about it then you have to spell them so forget about it so at least thank the immunologist for that now this is an important Concept in both immunology and hematology is the idea of the normal differentiation of hematopoetic stem cells and this is as again I'm sure all of you are familiar with this idea but you basically start with this plod nepotic stem cell and to a large degree this is like traines in this room you could be anything you want you can go into any direction that you want during this differentiation process you basically different stimuli factors micro environmental signals determine which direction it differentiates one group Becomes myoid of ayic cells dendritic cells another group becomes a lymphoid progenitor and these distinguish and differentiate into B cells T cells and K cells and ultimately become the effector cells the important point to appreciate it's a one-way Street you can go from PLO metapo stem cell lymphoid B cell t- cell but you can't go back you can't be a cytotoxic te- cell and say I really don't like killing things you know I'd Rather Be A B cell you know they secrete antibodies and they kind of like all in the background they let the antibodies do the dirty work but they're kind of like hanging out in a cool place I don't want to do hand-to-hand combat it's too late they can't change their mind you can't be a B cell and say the cytotoxic t- cells get all the glory you know no one really appreciates me uh it's too late and it's a one-way Street and again similarly you know at my stage of my career I can D differentiate I can't go on and become like a neurosurgeon U because I'm too differentiated but again this is an important constant to appreciate but what we will be discussing in subsequent lectures are the signals that's involved specifically in differentiation of T cells as well as the differentiation of B cells and sometimes how those can potentially go wrong any questions okay so now this is uh hystology seeing things is really so important in understanding things so this is a classic uh smear from peripheral blood and clearly the red cells are very very obvious but what what do you think this cell is any suggestions what oh well let's think this is an Immunology lecture so what cell am I going to show the antigen that it's pre-programmed to recognize and I always think of these like you think of these aliens they have these giant brains and like teeny teeny bodies because they're so mentally Superior uh and they probably don't have growth hormone and steroids where they come from so so to this lymphocytes all DNA no cytoplasm because it's not been activated and the body is being very energy efficient because why should it be having a t- cell or B cell that's not seeing the antigen is pre-programmed to recognize making anabi or making an immune response it's basically in a quient phase this is an uh an em of it electr micrograph showing again this is the nucleus there's not very very few organel here however once this lymphocytes activated now things change dramatically so you go from being all brain no organel to now in the case of a B cell you can see tremendous amount of endoplasmic reticulum we making a tremendous amount of RNA GGI we making a lot of secretory proteins and similarly effect the te- cells a lot of mitochondria now because this is going to be using tremendous amounts of energy secreting a large number of factors in order to become effor cells and again that's how you can see the cells been activated okay is that clear now how does the immune system provide this High degree of sensitivity and specificity to to attack all these different pathogens without attacking cell and this is a uh the clonal selection hypothesis which really provides our basis of our understanding of how the acquired immune system works and the postulates of this is that each lymphocyte beers expresses a single receptor that provides its specificity for whatever antigen is pre-programmed to recognize that's a very important point when I was first learning Immunology I really didn't have a good grasp of that because I thought well you have a B cell has thousands of antibodies surface maybe each one recognizes a different pathogen te- cells maybe each one recognizes a large number of peptides however with this is telling you is that for every B cell it may Express thousands of imunoglobulin molecules on its surface but every one of them recognizes the same antigen t- cell expresses thousands of t- cell receptors but each one only recognizes one specific pathogen so one t- cell May recognize a peptide from HIV but all the molecules recognize the same antigen another te- cell all of them recognize the same virus from for example varicelas from chickenpox what gets the process going is the interaction between the forign molecule and a lymphocyst that can recognize with a high degree of specificity and therefore turn that telon or that b cellon to allow it to mount an immune response and these as these cells proliferate it's like a Xerox copy the same way if you make a Xerox copy of something every copy looks exactly the same so to all the progeny of these cells for the most part I'll discuss for imunoglobulin how it actually changes but they recognize the same antigen that the initial cell recognized so now you're making thousands of copies but all of them recognize the initial antigen and therefore they mount an immune response against that pathogen and finally lymphocytes that recognize that have anthen receptors that recognize self you have to get rid of them somehow and exactly how that process occurs again I'll be discussing in a in a later lecture but these postulates the colonal selection hypothesis the Hallmark is single cell single antigen recognition really underlies our understanding of the acquired immune system B cells and t- cells and so now to kind of demonstrate it pictorially if this is the genes that code for IM imunoglobulin molecules or t- cell receptors the cells have exactly the same repertoire of genes but in the process I'm going to be discussing in a subsequent lecture they undergo a large number of rearrangements of the genes they Shuffle the genetic array in a way that one cell will now Express hundreds and thousands of molecules that have one specificity in this case this could be for example for tanus and another mole another cell coming from the same plur poent PR pre PR proger progenitor now expresses thousands of molecules that recognize a completely different antigen but now when these cells proliferate all the daughter cells of these uh either B cells and T cells will recognize exactly the same antigen that the primary cell that underwent the rearrangement recognized is that clear so now to again to now put it pictorial at the cellular level what is happening is that you have a plur poent B cell or T cell for example undergoing these genetic rearrangements which I'll discuss the mechanism subsequently but somehow now this leads to the generation of millions of cells and every one of those cells is expressing a unique receptor and the uniqueness of the receptor is demonstrated by the little changes in the three-dimensional structure of this molecule on the surface it's only showing one MO on the surface but there's thousands of molecules and think about it it's not the B cell and T cell is not a circle it's not a flat pie it's a three-dimensional ball and that three-dimensional ball is studded with receptors all which have the same specificity but this process occurs randomly when this is occurring these cells have no idea what they recognize they could be recognizing a critical P antigen like something from a pathogen that will be infected with frequently could be a cold virus it could be polio could be tetanus or they could be recognizing albumin or a self protein cardiol Lipan they had no idea what it's recognizing because this process occurs random genetic rearrangement so therefore you need to have a way of removing self-reactive lymphocytes so therefore a system will be utilized that these now get exposed to self antigen and when it binds to self antigen that's a terminal event elimin ating those cells so it's like filtering the immune system you before you drink water you're running through a filter to get rid of the stuff you don't want to drink and then you drink the water munism does the same way this is random but it needs to be filtered and you realize that every person's filter is different because my self is different from a lot of your self antigens so therefore I want to filter my cells differently than yours and we know that's the case because if I would give someone my cells then neither your your cells would kill my cells cuz see them as foreign and vice versa because we are different but within my own body this filtration system has eliminated all B cells and T cells that recognize my self antigens once that's occurred now the next step is you now have this large pool of we call naive lymphocytes these are those small unactivated lymphocytes they've never seen antigen and they're now waiting to see if they antigen is out there that they're pre-pro to recognize if it doesn't then nothing happens that cell will never undergo proliferation if it does if it binds to the forign antigen is pre-programmed to recognize as I showed you before the cell gets activated it gets stimulated and now proliferates to make thousands and thousands of copies of itself it's raised an army all of which now recognize this specific antigen and then it's able to eliminate the pathogen okay any questions now if we start now honing down on how this event occurs how do you recognize uh specific antigens the first thing again Immunology is very nomenclature driven so antigen is something you hear over and over and over again the definition of an antigen is very simple it's basically something the immune system can recognize that's a pretty simple broad definition uh the the other n you'll hear is immunogen has anyone heard immunogen right the definition of an immunogen is anything that the immune system that anything that can generate an immune response now is antigen equal to immunogen are they the same thing any suggestions ra you know raise people in the back right you always calling the people in the back cuz the people in front are brave people in the back you know it's like you know no they'll never see me I won't call it so anyone in the back are they the same no why not so all immunogens are prop antigens not the other way exactly all immunogens are antigens but not all antigens are immunogens why not because some antigens for whatever reasons are not strong enough to turn on an immune response and in fact that goes into vaccine design because a lot of what we do in vaccine design is take antigens that may themselves not be able to mount an generate immune response and make them antigen and in fact in a subsequent lecture I'll discuss how we do that with carbohydrate antigens to allow us to make vaccines against for example pacus or mocal diseases so again just to n clature is really critical so antigen is anything to be recognized by the immune response so that's pretty much everything but immunogens a smaller group those that themselves are strong enough that they could stimulate the proliferation of B cells and T cells okay now now when you home in within this anti there are small pieces of it that are called epitopes and these epitopes are really the smallest unit which a immune response could recognize and again an important point to realize is you could have a large antigen but that antigen may be composed of dozens or maybe hundreds of different small epitopes so you could have a protein and this protein could have multiple epitopes and each one of those epitopes can be recognized by different antibodies and it may very well be that recognizing one epitope may be a very good way of recognizing that antigen and eliminating it whereas recognizing another epitope may not be the best way and that may determine why some person's immune response may work better than somebody else's because they're recognizing the best epitope to to recognize to really eliminate the pathogen okay is that clear now this again is probably one of the most important slides in the entire field of Immunology because this is the recurring theme in terms of the ability of the acquired B cell and T Cell immune response to remember it's been infected and mount a much more rapid and vigorous response the second time it's been infected so in this cartoon uh the Y AIS represents antibody levels but the same thing would be true if You' count te- cells in a log arhythmic scale and and the x-axis represents the time that the immune system is undergoing its prolifer is responding and what you're seeing here is that you basically are exposed to antigen a for the first time indicated by purple what we see is is that there's this lag phase so for a week nothing happens you're not making antibody then all of a sudden by about a week you start making antibody by 14 days you pretty much have made your peak in this case antibody response and then as you clear the pathogen the antibod response goes down and you see that Illustrated here now let's say weeks or months later months later you're exposed to antigen a a second time but you're also exposed to antigen B at the same time you're getting infected you're getting a cold at the same time that you're exposed to chickenpox whereas you've been exposed to chickenpox here what you see is that the antigen you've been exposed to you now have a much more rapid response within a day or two to you have rapid ramping up of your antibody response and you're also starting from a baseline level that's much higher and you're going to levels that are hundreds and thousands times higher than the first time you've been infected well let's analyze what this means at if at Time Zero you infected with a pathogen 7 days you're not making antibody what do you think the pathogen is doing do you think the pathogen is saying I'm not going to proliferate because it's not a fair fight you know I'm going to really wait until those antibodies get revved up because I fight fair you know you know the the rules of pathogens we have certain rules we keep to like uh the Marcus of Queensberry for boxing you know you can't box below the belt etc etc uh pathogens don't do that you know pathogens the minute they hit you they start replicating like crazy and you're not getting back at them and that's why you get really really sick in general the first time you're infected you're sick as a dog about a week after the infection and then slowly it takes another week till you start getting better because you're mounting your immune response however the next time you're infected you don't get infected again so for Chickenpox for example you could be exposed to chickenpox every day and except maybe for a couple of exceptions you may get one or two pox you won't get infected why because as soon as a few viruses are in your body you have this rapid ramping up of your immune response and you basically wipe it out before it has a chance to get a beach head okay and in essence that's what we do with vaccines what we do with vaccines is we artificially educate our immune system we basically say you know what instead of letting you get infected and getting sick as a dog to educate your immune system let's design a way of doing exactly the same thing with an attenuated virus or a dead virus virus to rev you up so then the first time you actually see the real pathogen you're ready for it and you're able to eliminate the infection and not get sick again this is this is a critical concept to understanding how the acquired immune system works it's also how we know whether someone's been infected because we look at antibody tiers if there are no antibod tiers against the pathogen we say the person hasn't been infected if there are antibody titers where did they come from they only came because the person was exposed to the pathogen so that's how diagnose infection HIV you look for antibodies against HIV for flu anything you could imagine that's how you make the diagnosis okay any questions so we'll get back to talk now so now how what is the and this is actually um um slide you don't have I just um copied it but now so now you've seen the in infection now you're getting the secondary response what's the basis for for H H why is it how is it working and the answer is if you remember the first time you see anthen I'm sorry before you see antigen you have thousands of cells and you may have only one cell for example that recognizes varell Oster antigen chickenpox antigen and that you need to expand that cell during your immune response so that now you have thousands and millions of those cells but after you've cleared the infection you don't get rid of all these cells and get back to square one where you have one cell that recognizes it you keep thousands of these cells as memory cells you have those cells stocked in lymph node throughout your body in mucosal tissue throughout your body so therefore the next time you see the same pathogen instead of having to start from very few a handful of cells and expand those which takes time now you already have thousands and millions of cells all distributed throughout the body ready to rapidly Mount the immune response that's why the second time you see it is that much more dramatic is that clear and in terms of vaccine design what's a better vaccine a attenuated live virus vaccine or a killed virus vaccine who says attenuated virus raise your hand now you could you know you got to participate you know don't uh you know don't be passive and you know Victoria's not in the back taking notes and seeing who voted what you know and saying you know okay we're going to Mark you you know at the end of when you get your certificate in the back it'll say you got five questions wrong and three questions right so you got to go with it because this is how you think okay so what do you think is going to be more effective an attenuated virus which gives you some kind of lowlevel infection raise your hand okay and a dead virus raise your hand see a lot of people not voting let's you let's do this again you know we're GNA get you you know you got to be active Okay who here thinks a attenuated virus vaccine is going to be more effective raise your hand okay and who thinks a killed virus raise your hand okay so only one person didn't vote so okay so who why is it why is it attenuated virus vaccine going to be better tell me like the real virus excuse me it's more like the real virus and and therefore what's it going to do it's going to make a lot more proliferation occur because you're getting a real infection a real inflammation and therefore you can have a lot more memory cells throughout your body and what does that mean that the next time you get infected you're more protected most importantly it lasts longer so if you get vaccinated with a with a dead virus it's good for maybe 10 or 20 years if you get vaccinated with an attenuated virus it may be good for 20 or 30 years if you get the natural infection you probably protected for the rest of your life you don't need booster shots okay any does that make sense okay so now if we think in terms of the structure of the an of the these molecules that recognize antigen the theme behind it is it consists of two parts one part is the variable region that recognizes the antigen and this is different from antibody molecule to antibody molecule this is different from t- cell receptor to t- cell receptor but at the same time there's a constant region which is the the same for all imunoglobulin of the same is type or all t- cell receptors and the way I always think about it is you ever see those screwdrivers where you have the same handle but then you could stick in a Philips head the Flathead the wrench and everything like that has different specificity in terms of what you add but the underlying framework of the constant region the handle of the screwdriver is exactly the same why is that because the screwdriver has to fit in your hand and this constant region is designed to fit into your hand this is what's allowing it to do different things so to for t- cell receptor an antibod molecule this constant region has to have the same function for immunog globular molecules it has to bind to FC receptors for t- cell receptors it has to trigger the same transduction pathway and therefore you want it to be the same for all molecules so that's why you have this mixture of the variable region which recognizes antigen and the constant region which provides the effective function of the imunoglobulin or t- cell receptor molecule and antibodies and te- cells use different have different structures this is an antibody molecule we'll discuss this in Greater detail and this is a t- cell receptor again the same concept variable region constant region variable region constant region but using completely different structural motifs and again when I first learned Immunology it took me a while to understand this because I thought well t- cells recognize uh antigens and B and molecule so they must be the same kind of molecule completely different the same job they do but they do it in very very different ways and again we'll discuss how the structure has occurs so there's absolutely no similarity between the sequence of the imunoglobulin variable region and the sequence of the t- cell receptive variable region because just different proteins but they do exactly the same thing have a high level specificity and share a similar approach in order for generating this high level specificity now for look in the bloodstream and we ask the question what kinds of lymphocytes are there in the bloodstream this is teaching us you know what what our immune respon is like so the most highest level of lymphocytes in the in in the peripheral blood are lymphocytes as you know all know predominantly CD4 positive t- cells so about almost a 2 to1 ratio you have about twice as many CD4 cells as cd8 cells about 10% of your circulating lymphocytes are B cells and a small smaller number are NK cells and even small very very small number Gamma Delta T cells which pretty much I'm not going to discuss in detail maybe I'll discuss during the mucosal Immunology uh lecture so this is what you have circulating in your bloodstream if you think in terms of the immune system Anatomy is Destiny what you see is basically uh how the immune system is structured is is is based on where in the anatomy it's located and if if you recall I talked about how you have these generation of these antigen specific receptors that occur in an environment where they need to eliminate self-reacting cells where does that occur for t- cells that occurs in the thymus for B cells that occurs in the bone marrow so in te- cells you're generating all those t- cell receptors in the thymus they are not allowed to leave the thymus until you've eliminated all self-reactive te- cells bone marrows is where B cells mature so t- cells thus B cell bone marrow the B bone marrow you the B cells can't leave the bone marrow unless any self-reacting B cells have been eliminated and now once those elimination occurs and this is called the primary lymphoid tissue now it leaves get into the secondary lymphoid tissue lymph node spleen where now these cells that no longer are self-reactive are waiting to see if the antigen they're pre-programmed to recognize ever appears okay is that clear that makes sense and that's how you protect your body from ever having self-reactive cells Mount the response against that the lymph node again is very tightly regulated anatomically and B cells are located in the cortical region the outside of the lymph node and after a B cell stimulated by antigen it undergoes proliferation and these millions of colonal cells form a germinal center and again it's not slice it's a ball of cells all of which recognize the same antigen par cortical area are where te- cells are located and there's an interface where B cells and t- cells can interact so t- cells can help B cells proliferate and we'll discuss in Greater detail class switch after the undergo activation t- cells leave the lymph node migrate into the peripheral blood B cells frequently actually can stay in the lymph node making antibody in the lymph node and that antibody because that's the effector molecule for B cells then gets into the circulation and also will fight infection okay is that clear now if a we'll discuss in great detail if a B cell or t- cell doesn't see the antigens pre-programmed to recognize it may move to a different lymph node keeps waiting until it sees what it's pre-programmed to recognize and for the overwhelming number of B cells and t- cells they may never see an antigen they pre-programed to recognize and they just they just die undergo apotosis and you keep making new ones and new ones it's like the body is buying lottery tickets trying to say which is the winning receptor that's going to recognize an antigen that's going to be important for being infected and again spleen has a little bit different structure but again the same concept where you have B cells and T cells that are architecturally localized to very discrete locations the mucosal system which I'll discuss in an entire an entire lecture again same idea you have this is the epithelium and cells are areas where antigen can pass through here are dendritic cells but again B cell Empire's Pates are located in these germinal centers and follicles and you have te- cells located around them where they can interact after they're activated again they leave through the eer Loa and get into the circulation B cells for example especially class switch to IGA secrete IGA and the IGA passes across the epithelium into the uh Lumen of the intestine or whatever the broni or wherever the mucosal system is again to to fight whatever pathogens are on the other side of the epithelium now B cells and t- cells recognize different antigenic context so antibodies for example recognize sble three-dimensional proteins that's the epitope that they recognize they'll discuss that in Greater detail whereas it turns out that what t- cells recognize is a peptide that's digested from the initial protein that therefore gets presented in an MHC molecule and this is a lot of multiple different ramifications again I'm going to be discussing that in the next lecture in much greater detail and again this is imog globul molecule structure again I'll be seeing a lot of it almost all of you are very familiar with it the point I want to make though is that you have this heavy chain and you have this light chain mobular molecules consist of two heavy chains linked by disulfide bonds two light chains linked to the heavy chain with disulfide bonds but the critical point to appreciate is that the antigen receptor of the imunoglobulin molecule is provided by contributions from both the heavy chain and by the light chain what that allows you to do is have a lot more diversity because you can have the exact same heavy chain but by matching it up with different light chains you could recognize a completely different antigen you could take the same light chain match it up with a different heavy chain again recognize a completely different antigen so this process of having two chains contribute to recognition of antigen dramatically increases your capacity to recognize antigen and as you know t- cells function as cytotoxic te- cells in this case a cell is infected by virus it presents peptide from that virus to the cytotoxic T Cell it kills the t- cell these are cd8 molecules for the most part T helper cells are very very different t- helper cells as a name implies they they are very optimistic cells cytotoxic te- cells are pessimistic cells they basically say you've been infected you can't be saved you must be killed you know it's a pretty depressing view but that's that's what they go through life doing T helper cells as the name implies they're optimistic cells they believe cells can be infected they still can be saved and that may be true in this case this is example of a maccrage infected with tu microbacteria tuberculosis it's it's present in the vacu it hasn't escaped into the cytoplasm so it's not a terminal infection the helper cells discussed in in in subsequent lecture activates the maccrage makes it a more efficient killing cells makes all these Lymes making tons of antigen that now pour into these vacul kill the tuberculosis and clear the infection so even though this cell has been infected the helper cell now allows it to clear the infection ction and live and continue to function so these are two limbs of the immune system clearly yeah there are cells that can't be saved then you want cytotoxic tea cells to get rid of them but there are cells that can be saved and then you want te- helper cells to help them enable them to clear the infection so now to to kind of conclude so why do you only get some infections like chickenpox and zoster only once well now you appreciate the secondary immune response much more vigorous you have memory cells that can recognize it more rapidly how do you generate a system uh well we really haven't answered that question yet so you have to come back for lectures three and four but we'll talk about we just know that there are these genetic structures that get reshuffled to generate antigen specific receptors subsequent immune response more rapidly because you have a lot more memory cells that are available distributed throughout the body to respond to the immune response and therefore you have a more rapid response at higher levels how does the imun system provide High degree of spe sensitivity and specificity by having unique receptors and one cell expresses thousands of receptors for a specific antigen and the summation of all these cells allows you now to recognize millions of antigens very very specifically and why are T cells and B cells effective again we haven't completely answered that question yet but just think about it antibody molecules are soluble they're going to be very good against free floating antigens like bacteria tea cells seeing peptides coming from outside from inside the cell they're going to be very good for intracellular pathogens like viruses and t- helper cells because they're optimistic and they enable macros to be better kill killers they also more effective against intracellular bacteria so this concludes lecture one and hope it's been informative and we'll come back for lecture two in about 45 minutes thank you very much
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01_Biochemistry_Introductory_Lecture_for_Kevin_Aherns_BB_450550.txt
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Captioning provided by Disability Access Services at Oregon State University. Kevin Ahern: Let's see. How about now? Can you hear me up there? You can. Okay. Student: No. Kevin Ahern: No? More volume? Less volume? Student: More volume. Kevin Ahern: More volume! How's that? Higher! Lower. [students chuckling] What is this? Are you guys ready to start? The term isn't ready to start. This is awesome! Welcome! How are you guys doing? What do you think of this fancy, new auditorium? Student: It's awesome. Kevin Ahern: I'll tell you something. Look at this. Dead. Something I like is I can actually walk up here and the light's not hitting me in the face, because I was usually blinded. Now I can really see if you guys are sleeping or goofing off back there. So I know, right? So welcome to BB 450/550. I'm very happy to have you here. My name is Kevin Ahern, and I will be your instructor this term. And what you see on the screen is a page that you'll probably see a lot of during the term, and it gets updated pretty regularly. I'll tell you a little bit about myself and tell you a little bit about the course and also maybe even learn a little bit about you. So, it's a little warm in here. They're opening some doors, so hopefully we'll get some air in here, which would be nice. I've been teaching this course for, I don't know, seven or eight years, and have gotten to be very fond of the course and very fond of the number of students and the relationships I've had with students that have taken the course. I really am a person who likes that sort of personal connection. I like to interact with you. I like to help you in any way that I can. I also like to see you, of course, be responsible and do what you can, on your part, but I can assure you I will also do whatever I can, on my part, to help you to get on top of this subject. So, it's a difficult subject. I know it's a subject that many of you have a lot of anxiety about. And I feel bad about that, to be honest with you, because I think that it's a subject about which most students find that there really is a lot of really interesting stuff in it, but if you come in it kind of going “eeeeeeehh”, you know, that's kind of a barrier to learning, okay? So I want to do what I can to cut through that. I keep a lot of office hours. And if you look at my schedule, you'll see I don't even keep, I don't even list them. My policy is, when I don't have another meeting scheduled, I'm generally in my office and you are welcome to come any time. Okay? Now, you might check to see, if I have a meeting scheduled, I'm not going to be there, obviously, when you come by. But if you do that, I'm more than happy to meet with you. I'm more than happy to help you in any way that I can. That's part one of the things that I think is very important for me to do with you, okay? If you want to set a meeting with me where you want to be sure I'll be there, of course, you're always welcome to do that. Send me an email. I will be happy to schedule that with you, as well, okay? So it's important that I be available to you. And it's important that I be helpful to you. If I'm not being available to you, if I'm not being helpful to you then, of course, you'd let me know that and we'd deal with that. Biochemistry is a subject that most students take, not because they want to, but because they have to. "I gotta have it for a biology degree." "I gotta have it because I want “to go to medical school." "I gotta have it because I want “to go to dental school." And so that's the way that many of you come into this class thinking. And I will also tell you there are a lot of urban legends that are out there. I won't tell you that it's an easy course, but I will tell you that not all of the urban legends are true, just as, I'm sure you know, that not all the things that you read on the internet are true. So, as I get wind of those, I do try to shoot them down. I'll tell you the one that got started last year was that the averages in this course are never above 50. Well, A, if that were the case, you can't flunk everybody, right? B, it's not the case. So in the entire time I've been teaching biochemistry, which goes back to 1995, I've actually had three exams where the average was lower than 60. Urban legends aren't always true. So don't freak yourself out with what's basically a bunch of bullshit. I kind of loosened it up, right? Bullshit, right? I'll say it again. How 'bout some bullshit, right? You guys like bullshit? Let's talk about some bullshit today, okay? [class laughing] So anytime you have a question, please come see me. That's number one. If you don't want to come see me, you can email me. I don't care. It really doesn't matter. Before the first exam, I'll give you my cell phone number. You can call me. Alright? I have no problem with that. You can call me. You can come see me. You can do whatever works for you, because my job is to help you to learn biochemistry, because biochemistry is going to be important for you in whatever you want to go and do. Medical schools, microbiology programs, dental schools, graduate schools don't make biochemistry a requirement to be mean to you. They make biochemistry a requirement because it's going to be a very important part of your career. So it's important for you to get these things down that I'll be talking about today, and tomorrow, and the next day, and the next day. Okay? Alright. So, I videotape all my lectures. I try to get the videos posted within a couple of, certainly within a couple of days of the video having been made, and they're available for streaming. And you don't even have to come to class. In my experience, and I have done the experiment to test it, those who come to class do better. I have some very solid data for that. But it's your tuition money. If you decide you want to roll the dice and take the chance, and maybe you don't come to class and you do well, you can play that game. But I don't think you want to play that game with your career, in general. I do reserve the right to give pop quizzes if it looks like, well, people are really not taking this seriously. So a pop quiz is a possibility. I don't do it to be mean. I do it to encourage you to participate and come and listen. The best way to use the videos are to supplement the lecture. I tend to talk fast. You've already seen I like to pace. If you want to get a really humorous thing, watch my video and speed it up about three times, and you'll see I'm playing tennis, back and forth, right? So I pace. I will try to slow down. If I get going too fast for you, all you have to do is say, "Hey, Kevin! “Slow down!" And by the way, I like to be called "Kevin." Please don't call me "Dr. Ahern," okay? I find that really is a very stuffy term and it gets between me and you. So please call me "Kevin." Alright? Okay. It's important for you to get on top of this stuff. And it is a subject that is rapidly expanding. Our knowledge at the molecular level of what is happening in cells is exploding. There is nothing in all of the sciences that is as exciting right now as what's happening in biochemistry. That's absolutely true. We are in the middle of what's called the "biological revolution." Now, the biological revolution happened as the result of several things, the most recent of which was the ability to determine large quantities of genomic sequence information. As a result of that, we have an incredible amount of information about cells. You can't pick up a newspaper today and not see some exciting new biotechnology finding. What I hope you will get, as a result of this course and BB 451, is an understanding of the significance of some of these findings, and they are absolutely remarkable. So I hope that one of the things that I leave you with is an interest and an enthusiasm for the subject, and I hope to dispel any fears that you have about the subject. I know I'm coming back to that, but that's what people have. Okay? If you need a tutor, if you need assistance, I'm happy to help you to get that. If you want to come see me, I'm happy to help you work with me. So, whatever works for you, that's what I want to make sure that you do. I want you to play an active role in your own education. I'm not going to pound you on the head and put the information in. You're the one that's gotta do that. Whatever way works for you, works for me, short of looking at your neighbor's, you know, okay. Alright? Are we clear? Alright. So, first expectation I have of you, you'll read the syllabus. The syllabus is required reading. You can download it. You will see a question on the first exam come straight from the syllabus. You are responsible for reading and knowing what's in the syllabus. Okay? That's given. All of the videos this term I'll be using, I'm uploading to YouTube. YouTube videos are a little hard to download. If you're having some real issues and you really, really, really want to have a download, I will make some other options available to you. I also post my videos to iTunes U, and iTunes U videos are very easy to download. So if you're having trouble with YouTube videos, you can go to iTunes U and pull all of my videos down. And, in fact, there's a whole bunch of them there, right now, already. They're in a podcast format. Every time a new one's available, you get a notice, etc., etc. So if you want to have a downloadable version, iTunes U is probably the best way to go. Okay. Well, that's enough preliminaries. That's enough about me. I want to ask a little bit about you. So how many people are anxious for this term? Be honest, or I'll cut all your arms off. [class laughing] Alright. That's about average. How many people are looking forward to the term? Okay. Some of you are probably the same people, anxious and looking forward. Sometimes anxiety actually for people is adrenaline, you know? People who have to go on live television, they get that adrenaline rush, you know, and a lot of anxiety, but they go out and they do great stuff because that adrenaline makes them do great things. And adrenaline can do great things for you and it can do very bad things for you. Okay? We hope to make those things be good for you. Hopefully, we reduce that adrenaline, right? We get you into a place where you can be your best and do your best without having to be anxious about things that you have to do. Okay? Alright. What's the worst comment you heard about biochemistry before you took the class? Anybody. Yeah? Student: I heard that you tell people that you don't need to know it and then you test them on it anyway. Kevin Ahern: Oh, yeah, yeah. Yeah. You know, you can't teach a class in the university without hearing that comment, right there. "He didn't say we had to have it, “and then he said we had to have it!" I can tell you right now, okay? I'll tell you what I tell everybody. I write, at the end of every lecture I give, I write a series of what are called "highlights." You see them up there, highlights here. After class, there will be a link there, and it'll be my summary of what I talked about today. I can assure you that I write my exams looking at the highlights. You're looking at the same thing I'm looking at when I'm writing the exams. So I hear that, but I hope, I certainly hope I'm not asking you something that I haven't talked about. I hope that that's an urban legend. Okay? Because it's my aim, I have no desire to be tricky. I have no desire to be confusing to you, in any way. I want to test your knowledge. You had a comment here, is that right? Student: Yeah. I heard you force a curve, a bell curve. Kevin Ahern: I force a bell curve? Student: You only give like this many A's, and this many B's? Kevin Ahern: Ah, okay. I haven't heard that one, but, okay. So the comment is that I force a bell curve. I force, "You're only going to have 10% A's, “and you're going to have 20% B's, “and the rest of you are going “to get C's, D's and F's." I don't do that. I do not do it. Okay? What you will see is that the bell curve creates itself. It honest to God does. I don't force it. I post, at the end of every exam, I will post the distribution of the grades, and you will see that bell curve. And the only way that I would force that would be if I made those numbers up, which I don't. So, no. I grade according to that distribution of what's there. The other part of that is I don't have a fixed number of A's, B's, C's, D's or F's. So good. Student: I heard that 451 is significantly harder than 450. Kevin Ahern: He heard that 451 is significantly harder than 450. I would say, if you asked the average student, the average student will say it is much less difficult than 450. Most people find 450, at least in my experience talking to them, most people find 450 to be more challenging. You'll find more math in 450, and it's because of that 450 has a recitation, and because it has a recitation, it has a recitation because of the math. There's no math in 451, so most people find 451 actually to be easier, in that sense. But I can't comment for everybody. You may have talked to someone who thought that. What else? Yeah. Student: I heard that if a cell phone goes off you lose 10% of your grade. Kevin Ahern: If a cell phone goes off, you lose 10% of your grade. Yeah, I've been known to get a little wild on cell phones, I suppose. Let me ask you guys, what do you think about cell phones going off in class? Student: Irritating. Kevin Ahern: Irritating. Annoying. What else? Student: It's distracting. Kevin Ahern: So what would you do if a cell phone went off next to you? Student: I'd want to throw it against the wall. Kevin Ahern: Okay. I haven't done that. [chuckles] She wants to throw it against the wall. I had a professor one time that did that. He had this fake cell phone that he picks up. Okay? And he's got it, so he pulls it out of his pocket and he grabs the other student's phone, and he throws it against the wall and everybody thinks he's thrown it, you know? Evil professor from hell. I'll tell you a really cool trick I heard a professor pull one time. He's working on a blackboard. Back in the, you guys may not know it, but they used to write with chalk on blackboards, you know? [class laughing] It's a long time ago. People used to write with chalk on blackboards. And so it's the first day of class and so he's got a bunch of freshmen, and he's sitting there and he's writing up on the board. And what the freshmen don't know is that he stuck into his pocket some candy canes that he has licked the color off of, so they look white, okay? And so he's getting up here, and he's talking, and he's doing this, and he's putting his hands, and so forth, and of course, he reaches in and he grabs one of the candy canes, and nobody knows this. And a student asks him a question, and he says, "Yeah, that's a really good question." [chomping noise] [laughing] If you guys ever go teach high school or something, that would be a really cool trick to do. I'm not sure how well it works at college, but it'd be a very cool trick to pull. Student: We had a chem teacher that would take a hand and stick it in liquid nitrogen and smash it with a hammer. Kevin Ahern: Take a hand, stick it in liquid nitrogen. Student: So it would look like it was his hand. Kevin Ahern: See, these are instructors from hell. I mean, I hope I'm not this bad, hopefully. The only thing I'll do is I'll make you sing. So you'll have to sing. You will have to sing for your supper in this class, okay? Okay. What else? What other comments? This is the quietest you guys are going to be, all term. No other urban legends out there? You want to make one up? How come nobody ever makes up an urban legend and says, "Oh, man! “That Ahern, man, “he's the greatest guy in the world!" You never hear that one. You know? It never, ever happens. What is that? I should start that one. Facebook group, you know? "Ahern's awesome!" What's that? Student: What are the differences between you and the other biochem teachers? Kevin Ahern: What are the differences between me and the other biochem teachers? Very good question. I'm much better looking than they are. [laughing] You will have the same instructor. I teach both 450 and 451. If you take 450 next term, which I hope you're not planning to do, but if you take 450, usually those aren't done by choice, then there's somebody else that teaches that. But I'm much better looking than he is. You think I'm joking. You say, "He must really be ugly, huh?" [laughing] What else? The more we talk like this, the less we have to talk about biochemistry. You guys are just ready to dive into biochemistry? Is that what it is? It sounds like a fly flying around, doesn't it? It's my heart. The pacemaker's just going "errrrrrr". [laughing] Okay. You guys ready to dive into biochemistry? Student: Yeah. Student: I think so, yeah. Kevin Ahern: You'll regret saying that. Oh, there's a comment over there. There we go. Student: What's the weirdest thing that's happened in your class? Kevin Ahern: The weirdest thing that happened in my class... I can think of two things. Okay? Number one, I remember being down here lecturing one day, and I hear this sort of shriek from the back. And the restroom down there has started leaking water and there's literally a wall of water that's moving down this way. It was not pretty. The second one was a little bit more humorous, and it was on Halloween. I had a couple of young men who decided they would disrupt class. Well, I have been known to be the professor from hell if people disturb other people's learning, whether it's with cell phones or whatever, so I don't want to have students' learning get disrupted. So these two young men come in and they dressed as really old guys. It's a beautiful costume. Okay? I'll give them credit. But in the middle of my class, they walk down, and they come trotting down, and they're doing the whole thing, you know, and of course, every eye in the class is on these guys coming down here to come and sit down in the front row. And, by the way, you're welcome to do it, just don't disrupt the class. That's fine. Well, they made a big production of it, so they kind of irritated me. So I looked over at 'em and I said, "Well, welcome." [imitates old men muttering loudly] I said, "I'm very happy that you gentlemen are here. “And because you gentlemen are here, “I have an announcement to make." [imitates old men muttering loudly] "I'd like everybody to look at these two gentlemen. “And because these two gentlemen are here, “you guys are going to have a pop quiz today." [scattered gasping] [chuckling] Oh, did the attitude in the class turn bad. And it turned ugly! And these two guys are sitting here, and they are the focus of every eye in the room. [laughing] Needless to say, they got their butts out of here. So I said, after they left, of course, "And here's your pop quiz. “Please sign your name to the piece “of paper and turn it in," which meant that they got extra credit. Well, of course, everybody in the class was happy, these two yo-yos who were here got chased out, and I was happy because I don't like people disrupting my class. [laughing] Okay. So those were the two weirdest things that have happened to me. Student: You don't use Blackboard? Kevin Ahern: I don't use Blackboard for this class. Blackboard is an absolute kludge, okay? If you want to see a clinker, use Blackboard. I think the web is much more efficient, and I can get things to you faster, easier, without problems, by doing what I do. So, no. I like HTML a lot. I can lecture from this. I can't lecture from Blackboard. I mean, I'm sorry, unless you're brain dead. Yeah? Student: So you'll post grades on here, then? Kevin Ahern: I will post distributions. I will not post your grades. You have to get your grades off of your exam. I won't post your grade online, no I won't. But you will see the distributions of grades, yes. Because of privacy issues, I can't post names and grades, obviously. Okay? Other questions? Yeah? Student: This is kind of a quick one. I'd seen where there was a new Advanced Light Source going online at one of the research universities. Do you know if that's been in use long enough for any of that material to make it into the literature that we're going to be using? Kevin Ahern: I'm not even sure of what you're talking about, so I guess the answer is no. [laughing] But if you want to talk to me about it later, I'd be curious to hear what it is. Student: It was in "Popular Science" recently. Kevin Ahern: Okay. News to me. News to me. To me, this is great, because with the real projection, like I said, I didn't used to be able to see. There would be a projector sitting right there and I'd just be blinded by the stuff here. Yeah. Comment back there? Student: I was looking at the bookstore and it said the online class you teach is using the seventh edition? Kevin Ahern: Thanks for bringing up the edition. No, that's not true. There was a little confusion. My syllabus originally said "Sixth Edition." We actually use the seventh edition. The question is, "Can I use the sixth edition “and get away with it?" The answer is probably, yeah, you can. Things don't change that much from one edition to another. I can't assure the problems line up appropriately, but with your TA and/or me, we can help you to do that. So I think textbooks are an outrageously expensive item, and I don't like supporting the textbook publishers, to be honest with you. So the online class is always behind the classroom class, because in the online class, they see a whole term's worth of lectures like today. And those lectures obviously aren't from this term. They're from a previous term. So they're using the textbook. You guys are the first one to use seventh edition of the textbook. So that's what's up with that. My advice to students is, wait on the textbook, if you want. See if you even need it. Some people feel they don't even need it. Student: Do you know if they're still selling it? Because I just got mine, like, today, and the sixth edition is still under, like, the shelf for this class. Student: It's probably for the online version of the class. Kevin Ahern: That's the online, yeah. So the online is sixth. This classroom version is seventh edition of the textbook. Okay? So I provide a lot of materials. I actually am in the process, and I'm not sure I'll ever do it for this class, but I'm in the process of actually writing my own textbook. I hired a student last year to create all the figures that I needed for a textbook I will give away, for my BB 350 class, so the students don't have to pay $200 for a ridiculous textbook to a publisher. I really think that's outrageous. Paper is not that expensive! Student: I agree. Kevin Ahern: Okay. It's not. So whatever I can do to work around that, I do try to do that. Okay, so, as I say, I'm paid to bore people. Hopefully, you guys are still awake. We should get started. Let's do that. Okay. So I'm going to say a few things, in general, about biochemistry here, that I'm not going to hold you responsible for. And I will tell you when I'm going to get going about something that you are responsible for. So, just in general terms, and you're not responsible for this, but, in general terms, biochemistry is the science of the molecular basis of life. That's what biochemistry is. We don't think about that so much. We don't think about biochemistry being that new. We think about biochemistry, "Well, it was always around, “like anatomy, or physiology, “or biology, or chemistry, or something." But it wasn't. Okay? The roots of biochemistry literally date to the 1930s. The modern roots of biochemistry date to the 1930s, when a man named Schrodinger said, for the first time, that the basis of life is not cells, it's not tissues, it's not organs. The basis of life can be found in molecules. And it was that fundamentally different way of looking at things that got people trying to understand the molecular reactions, the molecular basis of life in cells. It led to the discovery of the structure of DNA, which is truly where the roots of the modern biological revolution can be traced. Everything that we have, with respect to genomic sequence, to the topics of genomics, proteomics, all these various "omics," all date to 1953, when Watson and Crick stole data from Rosalind Franklin to show the structure of DNA. I've got a limerick I wrote about that. I'll share it with you guys when we get to that point. They stole it. They acknowledge they stole it. But the point is that, because of a variety of things, we knew the structure of DNA, it was quite clear that the information that we needed to know to understand that molecular basis of life came from that, as a result of that, okay? Here's a figure. Maybe it's not. The tree of life, okay? We think about three major branches of the tree of life in this class, and we'll talk mostly about bacteria and eukaryotic cells. Bacteria being grouped in the category we call "prokaryotes," and the higher cells, like people, plants, dogs, cats, fleas, multicellular organisms will all fit into "eukaryotes." And some unicellular organisms, like yeast, will also fit in the eukaryotes. All of the prokaryotes are single-celled. They're not multicellular organisms. So we'll see that that division between prokaryote and eukaryote is an important one, and we'll see that, even though there are differences that are there, at the molecular base, they're not nearly as big as you would think they are. I'll give you an idea. Virtually every cell on the face of the Earth has a common set of pathways that are virtually identical. Okay? We burn glucose in the way, identically to the way that the E. coli bacteria in our gut burn glucose. We make proteins fundamentally the same way that the E. coli in our gut make proteins, okay? The molecular foundations of life are remarkably similar. There are differences, and we'll talk about some of those differences, but the fundamentals are absolutely written in stone, as it were. And that's a really interesting thing. It's actually a good thing. One of the good things, it's a simplifying feature of biochemistry. We used to have a clock in here that you could see, but now I have my watch and what's on the screen. Okay, DNA. Memorize those structures for the first exam. Okay? That was a joke. Alright? [nervous laughing] In general, I try to minimize the number of structures that you have to memorize. I will tell you every structure I expect you to memorize, and I think it's fairly small. It's not a large number of structures you're going to memorize. I would much rather have you spend your time understanding concepts than memorizing structures. There are going to be some structures that you will memorize, but not a lot. And I will always tell you what they are. I can assure you of that. You will never have a structure that you have to memorize that I don't tell you about. Covalent bonds. You've had organic chemistry. Covalent bonds are fundamental to the molecular basis of life. Covalent bonds involve reasonably equal sharing of electrons. They're not like ionic bonds where one, like sodium, pretty much gives up its electron to chlorine to make sodium chloride. Covalent bonds, there's a sharing. And though that sharing isn't always equal, and there are consequences of that, it's much more equal than what we have in an ionic bond, where basically one atom gives up its electrons to another atom. Because of that unequal sharing of electrons, we see inequalities in terms of charge. This is a prime example of that. Nitrogen tends to hold electrons closer to itself than hydrogen does. So when nitrogen and hydrogen come together to make a covalent bond, the nitrogen ends up being slightly negative and the hydrogen ends up being slightly positive. I know that's freshman chemistry, but I find that people in freshman chemistry, you know, didn't learn it. Was it your fault or was it the professor's fault? I think it's the damn professor's fault, okay? We can stand around and bitch all we want about people not learning something, but if we don't, ourselves, make sure that learning things is critical to going forward, how can we complain? So one of the things I want us to all start on the same page about is something that I expect that we will all understand. That is that uneven sharing of electrons leads to partial charges. We will see this gives rise to what are called, in biological molecules, "hydrogen bonds." And hydrogen bonds are remarkable things. You're going to hear "hydrogen bonds" over and over and over, as we talk about structure this term, because they're incredibly important and they're incredibly weak, at the same time. And one of the reasons that they're important is because they are weak. If we compare the energy of a hydrogen bond to the energy of a covalent bond, there's no comparison. Hydrogen bonds are much weaker. That means it's much easier to break hydrogen bonds than it is to break covalent bonds. You say, "Why is that good?" And I say, "Well, the answer's partly on the screen." That's a base pair. Everybody learned A pairs with T. In your basic biology classes, you learned that they have two hydrogen bonds, and if we have GC, we have three hydrogen bonds. And you can do the math and figure that three hydrogen bonds take more energy to break than two hydrogen bonds, right? But they're still relatively easy to break. Why is that important? Well, think about what a cell has to do. A cell has to replicate its DNA. A cell has to make RNA from DNA. And both of these processes require pulling strands apart. Do you want to pull apart covalent bonds or do you want to pull apart hydrogen bonds? Ahh! But if they're so weak, then how do the strands stay together? There's safety in numbers, folks. There's safety in numbers. Millions of hydrogen bonds held together hold DNA in a double helix. We can take apart short stretches quite easily. But taking apart the entirety of a million-base-pair or a billion-base-pair chromosome takes an enormous amount of energy. Cells don't bother with that. So this weakness of a hydrogen bond, as we will see, is critical, not only for the structure of DNA. It's not just an obscure thing. But we'll see it's important for the structure of proteins. And we'll see that hydrogen bonds help to stabilize the structure of proteins. And because they stabilize the structure of proteins and they're weak forces, they can be fairly easily disrupted. Hey, that's kind of good! You know why that's good? Because we like to kill bacteria in our food. We can cook it and destroy the structure of the proteins in those bacteria and kill the bacteria. If those are covalent bonds that are holding those protein structures in place, folks, we couldn't kill bacteria. We probably wouldn't be here. Cooking provides enough energy to destroy the hydrogen bonds in the protein so that those proteins don't function, and we kill bacteria by cooking. We can kill bacteria by washing our hands with soap, because we're doing the same thing. Those hydrogen bonds can get broken with the interactions that we're giving to them. We'll talk more about that. So there's a real beauty to hydrogen bonds. I want you to understand that. Now, we could spend a lot of time talking about donors and acceptors, and blah, blah, blah, okay? To be honest with you, I don't think it tells you much. You can memorize that, if you want to. I'm not going to ask you this. Okay? What I told you to start with was the most important thing. The most important thing was that there's uneven sharing of electrons. Nitrogen has a greater electronegativity. Remember electronegativity from freshman chemistry? Right? Greater electronegativity, stronger affinity for electrons. Nitrogen and oxygen have greater electronegativities than does hydrogen. Therefore, when oxygen or nitrogen is bonded to a hydrogen, oxygen and nitrogen will be more slightly negative. That's that little delta sign that you see there. It means it's partially negative, not fully negative. And the hydrogen will be partially positive. Well, just like a full positive can be attracted to a full negative too, can a partial positive be attracted to a partial negative. So can a partial negative repel a partial negative. Alright? These are all important things to understand. Now, these types of structures that you see here, the N with an H, the O with an H, are very, very, very common things that we see in biological molecules, proteins, DNA, fats. There's our base pair. Here's some examples. Again, don't memorize this. But you see examples about how partial positives can interact with partial negatives. Look at the hydrogen on the water, partially attracted to the oxygen on the carbonyl group, okay? Very, very important. [rustling] Oh. [rustling continue] What did I do.... ? [rustling] [scraping] [bumping] I'm not sure what I did here. [noises continue] [silence] Well, it shut up. Now. I can work on it. [rustling noises] Oh, I see what happened! Oh! It's going to tear my favorite tie. I don't want to do that. [rustling continues] Okay, now we're back on. [laughing] There we go. Alright. [class laughing] You guys were hoping that everything doesn't work so that we'd call the day off, right? Alright. Okay. So, we're about at a point where we should start understanding material. So you're responsible from here forwards, okay? Here is a laser pointer that doesn't work. Okay. Well, I guess it wasn't even worth all the effort. Alright. Hydrogen on water, partially positively charged. Oxygen on a carbonyl group, partially negatively charged. And yes, oxygen has a greater electronegativity than carbon does. Okay? They're attracted to each other. There's a force that holds them together. If we want to pull them apart, we have to provide energy to pull them apart. That's why we have to cook food. That's why we have to do whatever we do to break those types of bonds. These types of bonds are everywhere, we find in proteins, carbonyl with water, we find water with an amine. Amino acids get their name by virtue of the fact they have amines in them. Quite a variety of structures that we have that's there. Okay? Now we want to talk about van der Waals interactions, and the main thing I want to talk about van der Waals interactions is just related to this one figure, right here. Van der Waals interactions tell us that if you try to put two nuclei too close together, they will fight it like crazy. There's an ideal distance. You put them too close together and the energy that it takes to put them together goes to the power of 12 as a function of distance. You try decreasing that distance beyond this point at the bottom of the curve, and you see it ain't gonna go. Atoms are just like relationships, right? You guys ever had the relationship, you know, where things are going really great? And then, after a while, van der Waals kicks in, folks, because what happens? "I gotta have my space.” Right? “I gotta have my space!" You can say, "Wow, van der Waals interactions apply to relationships," as you're crying your way home to mother or something, right? "I gotta have my space." Atoms have to have their space. If we try to put two atoms too close together, somebody's going to go home crying to mother. It ain't gonna go. Alright? So it's very important to remember that atoms have to have sufficient space. That drives everything that they do. Hydrogen bonds. You have water, the really bizarre properties and the wonderful properties that water does. We think of water and we just think, "Oh, it's a liquid." That liquid has zillions of hydrogen bonds that allow it to be a liquid at room temperature. Water has an atomic weight of 18. It's liquid at room temperature. Methane has a molecular weight of 16. It doesn't have hydrogen bonds. It's a gas until way below zero. Carbon dioxide has a molecular weight of 44, greater than that of water. It's a gas at room temperature because it doesn't have hydrogen bonds. Hydrogen bonds are a glue. [phone ringing] Oh, please. [laughing] Not a good way to get started. Okay. They're a glue that literally holds things together. Right? That's important, very, very important. Hydrogen bonds give water the property that's necessary for life on Earth. How about other effects? Well, other effects are very important, as well. What I've been talking about are hydrophilic things, thing that like water, things that interact with water. To interact with water, things have to be either ionic or have some partial charges to them. Well, what about the things that don't have much ionic character and don't have much in the way of partial charges? These guys are what we call hydrophobic. Take oil. Take water. Shake 'em up! Shake 'em up! What happens when you shake up oil and water? Student: They separate. Kevin Ahern: It looks, at first, like they sort of mix, right? And you set it down on the table and in a very short period of time, they separate. Do you know why they separate? They don't like each other, but there's a more important reason. Yes, sir? Student: [inaudible] Kevin Ahern: There's no hydrogen bonding, but there's another, more important, reason. Yes? Student: When they aggregate into a larger single unit, they have less surface area for more volume. Kevin Ahern: Ooh,ooh! I like this! This is good, okay? Exactly what he said. When they separate, they have less surface area that's interacting between the two. When I have little droplets, I add up all those surface areas, there's a hell of a lot more interaction between the water and the droplet than there is when they separate. Then we only have that immediate interface that's there. And if you add up those surface interactions, they're way greater when we have little droplets than when the layers separate. Okay? That's kind of cool. These things all give rise to something we'll spend a couple of lectures talking about, and that's an absolutely phenomenal process called protein folding. Protein folding, as we shall see, arises from a variety of chemical interactions. They include hydrogen bonds. They include ionic bonds. No, you don't need to write that down right now because we'll talk about them later. They include hydrophobic interactions. They include metals, in some cases. They actually, even, in a few cases, include covalent bonds, as well. Now, that's important, excuse me, because structure is essential for function. You know that. Somebody takes your bicycle wheel off, you don't have the structure, the bicycle isn't going to function. Right? Okay? It's important to recognize that structure is necessary for function. If we disturb the structure of a protein, we disturb the function of a protein, and, hey, that's what we talked about when we're cooking bacteria to destroy their protein structure and destroy the protein function. Things that disturb protein structure interfere with the ability of the protein to perform its natural function. We'll talk a lot about that as we get talking about proteins. Now let's see where I am. I've got about seven minutes left. We can actually talk very briefly about what I'll spend an entire period talking next time, and that's pH in solutions. Now, one of the places where your chemistry teachers didn't give you your money's worth in chemistry was teaching you about pH in solutions. So I'm going to spend some time getting, hopefully, everybody up to speed regarding pH and solutions. pH “Oh, yeah, that's the measure “of how strong an acid is,” right? pH is a measure of the hydrogen ion concentration, the measure of the hydrogen ion concentration. pH, you probably remember, is defined as the negative log of the hydrogen ion concentration. But in order for you to understand that, you have to understand what it means. What is the hydrogen ion concentration? Concentration is measured in moles per liter. And here's where 50% of you in the class, to my surprise, will not recognize the difference between that and moles. And I'm not saying it's your fault. I'm not saying it's my fault. But I'm saying, somewhere along the line you didn't get that hammered into your heads. Concentration is moles per liter. Remember that. Moles is a quantity. You could say, "Okay, I'll memorize this," but it has to be meaningful to you. So the way I'm make it meaningful, well, what's the difference between, "I'm going 60 miles an hour," and "I go 60 miles"? Big difference, right? If I go 60 miles an hour for a while, and then I go 40 miles an hour for a while, and then I go 30 miles for an hour, did I go 130 miles an hour? Of course not. If I know how many hours I traveled each one, I can determine how far I went, right? Isn't that just like concentration? If I know how many liters of a solution I have that's 1 mole per liter, I know how many moles I've got, because I can multiply pretty well, right? If I know I have 6 moles, and somebody put it into 113.4 liters, I could calculate the concentration by taking the moles divided by the liters. Don't confuse those two. You're gonna do it. You're gonna do it! If you're having problems with concentration, I know many of you will, come see me. I'll be happy to talk you through it. That's something your freshman chemistry teacher should have pounded into your head and didn't. You should go back and pound on your freshman chemistry teacher's head. "Why did you give me that grade?!" No, you won't go do that, I know that. Alright, pOH is the negative log of the hydrogen ion concentration. That means if I know the hydrogen ion concentration, I take the negative log of it. pOH. What is pOH? It sounds kind of like "pissed off" or something, doesn't it? P-O-H. "I'm P-O-H today," right? pOH is the negative log of the hydroxide ion concentration. It follows. If pH is the negative log of the hydrogen ion concentration, pOH is the negative log of the hydroxide ion concentration, okay? Same thing. Now, there's a relationship that pops up. You learned this in freshman chemistry. It's very useful. And that is that the pOH plus the pH of a solution always equals 14. If the pH of a solution is 6, its pOH is 8. So a solution that has a pOH of 8 is equivalent to a solution that has a pH of 6. They're the same thing. Okay? Now, one of the things, and this is where I'm going to finish today, one of the things that your freshman chemistry class didn't teach you very well is that they taught you reasonably well about strong acids. I've got a 1-molar solution of HCl, okay? If I have a 1-molar solution of HCl, it means I've got 1 mole per liter of HCl, and when I put HCl in solution, it completely dissociates. It comes apart. It means I have 1 mole of hydrogen ions and 1 mole of chloride ions. And I have zero moles left of HCl, because they've completely come apart. It's a strong acid. Strong acids, you're going to hear me say this over and over until you get nauseous, strong acids completely dissociate in water. I put HCl in water, I end up with completely H+ and completely Cl-. Not all acids behave that way, meaning that not all acids are strong acids. Okay? Well, what happens if it doesn't completely behave that way? Well, here's a weak acid we'll talk a lot about, acetic acid. Okay? Look at that. HAc goes to H+ and Ac-. You say, "Well, look! “It just did it!" Not quite. I put a million molecules of HCl in solution, I get a million H+'s and I get a million Cl-'s. Okay? If I put a million HAc's into solution, I'll be lucky to get a thousand H+'s and a thousand Ac-'s, which means I have 999,000 molecules of HAc that didn't do anything. It's a weak acid. With weak acids, we have mixtures, of dissociated and undissociated, the undissociated being the HAc, the dissociated being the H+ and the Ac-. Now, why do I tell you that? Is it to give you something to, "Oh, my god, “I've got something else I've got to memorize."? No. Most acids that are in our body are weak acids, folks. Most acids are very, very weak acids. Amino acids are weak acids. We'll see this. Proteins are full of weak acids. DNA is full of weak acids. Your cells are loaded with weak acids. We have to understand weak acids. Strong acids are very easy to understand. Weak acids we'll spend more time doing. [END]
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Medical_Lectures
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08_Biochemistry_Hemoglobin_Lecture_for_Kevin_Aherns_BB_450550.txt
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Captioning provided by Disability Access Services at Oregon State University. Ahern: We are moving rapidly through stuff and I finally sat down and looked at when the exam is and the exam is this Friday, so I guess... [classroom chatter] Ahern: It's not? The exam is a week from Monday. I'm guessing you guys wouldn't be ready for it this Friday? No? There will be fewer things on it. Think about that. A smaller nice, little tidy exam. Student: And the next will destroy us. Ahern: And the next will kill you. [laughs] Or you will kill me, I don't know. The second exam is not cumulative. The final is cumulative. I just have a couple fairly minor things I want to say about characterizing proteins and I do that in recognition of the fact this is not a biophysics class. The things that are there are really more biophysical in nature than they are biochemical in nature. I want you to be aware of what we can do with certain techniques. One of the techniques, last time I talked about MALDI-TOF and MALDI-TOF is a fantastic method as I say that I realize I didn't post that figure for it. I will get that figure posted showing you the set up for a MALDI-TOF instrument and you can hopefully see it better than my words can describe. I will get that posted for you. What I want to talk about today are a couple other very powerful techniques that come to us from biophysics. I'm not going to go in depth but you should know the basics of them and you should be able to understand why they're useful for us. The two techniques I want to talk about are X-Ray crystallography and nuclear magnetic resonance. You've probably had some exposure at least with nuclear magnetic resonance, I'm guessing in your organic chemistry class. Has anyone had exposure to X-Ray crystallography before? Little bit, okay. I'll start with X-Ray crystallography. X-Ray crystallography, both these techniques, by the way, are extraordinarily useful for helping us to understand relative positions of nuclei in space. I always tell the story when I give a tour of our facilities in ALS in Biochemstry and biophysics to students that using the tools of X-Ray crystallography and nuclear magnetic resonance, biophysicists can determine the position in three dimensional space of every atom that might exist in an enzyme that has 10,000 or 50,000 atoms. That's a really remarkable thing because knowledge about structure leads to understanding about function. We've heard structure function before. When we think about how drugs get designed, the design of drugs is happening increasingly as a result of the molecular knowledge of the structure of the proteins that they're targeting. If I know the position or I know the structure of the active site of the enzyme, the place where the reaction is catalyzed, I know the dimensions of the molecule I need to design to plug up that enzyme. That knowledge of structure is really valuable for us to have for whatever purpose. And there are purposes aside from designing drugs as well. X-Ray crystallography arises from the fact that X-rays get diffracted which means they get bent when they encounter electron clouds. That diffraction process is depicted here. To do X-Ray crystallography, one has to have first of all a crystal and a crystal is a as you guys see crystals you don't think at a molecular level what a crystal is, but a crystal is a perfectly packed homogenous molecule that has a regular repeat to it. That is all the molecules in there are the same composition and they're organized in a regular fashion. That regular repeat is what gives rise to the crystal itself. The process of making a crystal for a lot of X-ray crystal analyses is actually the thing that takes the longest. A lot of different, there's no one formula for making a crystal. Different proteins crystallize in different to way. Suffice to say that making the crystal, which is already shown right here, can be a very and time consuming step and a frustrating step in the process. Once one has a crystal, one can take they crystal and put in the path of an X-ray beam. That X-ray beam will have its rays diffracted again according to the electron clouds that it encounters. The importance of the regularity of the molecules in the crystal are very important because those really add up and give us the diffraction patterns that we see. Now interpreting the diffraction patterns obviously isn't a thing we're going to do here, but sufficed to say that diffraction gives, this is what a diffraction pattern of a given crystal might look like and we say wow, there are some spots. These are the spots correspond places where X-rays got diffracted to. So a biophysicist can take information from a diffraction pattern and work backwards, ultimately to determine where the individual electron clouds where it caused the diffraction pattern to exist. So X-Ray crystallography is extraordinarily powerful because it does give us three dimensional information about the position and orientation of electron clouds and a working map might look something like what you see right here. Here are patterns of electron clouds and then within there we decide what are the individual atoms that correspond to there, you see different carbons and hydrogues and oxygens and so fourth scattered through there. And interpreting these patterns can again be a very time consuming process it's very computationally intensive. The result is at the end of this that one has that structural information that's very useful. Again, that's just a very cursory description of X-Ray crystallography. X-Ray crystallography has its advantages and it has its disadvantages. The advantage of X-Ray crystallography is if you get crystals, then you can determine these positions very nicely. Sometimes you can't always get crystals and that's one limitation. And the other is crystals may or may not correspond to the natural, whatever that means, shape of the solution. So we think about the enzymes we have in our cells, most of them are dissolved the in water of the cytoplasm. So when I'm making a crystal, I'm basically taking it out of the solution so one thought is well is this reflecting the actual structure it has when it's in the solution? So to partly address some of those concerns, this additional technology of nuclear magnetic resonance is very useful because nuclear magnetic resonance analysis allows one to determine molecular structures in aqueous solution. They work in different ways and nuclear magnetic resonance relies on the fact that certain nuclei have spins that are characteristic of them and those spins can be altered in the presence of an electromagnetic field. So understanding the energies it takes to alter those spins of given nuclei for example protons. Protons are very commonly used in analysis to understand the changes in those spin gives us some knowledge about the structure. I'll just show you a very brief example here. Here's a nucleus that has a characteristic spin. There are two possible spins that can exist. One has a slightly higher energy than the other and the difference between that energy is what is excited using the electromagnetic radiation. It turns out the different nuclei have different spins corresponding to the electronic environment in which they find themselves. This depicts the nuclear magnetic resonance signal of a very simple molecule. This is ethanol. We can see that ethanol has three different kinds of protons in it. It has methyl protons that are farthest off in the end. It has methylene protons in the middle here and it has hydroxyl on the end. These give rise to characteristic signals. These signals have known positions, we know where hydroxyl protons arise, we know where methyl protons arise, etc. And so we can examine the spectrum that comes from this. The spectrum is called the chemical shift which I won't go in to. This molecule has some hydroxyl, it has methylene, it has methyl protons. As you can imagine for a molecule like a protein, it's not not nearly as this. We might get very complicated spectra and in fact we do get complicated spectra. So this is a little bit more challenging to interpret. We're not going to do that obviously here. But sufficed to say that analysis of nuclear magnetic spectra does allow, ultimately a scientist, to allow which signal corresponds to which groups inside the protein. Now understanding the different kinds in this case of protons that exist in a protein is useful information but one of the things we're interested in as biochemists is how do proteins fold. Because remember that folding really gives the protein its characteristic shapes so we would like to get more information because the knowledge of different protons we have in the protein really isn't sufficient enough to tell us structure. One of the techniques done with that is an enhancement as it were of nuclear magnetic analysis. It's called the nuclear overhouser effect and it arises from the fact that, this clip doesn't work, it arises from the fact that nuclei, when they interact with each other, also have effects. In this case, here are two protons that, as a result of folding, have been brought into close proximity. And if they're brought into close enough proximity they actually do affect the signal of the other one. This requires a very sophisticated analysis called 2D and that's obviously a lot more complicated than 2D gel electrophoresis I'm not going to go into that. But sufficed to say that with this type of an analysis, one generates some even more interesting spectra but this information that we see here now tells us not only what kinds of protons that we have but how close those protons are to each other. That's very, very useful when one goes to trying to determine the overall structure of a protein molecule. So that can be very, very useful. Commonly these two techniques are used in combination with each other to help elucidate molecular structure. There is a biophysics course in 5 minutes. How's that? Questions or comments about that? Let's get away from biophysics and talk about, this is the lecture I'm going to give today is most of the most popular lectures I give throughout the entire term. It's the lecture on hemoglobin. And hemoglobin is, I hope to convince you by the end of the lecture on Friday, one of the most magical molecules in our body. It is absolutely incredible the abilities that are built in to the structure of this protein. I start talking not about hemoglobin but about a related protein called myoglobin and I introduced myoglobin to you before as a protein related to hemoglobin. It's found in our muscle cells primarily and there it serves the function of storing oxygen. It's a very good way to store oxygen. Hemoglobin is very good at delivering, that is picking up and dropping off oxygen. The difference you recall structurally I hope between the two proteins and that myoglobin has a single protein subunit. And hemoglobin has four protein subunits. So hemoglobin has quaternary structure, myoglobin does not. And this quaternary structure that myoglobin has is, that hemoglobin has is what gives rise to all of the properties that the molecule has. Well, you've seen myoglobin before, it's mostly alpha helical structures, it looks something like this. There's the amino terminus and there's the carboxyl terminus. And here you see alpha helix bend, alpha helix bend, alpha helix bend, a lot of alpha helices here. We see the amino acids, we see 146 amino acids. Myoglobin was I believe the first protein whose structure of this nature was actually determined and so that has some biochemical significance. Not of any concern to us at the moment. But the other concern for us is it has an oxygen binding group in it called a heme. So the heme, and yes, myoglobin has a heme just as hemoglobin has a heme. The heme is located right here. Yes, sir? Student: So wait, is this myoglobin or this hemoglobin? Ahern: Actually, I'm sorry, this is the beta chain. I have a link to it that says myoglobin. They're very, very similar so this is the beta chain of hemoglobin. We can think of this as myoglobin because as I said the structure is very similar between the two. Thanks for noticing that. Anyway, both myoglobin and hemoglobin have a heme. Hemoglobin you recall has four chains to call beta and to call alpha. This is one of the betas right here. Now the heme turns out to be really important for several reasons. The number one being of course that it's the place where the oxygen is bound by this protein subunit. There's the heme. Heme is a flat ring. It is something we refer to in these proteins as a prosthetic group. Sounds like a very mouthful name. Prosthetic group is simply a molecule bound to a protein that helps the protein do what it does. It's a non-amino acid. So it's a non-amino acid bound to a protein that helps the protein to function. The heme group of hemoglobin and myoglobin, the two are essentially identical and they're very, very similar in structure to chlorophyll. The electron gathering component of chlorophyll that we find in plants. The difference in plants that instead of having iron in middle, we have a magnesium in the middle. Student: You said this is planar? It has like 20 carbons and stuff in the middle. How is it possible? Ahern: How is it possible? Well, if we're talking about an exact plane, there's nothing that's exactly flat. Generally, it's a flat structure. You can see when I talk to you about this that the places actually pucker. So it's planar, but I wouldn't say it's a perfect plane, no. Alright, that puckering that we will see is very, very important. It's actually seen right here in this figure. What I'm getting ready to tell you about here occurs in both myoglobin and hemoglobin but the impact is felt in hemoglobin because of its four subunits. What you see happening on the screen happens in both proteins. Let me describe to you what's going on here. If you we look at the deoxyhemoglobin on the left, that's the way it normally sits. Shannon says, "well that's not exactly planar." And I say well okay, look. It is slightly puckered. We can imagine it being a little concaved downwards like my hand is. When the oxygen binds and we see oxygen bound over here, there is a very, very tiny change. So instead of being slightly puckered, it flattens a bit. Why does that happen? It happens because the oxygen pulls up the iron atom. The iron atom physically gets lifted. This change is minuscule. We're talking fractions of angstroms. Very, very tiny change. Yes, sir? Ahern: His question is, "Is this just the heme group?" It turns out this movement affects a lot of things. It's a very good question. For the moment, we're thinking only about the iron atom. The heme group itself is not moving. It's the iron atom that's moving. So the iron atom moves up a very tiny fraction of an angstrom. And if you look at the structure, you'll notice that the iron atom is not floating freely there. It is in fact attached to an amino acid beneath it. This amino acid that it's attached to is a histidine. Now, if I pull up on iron and iron's attached to histidine, you can do the math and figure that the histidine is probably moving a fraction of an angstrom as well and you'd be exactly right. And you'd say histidine is attached to another amino acid in the protein, is it moving also? Yep, so the foot bone, the toe bone's connected to the foot bone, and the foot bone, this isn't going to be a song by the way. The foot bone's connected to the ankle bone, and the ankle bone is connected to the shin bone and by pulling on the toe, I'm ultimately going to affect the hip. Even if it's by a very tiny amount. And this very tiny amount, I can't emphasize enough the importance of this very tiny change because I'm going to hopefully convince you by the end that the result of this very tiny amount of movement allows us to be animals. Without this movement, animal life is essentially not possible. This is a scary thought. Why is it that this makes animal life possible? We'll talk about that. Hemoglobin of course doesn't exist as a single subunit, it exists as 4 subunits. All 4 subunits have a, each subunit of the 4 has a heme group of its own. So when this guy binds an oxygen, and by the way, because it's a schematic, they're not showing the connection, but in each case it's connected to a histidine. When this guy binds to an oxygen, let's see I've got this hemoglobin that's got no oxygen whatsoever. This guy binds an oxygen, it's going to cause the iron atom to move a slight distance, it's going to cause that histidine to move a slight distance. It's going to cause that entire chain to very slightly shift. That very slight shift changes the overall shape of this subunit. And guess what? That shift affects how it interacts with its adjecent subunits. And the adjacent subunit now becomes more favorable for binding oxygen. So this one, when we have hemoglobin that's empty of oxygen, it's not real keen on binding oxygen, but once one of them binds oxygen, these changes get communicated between the subunits and each additional oxygen is increasingly favored for binding. This phenomena I've just described to you is called cooperativity. The bonding of one molecule to a protein affecting the binding of others. In this case, it's positive cooperativity. It's favoring more binding. Now this is really important. We have to, we are animals, we are moving creatures, we have to have an adequate oxygen supply. Plants don't have this issue. Plants don't have to get up and run around and jump and go chase things or run away from things. Their oxygen needs are more constant. Ours are rapidly changing. We need oxygen, we need it now. When our hemoglobin gets to our lungs, it doesn't have an awful lot of time to be there. We want it to load up on oxygen as much as it can and take that oxygen out to the tissues where its needed. Cooperativity as we will see, plays a very big role in the loading up of hemoglobin. So we're loading up hemoglobin. If we can't load up hemoglobin, we don't have enough oxygen, we can't go run, we can't go escape, we can't do things that animals do. Very, very important. The other thing I want you to look at in this structure, it's actually easier to see right here is that when we look at hemoglobin from above as we are in this case, we see that hemoglobin is shaped sort of like a doughnut and there's a little hole in the middle. That little hole turns out to be extremely important. Extremely important. So I'm going to talk back and talk about that hole but before I do that I want to tell you a little bit about the needs of oxygen in the cell and how hemoglobin helps to supply those. Questions on this before I move forward? Everybody understands what cooperativity is? Yes, sir? Student: Does the inverse occur when it unbinds the oxygen? Ahern: Good question. Does the inverse occur when it unbinds oxygen? The answer is to some extent, yes it does. Loss of one will favor the loss of additional ones. So it would be a negative cooperative. So you start to see where this is heading, right? If we look at the oxygen binding of myoglobin, this is a plot. Again, whenever I show you a plot, I always want you to know what the axes tell us because without the axes, the plot has no meaning. This is the fractional saturation meaning what fraction of all the myoglobin molecules in the solution have an oxygen bound to them? Myoglobin can only bind one oxygen per protein because there's only one subunit and each heme only binds one oxygen. Either it's bound or it's not bound. What percentage of those guys are bound with oxygen? What we see that it takes very little oxygen. This is very low, the pressure of oxygen on the X-axis. Very little oxygen for us to get 50% saturated. What does that mean? It means that myoglobin when it has the chance is grabbing a hold of oxygen. It's very good at storing oxygen. It grabs it, it holds onto it very well. Well that's nice, but it's not ideal for delivering oxygen because if myoglobin didn't give up its oxygen til the oxygen concentration got very low, it could travel all the way through the body, get all the way to the lungs and it hasn't given up its oxygen. "No, it's mine. I'm the big kid I get the quarter." Right? "I'm not going to give this up for you." Myoglobin only gives up its oxygen when the oxygen concentration gets very low. How many people have UPS on their computer? Anybody know what a UPS is? It's an uninterruptible power supply. It's there to give you power when the power goes off so you have a chance to shut down and save your work. Myoglobin is the UPS for your muscles. When you're working very hard, it's very easy for you to use oxygen faster than your blood can deliver it. Well oxygen is important. It's not essential, but it's important so that more oxygen your muscles have, the better off they are because muscles need it for contracting, we gotta run away from something, we gotta beat something up, we gotta do whatever, hopefully we're not doing too much of that. If hemoglobin can't supply all the oxygen that's needed, we want something there to back it up and this is backing it up. When the oxygen concentration starts getting very low, myoglobin says, "Oh, here's some oxygen." That's the only time myoglobin gives up oxygen. When the concentration gets very, very low. But it helps us when we need that. Let's compare that with hemoglobin. Ahern: That's the oxygen concentration. Oh, here? How much does it take to get half of it saturated, half of it bound to oxygen. Is P 1/2 just refers to 50% of it. So very, very low number there. If we compare this with hemoglobin, hemoglobin has a different looking curve. So the curve that corresponds to myoglobin is what we call hyperbolic. It's a hyperbolic curve. It's a hyperbolic function that will fit that curve. The curve that hemoglobin gives is called sigmoidal because it has sort of a S shape. It's sigmoidal. Look at this. At low oxygen concentrations, there's not very much oxygen bound. When hemoglobin travels through our body, it goes from places of high oxygen concentration, our lungs, high oxygen concentrations, essentially 100% of it gets bound with oxygen. As it travels through the body, the oxygen concentration starts dropping because the cells are using oxygen and hemoglobin is the only source, oxygen concentration drops and hemoglobin starts letting go of its oxygen. It's a perfect system for delivery. We see it cooperatively binding oxygen to begin with and we see cooperatively letting go of oxygen. Binding in a negative sense when the oxygen concentration starts to fall. Hemoglobin, because of cooperativity can satisfy an immense, or a diverse set of oxygen concentrations as they occur in our bodies. This is essential for an animal. I just cannot, I keep coming back to that, but I can't emphasize that enough. It binds like myoglobin at the very highest concentrations. It will get 100% bound. But as oxygen concentration falls, it's down over here. And that's pretty cool. Then by the time it's dumped its oxygen, it goes back to the lungs and it gets more oxygen. So let's think about that a little bit. No, I'm not going to ask you to draw this particular figure although you should be familiar with any of these figures. We see the differences in oxygen concentration, this figure is nice in that it shows us the concentration of oxygen roughly that occurs in tissues and the concentration of oxygen that roughly occurs in lungs. We see that again, way up here in the concentration of the lungs, myoglobin and hemoglobin are essentially the same. And then way out over here, when this thing gets out to the tissues, only what is 38%, no, I'm sorry, what is this, about 30% or so of the hemoglobin is actually still bound to oxygen. So that flexibility of hemoglobin for oxygen is very, very valuable for us and allows us to things like sitting here, or getting up and running if we have to go running. And both of those work. Okay, if I exercise, blah blah blah, if I rest, I have different needs, if I have lungs, same sort of thing. This shows us the quaternary changes that happen as a result of oxygen binding to hemoglobin. Quaternary changes mean the four subunits are actually changing on oxygen binding. Notice that doughnut hole that I had before. The doughnut hole has largely closed up. It turns out that these two different states of hemoglobin have names that we give. And we're going to hear more about these names later. They're called the R state and the T state. I need to define them for you. The one on the left is called the T state. The T stands for tight. I like to think about it as people I know who are uptight. People you know who are uptight, you can just sense it around them and they can't take anything more. They're very rigid. Give me some oxygen, "No, I don't want any oxygen!" Okay, tight structures. Ahern: What's that? Student: [inaudible] the right? Or the left? Ahern: The left? Yeah, that's tight, yeah. They're not very flexible. They have poor binding of oxygen. So when hemoglobin has no oxygen, it's in the T state. It doesn't want to bind more oxygen. On the other hand, once it's bound and it's gotten full of oxygen, its structure changes to what we call the R state. R is the relaxed state. The relaxed state, "yeah, come on, we'll take this, we'll have it, I can take a lot of stuff, man." Right? [class laughing] I grew up in the 60s, guys, you gotta give me credit for the language at least. So the relaxed state is a high affinity binding for oxygen. Once we put one oxygen on there, we start flipping it into the R state and we've got affinity to find more oxygen. We flip it into the T state, it doesn't want to bind oxygen. That's kinda good. If you think about it. Let's imagine that I'm a hemoglobin that's floating around and I've just gone to let's say the muscles where they're exercising pretty heavily and I dump all my oxygen. On the way back to the heart or to the lungs, I pass through the kidney. Do I want the hemoglobin taking the oxygen away from the kidney? That wouldn't be a good career move, right? So I only want it to flip when there's a high oxygen concentration. That's what's going to happen when it gets back to the lungs. So the T state and R state really serve the body's needs very, very usefully. There are a couple of ways of describing how this phenomena occurs. The way I've described to you is called a sequential mains. That is the binding of the first one affects the second one, affects the third one, affects the fourth one. And that's not shown on the screen, this is a different model. If I were to show what happens, we've got T state above and R state below. In the sequential model, one of these guys turns circular, it favors the next one turning circular, the next one turning circular, etc. That model's called a sequential model of binding. Changes in the structure of one changes the next one, which changes the next one, etc. It's sequential. This model you see on the screen is the opposite of the sequential, it's called the concerted model. These are models. Models are way of explaining things. This model says that we don't see one followed by the other, followed by the other, followed by the other. Instead, what we see is we're either in one state or we're in the other state and binding of things locks them into that state. Now hemoglobin is not a good model for this. We'll see in next week's lecture how an enzyme is a much better model for this. This model and I'll say more about this model next week so I'm not going to go into much here, but this model says that the changes happen all at once and they're independent of the binding of anything. We see this as back and fourth. But they get locked into one vs. the other based on what they bind. We'll come back to that next time. For right now, think sequential. Binding the first changes the second, changes the third, changes the fourth. Changing the T to R state changes significantly the binding affinity of hemoglobin for oxygen, which I showed you before and if we look at what hemoglobin would look like in the T state, this is what it would look like. If it were only in the R state, this is what it would look like and in fact, hemoglobin goes through a transition from T to R and that's what we're seeing, why this curve has a couple of shapes in it. We're seeing a change from T to R. That change is what we've already described as cooperativity and that cooperativity is favoring bonding in this case is oxygen. If we're getting more oxygen or favoring the release of oxygen if we're getting into lower concentrations of oxygen. This is what a sequential looks like. Nothing bound, first one changes this one, which causes this one to chances, which bind has caused this one to change, which caused this one to change its find. It doesn't matter if it binds to an alpha subunit or a beta subunit. It doesn't matter. They're essentially the same as far as this molecule exists. Yes sir? Student: Shouldn't the [inaudible] under K4 be switched where there's a higher affinity to drive further to the right? Ahern: All of these depend on oxygen concentrations, so you're exactly right. In the concentration of the lungs, even though you've got a lower going to the right, there's enough oxygen concentration to drive that. Student: You're showing a sequential increase in K1 and [inaudible] the last one is shorter. It's counter-intuitive. Ahern: No, because it doesn't want to bind that first one. I agree that this is important for the releasing of oxygen. This, from what I've told you sounds a little odd in terms of putting that last one on there. But the reason this is the case is because that first one doesn't want to bind. And that's because this guy here is in the T state. That should answer your question. Student: Once you have the three on there, it seems like it should be more of a push in the equilibrium to push [inaudible]. Ahern: Right, so his point is that this equilibrium is favored actually in the leftward direction and that would be true if it weren't in the high oxygen concentration in the lungs. The lungs are loaded with oxygen and that drives it to this state. We want this guy to dump off oxygen once it gets out of the lungs. That's why the arrow is back to the left. Where am I at here? That's the basics of hemoglobin but there's so much more that's built into this molecule. The first one I'm going to show you right here is a really interesting and cool molecule called 2,3-BPG. We'll talk more about this molecule later in this term when I talk about glycolysis but this molecule turns out to be a fascinating molecule. You don't need to know the structure but you definitely need to know at least this part of the name: 2,3-BPG. It's real name is 2,3-bisphosphoglycerate. I need to tell you why this molecule is important. If my microphone works that is. Why this molecule is important. This molecule is a molecule that is released by rapidly respiring cells. If I'm a muscle cell and I'm doing my business I'm making 2,3-BPG, we'll see later it's actually a byproduct, but that doesn't matter for our purposes right now. Actively respiring cells release 2,3-BPG. So my muscle cells may have a lot of 2,3-BPG, my nose cells may not have so many. Okay? With me? Unless I'm sneezing, I've got that cold everybody else does, I might have more 2,3-BPG. It turns out that 2,3-BPG affects hemoglobin. If we look at hemoglobin in the presence of 2,3-BPG and red, we see that it binds less, and they're not exaggerating the S so much here, they've sort of drawn this to make their point. The point is that in the presence of 2,3-BPG, hemoglobin holds onto less oxygen. So 2,3-BPG it turns out causes the hemoglobin to release oxygen. How does it do it? It's very simple. 2,3-BPG has a shape that fits exactly in that doughnut. It fits exactly in that doughnut and when it fits into the doughnut, it favors the conversion of hemoglobin from the R state to the T state. T state has low affinity for oxygen, guess what hemoglobin's going to do when 2,3-BPG binds to it, it's going to start giving up more oxygen and that's exactly what this curve is telling us. This turns out from a bodily perspective to be very useful because when I've got actively respiring tissue and I've got a lot of 2,3-BPG, what's 2,3-BPG going to do? It's going to bind hemoglobin and hemoglobin's going to say, "Okay, flip into the T state, I'm going to let go of the oxygen." And as concequence of letting go of oxygen the tissues that need the oxygen get it. That's great. But wait, there's more. But wait. If hemoglobin has bound to 2,3-BPG, it's going to be in the T state and when it gets back to the lungs, it's still got 2,3-BPG, it doesn't want to bind to more oxygen, I've got trouble. Well fortunately, remember these are not covalent bindings, fortunately 2,3-BPG fits in that pocket. But it goes in, comes out, goes in, comes out. Like any binding that occurs, binding and letting go happens all the time. On the way back to the lungs, 2,3-BPG, when it gets off of the hemoglobin can get grabbed by cells and be metabolized. So as hemoglobin is making its way back to cells in most people, the cells are grabbing it, burning it up, and hemoglobin gets back to the lungs and it has no 2,3-BPG in it. If it had 2,3-BPG in it, you wouldn't bind as much oxygen. Now all of you pre-meds, everybody looks up at this point, smokers are full of 2,3-BPG. Smokers are full of 2,3-BPG. The reason that, one of the reasons that smokers huff and puff going up stairs is that 2,3-BPG doesn't get all the way broken down. The hemoglobin gets back to the lungs, uh oh. My oxygen carrying capacity is lower, that's why smokers huff and puff going up stairs. They've got too much 2,3-BPG. The next question is why do they have more? And that we will save and talk about when we talk about glycolysis. Suffice to say, they have much more 2,3-BPG in their blood than do non-smokers. Yes sir? Student: So is that a large contributory factor to say COPD? Ahern: COPD being? Student: Chronic obstructive pulmonary disease. Ahern: Is it a contributor to COPD? Not to my knowledge. There are other things that give rise to that. I'm not a medical person so I can't tell you that. But sufficed to say that the primary physical observation that you could make with respect to hemoglobin, respect to BPG in smokers is that they huff and puff. They puff and then they really huff and puff. If you smoke, quit doing it. Now you know at the molecular basis why smokers are having a hard time going up stairs. Their hemoglobin is stuck in the T state and they can't get it out of that T state very well. There we go. There's your doughnut and there's the binding. We'll need to worry about the various other stuff that's here. Now, hemoglobin is some pretty cool stuff. There's a problem, though. We are not chickens. And some would say that's probably good. But chickens lay eggs and the fetus that develops inside the egg has its own resources and doesn't have to rely on mom except to the point where the egg is laid. Mom, however, is the source of nutrients in mammals for food, for water, and for oxygen. Now there's a problem. The problem is what if mom's hemoglobin is competing with the baby's hemoglobin? They both have oxygen, why should the baby's hemoglobin have to fight with mom for that? What turns out that fetuses have a modified hemoglobin. They have a different hemoglobin than adults do. They have something called fetal hemoglobin. Adults have alpha-2 beta-2, meaning we have two alpha subunits, two beta subunits and that makes up four subunits which you saw. A fetus on the other hand has two alpha sub units and twp slightly different gamma sub units. So you've got alpha-2, gamma-2. Those two gamma subunits give the hemoglobin of a fetus a very, I shouldn't say very, but a slightly different property than mom's. Do they have cooperativity? But they have a greater affinity for oxygen than mom's hemoglobin. They can literally take oxygen away from mom. Talk about a little parasite. [class laughing] A little parasite sitting there sucking my oxygen, right? How do they do it? The gamma subunits in addition to having a slightly different structure cause the hemoglobin that they're in to not have a doughnut. That little doughnut hole where the 2,3-BPG fits doesn't fit 2,3-BPG anymore. The fetal hemoglobin essentially stays in the R state all the time. Essentially stays in the R state all the time. Now, you say that's great, so obviously it can take oxygen away from mom and yes, it can, and yes, it does. But we just saw how the T state helped to release oxygen, right? Is the fetus starved for oxygen? What do you think? No? Okay, there's a no, there's a no. Nobody thinks yes? No, it's not. Why? Why is it not starved for oxygen? Student: Higher net oxygenation level? Ahern: Higher net oxygenation level. It does have a higher net oxygenation level, but it also has trouble releasing oxygen so the answer is no. The answer is simpler than you think. Connie? Student: I mean it just sits there, right? Ahern: Okay, and that's the answer. It just sits there. It doesn't have widely varying oxygen needs. Mom goes and climbs the stairs, she needs more oxygen than when she's sitting around in a chair. All the fetus does is kick. [Class laughing] So it doesn't have widely varying oxygen needs. It needs a relatively constant supply of oxygen and because it does have a high oxygen carrying capacity as you noted, there's enough release so that it can satisfy those needs. If it had very diverse and very challenging needs, then you betcha there would be an issue. Other questions? I'm going through this kinda quickly. Shannon? Student: So if you're like a mom and maybe you have a very low blood concentration, like iron concentration, would it be smart to take supplements for that? Ahern: If you were a mom and you had anemia, is that what you're saying? Student: Yeah. Ahern: Would it be smart to take, I don't know, iron supplements or something like that? Yeah, for people that are anemic, that can be a consideration with mom but again, I'm not a physician but I'd imagine that yes, they would use that. Yes? Student: So when the fetus is born, it becomes...[inaudible] Ahern: Yeah, yeah. Ahern: Yeah, when the fetus is born, it has got fetal hemoglobin. So that change over happens in the first year or two where the gamma sub units stop being made and the beta subunits start being made. And so the fetus transitions to adult hemoglobin fairly early in its life. But you're exactly right, yeah. Student: Does it change based on how active the baby [inaudible]? Ahern: Does it change based on how active the baby is? I honestly don't know the answer to that question. I don't know. As far as I know, it's simply a developmental thing. Your question is whether it responds to environment and I don't know the answer to that. Well we've gone through a lot I thought we should finish with a song today. What do you guys think? We haven't done a song in awhile. I have a cold so I think it will be worse than usual so I want you to sing really loud today. And by the way, I have an idea I will do with my classes. If you guys sing loud, I assure you you'll have an extra credit question on the exam. But if I can't hear you... Everybody ready? Okay, everybody sing. "Biochemistry, biochemistry, "I wish that I were wiser. "I feel I'm in way over my head. "I need a new advisor. "My courses really shook me. "Such metabolic misery, biochemistry, biochemistry. "I wish that I were wiser. "Biochemistry, biochemistry, reactions make me shiver. "They're in my heart and in my lungs." "They're even in my liver. "I promise I will not complain. "If I could store them in my brain. "Biochemistry, biochemistry, I wish that I were wiser. "Biochemistry, biochemistry, I truly am in a panic. "The mechanisms murder me. "I should've learned organic. "For all I have to memorize, I outta win a Nobel prize. "Biochemistry, biochemistry, I wish that I were wiser." Alright, guys. See you Friday. [END]
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Medical_Lectures
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Hemostasis_Lesson_4_Tests_INR_PTT_platelets_fibrinogen_Ddimer.txt
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this is the fourth video in this series on hemostasis and today i'll be discussing tests of hemostasis the learning objectives are first to list the conventional tests of hemostasis and to describe how they are performed these will include the platelet count pt and ptt next to list the common ideologies of abnormal results of these tests then to describe how to approach an abnormality of pt or the ptt and last to interpret typical coagulation profiles this video will start to introduce a few medications and common disorders of hemostasis such as von willebrand disease hemophilia and the antibody syndrome however i won't be talking about these conditions in any depth here rather they'll each be discussed in future videos in the series let me start by discussing the indications for hemostasis tests to identify which patients presenting with acute bleeding have a correctable bleeding tendency for example von willebrand disease hemophilia and disseminated intravascular coagulation just to name a couple to identify which patients presenting with acute clots have a correctable clotting tendency such as anti-phospholipid antibody syndrome to monitor response to anticoagulant medications such as warfarin and heparin to aid prognostication for patients with liver failure for example the pt test is used to predict mortality both in alcohol hepatitis as part of a calculation called the discriminant function and in cirrhosis as part of the meld score the screen for disseminated intravascular coagulation commonly known as dic in patients with severe sepsis and last to screen for occult bleeding disorders prior to surgery or other invasive procedures this last indication is controversial and i'll talk more about it at the end of the video i think about hemostasis tests in two categories first are the common tests these are the ones that i see ordered all the time in the hospital they include the platelet count pt and inr the aptt which is more commonly referred to as just ptt fibrinogen and dimer then there are the uncommon tests which are ordered only in very specific situations the only uncommon test i'll discuss more in this video is the mixing study which is important in evaluating an increased pt and particularly ptt however there is also the thrombin time which is a measure of the function of the common pathway although textbooks generally make a point of discussing this test in over 10 years of internal medicine i've never seen it ordered once or recommended by a hematologist in real life and therefore won't be talking about it next is the activated whole blood clotting time test of anti-10 activity in the echoin clotting times these are all used exclusively to measure anticoagulation effect from specific medications and will be discussed in the anticoagulation video then there are various tests of antiphospholipid antibodies qualitative platelet function and heparin-induced thrombocytopenia these three will be covered in future videos as well at the same time as the relevant conditions are discussed not listed here is an archaic test called bleeding time in which a standardized nick is made in a patient's skin and the time it takes for it to stop bleeding is measured although mention of it still creeps into some review books the test suffers from poor reproducibility and is essentially no longer performed the first test to discuss in depth is the platelet count which is both the most basic and also the most common of all hemostasis tests platelet counts are tested on machines called autoanalyzers as part of the complete blood count which is a broader panel of hematology tests that also includes hemoglobin concentration and white blood cell count the test must be performed on blood collected in an anticoagulated specimen vial which is usually purple topped indicating that it contains edta there are many etiologies of quantitative platelet abnormalities a platelet count that is too low a derangement known as thrombocytopenia can be due to either decreased production of platelets increased destruction or hypersplenism also known as sequestration decreased production is most commonly due to bone marrow disease such as infiltration by cancer or systemic infection or due to nutritional deficiencies such as b12 and folate deficiency decreased production is typically accompanied by decreases in the other cell lines of red and white blood cells as well now increased destruction can be due to any of a very long list of medications along with alcohol also autoimmune disease such as idiopathic thrombocytopenic purpura and dic a platelet count that is too high usually called thrombocytosis can be reactive in response to infection surgery particularly splenectomy malignancy or acute blood loss much less commonly thrombocytosis is a consequence of a myeloproliferative disorder both thrombocytopenia and thrombocytosis will be the subjects of their own videos later in the series one additional quantitative platelet disorder that should be discussed is pseudothrombocytopenia this refers to a falsely low platelet count as a result of lab artifact it's relatively common there are two main etiologies the more common of the two is incomplete mixing of blood collection tubes which causes small clots to form which then traps platelets and confuses the auto analyzer a less common etiology is edta dependent agglutins in this case in order to get an accurate platelet count a different anticoagulant must be used pseudo thrombocytopenia can almost always be identified from a review of the peripheral blood smear as in this case when we can see the extreme platelet clumping the next test to discuss is called the prothrombin time or more commonly just the pt to measure the pt whole blood is collected into a sealed evacuated tube which by convention is blue topped signifying that it contains citrate the citrate binds calcium in the blood which prevents the blood from coagulating to maintain accuracy a standard amount of blood must be collected into the tube such that the ratio of blood to citrated solution is also standardized the whole blood is then centrifuged to separate the blood cells from everything else the blood cells are then discarded leaving a translucent yellowish fluid called plasma plasma is essentially the liquid portion of blood and predominantly contains electrolytes albumin immunoglobulins glucose and of course clotting factors to the plasma is added calcium and a substance called thromboplastin which is a mixture of tissue factor and phospholipids if we look back at the coagulation cascade from the video on normal hemostasis we will be reminded that tissue factor is present here as the primary trigger for the tissue factor pathway also known as the extrinsic pathway therefore when added to plasma along with calcium a fibrin clot will form macroscopically this changes the translucency and color of the plasma and will transform the free-flowing liquid into a solidified gel-like substance any of those changes can be used by either a machine or human being to determine when the clot has formed the pt is defined as the time in seconds for this to happen from the coagulation cascade diagram you can see that this is a measure of the function of the tissue factor pathway however it is also a measure of the function of the common pathway since deficiencies or abnormalities of thrombin fibrinogen factor 5 or factor 13 will also lead to delayed clot formation the normal range for the pt is about 11 to 14 seconds but this has high variation due primarily to differences in the manufacturer's thromboplastin but also due to differences in incubation time differences in the storage conditions of the samples and differences in the method of endpoint detection this is actually a big problem particularly for patients on anticoagulant medications affecting the tissue factor pathway such as a medication called warfarin for example the degree of anticoagulation a patient on warfarin might be experiencing could be labeled appropriate by one lab too much by another and too little by a third this inconsistency made it very difficult to publish standardized guidelines about how these drugs should be used and monitored and it made it impossible for a patient on warfarin to have their coagulation profile monitored by more than one lab however in the early 1980s to account for this variation a derived value called the international normalized ratio or inr was developed the inr is calculated as the patient's pt divided by a controlled pt raised to the power of something called the isi which stands for the international sensitivity index the isi is specific for each lot of thromboplastin reagent the isi of each lot is provided by the commercial vendors of thromboplastin but should be independently confirmed by the clinical laboratory this didn't completely eliminate the problem of standardization but it greatly reduced it a normal inr for someone not on anticoagulation is 0.8 to 1.2 so what are the etiologies of an abnormal inr well first is there such a thing as an inr that is below normal there is but an inr significantly below normal is very uncommon has been poorly characterized and is of unclear clinical significance in other words unless you're a hematologist you'll probably never need to worry about it however there are many causes of an elevated inr they can be subdivided into some anticoagulants those associated with a decreased synthesis of clotting factors which can be from chronic liver disease or vitamin k deficiency and those associated with an increased consumption of clotting factors such as sepsis and dic the next test to discuss is the activated partial thromboplastin time frequently called the ptt i don't actually know why everyone always drops the a when saying it allowed but they do this test is nearly identical to the pt with one important difference once the plasma has been separated from the cells instead of calcium and thromboplastin calcium in something called partial thromboplastin is added this partial thromboplastin consists of phospholipids without tissue factor and without tissue factor present an activating agent must be added in its place which is often silica or diatomaceous earth as with the pt the ptt is the time in seconds it takes for the fibrin clot to form the ptt measures the function of the contact activation and common pathways the normal range for the ptt is about 25 to 40 seconds but it has substantial variation across different reagents and testing systems unlike pt and the inr there is no equivalent standardization of the ptt and therefore each lab and hospital should establish their own normal range and adjust anti-coagulation protocols accordingly having been the one responsible for coordinating such an adjustment once at a hospital i can attest to the fact the process is haphazard and inexact which is why i personally have very limited confidence in the safety of infusions of unfractionated heparin which rely on the ptt test for titration this will be discussed more in the video on anticoagulant medications there are a fair number of causes of an elevated ptt they include anticoagulants again a genetic defect in von willebrand factor predictably called von willow brand disease hemophilia anti-phospholipid antibodies sepsis and dic the next test fibrinogen is not ordered nearly as frequently as the pt and ptt but it's still not rare to remind you fibrinogen is a glycoprotein critical for both the formation of the platelet plug as well as serving as a precursor for fibrin which is the penultimate step of the whole coagulation cascade fibrinogen is synthesized in the liver and its normal range is around 200 to 400 milligrams per deciliter most abnormalities of fibrinogen are quantitative however both thrombophilic and hemorrhagic versions of congenital dysfunction anemia also exist in which measured levels of fibrinogen are normal but these are very rare quantitative abnormalities of fibrinogen can either be high or low high fibrinogen can be seen as part of a general increase in so-called acute phase reactants which are a collection of compounds in the blood whose levels increase in response to non-specific tissue injury inflammation or infection other acute phase reactants include ferritin c-reactive protein c3 and c4 high fibrinogen is also frequently observed in normal pregnancy low fibrinogen is predominantly seen in two conditions liver failure and dic the last of the common hemostasis tests is the d-dimer as mentioned in the last hemostasis video the d-dimer is the most clinically relevant of the numerous types of fibrin degradation products it consists of two d-domains from adjacent fibrin monomers thus the presence of d dimer requires recent intravascular coagulation the normal range is less than 0.5 milligrams per deciliter which is sometimes expressed as less than 500 micrograms per deciliter there are many many causes of an elevated t dimer the most clinically relevant include venous thromboembolism such as a pe arterial clot anywhere dic severe sepsis malignancy recent surgery or trauma liver disease and once again normal pregnancy so those are the five main coagulation tests platelets pt or inr ptt fibrinogen and d dimer when one considers all five together the group is considered a coagulation profile and often can be collectively ordered as something called the dic panel let's review what typical coagulation profiles look like for some common diseases in advanced liver disease platelets will be low due to hypersplenism and splenic sequestration the inr will be high due to impaired production of clotting factors and the ptt can also be high or is occasionally normal fibrinogen will be low and the dimer will either be normal or elevated in dic platelets are low the inr and ptt are either normal or elevated fibrinogen is normal or low and the dimer is usually elevated although not included in this panel the other key finding in dic is a so-called microangiopathic hemolytic anemia identifiable by encountering red blood cell fragments called schistocytes on a routine blood smear in von willebrand disease platelets are usually normal but occasionally mildly decreased inr is normal ptt is normal or elevated and both fibrinogen and d-dimer are normal in both hemophilia and anti-phospholipid antibody syndrome often the only abnormality is an elevated ptt and in pregnancy platelets can be either normal or low the inr and ptt are normal and fibrinogen and d dimer are elevated the condition in which an otherwise normal pregnancy is accompanied by mild thrombocytopenia is called gestational thrombocytopenia which is important to distinguish from a serious complication of pregnancy called help syndrome the help is not like the word help as i can get some help but rather an acronym standing for hemolysis elevated liver enzymes and low platelets which are the three main features of the disorder all of the diseases in this chart are going to be discussed in more detail later in the series now among the tests i designated at the beginning of the video as uncommon there is one which merits discussion here that is the mixing study clotting tests can be abnormal due to either a clotting factor deficiency or due to the presence of an inhibitor of normal factor activity usually an antibody a mixing study differentiates between these two possibilities and it relies on the observation that clotting tests generally only require factors to have 50 percent or sometimes even less of their normal activity in order for the test to be normal so here's how a mixing study works some volume of the patient's plasma is literally mixed together in a one-to-one ratio of pulled plasma from normal individuals then the initially abnormal clotting tests are rechecked if the abnormal clotting test normalizes the patient must have a factor deficiency because whichever factor was deficient in the patient's plasma has been supplemented by the normal level of that factor in the normal plasma but if the clotting test does not normalize the patient must have a factor inhibitor because when factor inhibitors are present their concentration is usually high enough to inhibit the additional factors added to the mix from the normal plasma the one caveat to this relatively simple test is that some inhibitors require time to inhibit the clotting factors that have been added from the normal plasma therefore if the clotting times in the mixture are normalized when checked initially the test should be repeated and the mixed plasma allowed to incubate at 37 degrees for several hours before checking once more the lab should then report both the results of the mixing study when performed immediately and when performed after incubation so now let's put all this information together into a helpful practical algorithm for investigating abnormal coagulation tests that is focusing just on the pt and ptt there are many algorithms out there on the internet and they actually vary quite a bit but this is the one that makes the most sense to me and most conforms to my own clinical experience importantly this algorithm assumes the patient is on no anticoagulant medications and is not critically ill if either the pt alone or both the pt and ptt are prolonged then give the patient vitamin k if the test or test is correct you have your diagnosis vitamin k deficiency if there is not correction next consider whether the patient has severe liver disease if they do the prolonged clotting times are almost certainly due to that liver disease this isn't necessarily true 100 of the time but if a patient has both severe liver disease and another completely independent clotting disorder it will be very challenging to diagnose the latter and if that's the consideration you should consult a hematologist if the patient either has no liver disease or if the liver disease is mild enough that a secondary clotting disorder seems unlikely then perform a mixing study and if between the pt and ptt only the ptt is prolonged then go right to the mixing study from the mixing study if the prolonged clotting corrects next check the patient for von rulebrand disease and measure specific factor activity levels on the other hand if there is no correction next check for anti-phospholipid antibodies and specific factor inhibitors so i'm almost at the end but there are just two very quick practical topics that also need to be mentioned first is the interpretation of coagulation tests in patients with liver disease it's commonly believed that liver patients with abnormal coagulation panels have an increased risk of bleeding on the words i often hear the term auto anticoagulated particularly in the context of the decision to not place a patient with cirrhosis on dbt prophylaxis however the reality is much more complicated there are reduced levels of most pro coagulant factors that's true but also most anticoagulant factors as well there are actually increased levels of factor 8 and while there are frequently reduced platelet counts which predispose to bleeding there are also increased levels of von willebrand factor and decreased levels of atoms 13 which may predispose to clotting therefore liver patients with markerly abnormal coagulation tests may actually still be in hemostatic balance but it's a much more precipitous balance in addition to decisions about dvt prophylaxis the other scenario in which this comes up is when a surgeon or other proceduralist requires a liver patient to receive transfusions of ffp in order to drive the inr down below some arbitrary value prior to performing a procedure in the absence of other evidence that the patient is at high risk of bleeding this is probably not justified particularly if such a transfusion will either risk volume overload or will delay a critical time sensitive intervention finally i will end with a few words about the use of coagulation panels to screen for bleeding disorders in asymptomatic patients particularly those with no family history i started off this video with a list of common indications for coagulation tests and this is one of them but it probably should not be so first it's very common for the inr and ptt to be checked before routine surgeries and other procedures however the inr and ptt have low sensitivity and specificity for bleeding disorders the rate of clinically significant but occult bleeding disorders in otherwise healthy people is quite low and personal or family history of excessive bleeding is much more predictive of perioperative bleeding than coagulation tests therefore many experts recommend an emphasis on personal and family bleeding history when predicting operative bleeding risk and explicitly recommend against the use of coagulation tests for this purpose in my own practice i will still frequently order pre-operative inrs and ptts and otherwise healthy people but that's only because those performing the procedures usually insist on it this is not necessarily a battle worth fighting with your hospital colleagues so that concludes this video on the test of hemostasis the next three videos will discuss medications which alter hemostatic balance specifically anti-platelet drugs anticoagulants and fibrinolytic meds and the less commonly used pro-clotting meds you
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Medical_Lectures
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10_Biochemistry_Enzymes_II_Lecture_for_Kevin_Aherns_BB_450550.txt
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Kevin Ahern: Tailgating is always fun. The beautiful thing about tailgating is, it doesn't matter if you win or lose. [laughter] Before the game even starts, you're ready. It doesn't matter, right? [laughter] Last time I got started talking about enzymes, and I'll be talking about enzymes today and in the lecture on Wednesday. I've had a couple of questions about where the material's going to go and I haven't decided yet. It will almost certainly go through the exam on Wednesday, but it all depends on how far I get. So I don't have a set place, I just like to see how things are going and decide according to that. But it will probably go through the exam on Wednesday. I will announce on Wednesday where the exam will go to. In anticipation of a couple of other questions about the exam, number one, I will do a review session and that will likely be on the weekend. I haven't decide on a time for that yet, but I will announce that, as well. Yes, I will videotape the review session, so if you can't be there you'll have a chance to watch it, et cetera. Let's see, what else? How to study for the exam. My strong recommendation is that you use my highlights as an outline for your studying. It doesn't mean you should only look at the highlights but you should use the highlights as an outline. That lets you sort of broaden from there. I definitely do not recommend you study old exams. That's just a dumb idea. So study the material. The old exams that I provide are there for you to see the format of my exam, and it's very important you understand the format. So you don't waste too much time on a question, you don't waste too much time on a section, you need to understand that format. The format will not change from the way you see it on the exam, and I'll have more to say about that as we get closer to it. Those are just some general guidelines for you. A lot of you are coming and asking questions, and I think that's great. I love to answer questions. If you have questions, please come see me. Please come see the TAs. If you want a tutor, I can help you get a tutor. Remember that the TAs are free tutoring during their office hours. You don't have to go to just your TA's office hours. You can go to any TA's office hours and have, essentially, a free tutor, so take advantage of that. You're paying for it. You should get it. Last time, I think I finished at about this point, where I talked about maximum velocity and I talked about Kcat. Let me just briefly refresh what I said on that, and that is that maximum velocity is dependent on enzyme, on how much enzyme that we use, just like maximum car production is dependent upon how many factories I have producing cars. So it's a relative thing. I use more enzyme, I get a higher maximum velocity. I use a lower amount of enzyme, I have a lower maximum velocity. We need to take maximum velocity and turn it into a quantity that is independent of the amount of enzyme, and that's why we take the maximum velocity that we determine and divide it by the concentration of enzyme that we used to get that. So the maximum velocity divided by the concentration of enzyme gives us a very important quality called Kcat. Kcat is a quantity that is linked to the enzyme. It's a characteristic of the enzyme. It is not dependent upon the quantity of the enzyme that I use. So Kcat is also called "turnover number" or identical things, and they represent a number. So if I had a Kcat of 100,000, then it means that my enzyme is converting 100,000 molecules of substrate into product per second. Each molecule of enzyme is doing that, okay? So the turnover number gives me a relative measure of how fast the enzyme is working. If we think about an enzyme, there's a couple of terms that I want to define for you. One, I keep talking about substrate, and I sort of briefly mentioned it as an aside, but you should know that a substrate is a molecule that the enzyme is acting upon. A substrate is a molecule that the enzyme is acting upon. The place where the enzyme acts upon the substrate is known as the "active site." This is the place where the reaction is catalyzed. So the active site is a portion of the enzyme, a specific place in the enzyme, where it catalyzes the reaction. Now, commonly, the active site is internal to the enzyme. We'll see some examples of that actually next week, not this week. Molecules that will fit into the active site have a fairly specific shape. This is why enzymes, as I said earlier, will not work on all molecules. They work on specific molecules of specific shapes that will fit into their active site. Some enzymes have more flexibility than others do to accommodate other molecules. Some are extraordinarily specific. So there are some big differences between them. What you see here is an enzyme and it has some various side chains that are here. This enzyme probably works on a couple of substrates. Some enzymes may put two things together and make one. Some enzymes may transfer something from one substrate to the other inside of it, and some enzymes may simply take a substrate and rearrange it. So there's all kinds of possibilities for what an enzyme can do. Neal? Student: Are enzymes typically bigger than the substrate? Kevin Ahern: Are enzymes typically bigger than the substrate? Absolutely, absolutely. Well, let me back up on that. Yes and no, alright? So, yes, you should always think of them in those terms. But there are some enzymes that work on other enzymes. So if we define that other enzyme as a substrate, then I would say, well, not necessarily, but, in each case, when it's working on a another enzyme, it's working on a specific portion of that other enzyme. So, in essence, the answer is, yes, they will always be bigger than their substrates. Other questions? I got off on a big roll, there, didn't I? Student: Does substrate usually bind to the active site, is that... Kevin Ahern: So the substrate binds and will be held in at the active site, right. So we can think of a binding site, and people say, "What's the difference between a binding site and the active site?" Well, we can almost think of them interchangeably. The substrate has to bind and a portion of it is going to stick into the active site where the reaction's catalyzed. That's really what's going on. So I use the terms essentially interchangeably, "active site" and "binding site." Yes, sir? Student: You said that commonly the active site is internal, like inside the enzyme. Does that mean that there has to be a very specific molecule to go inside the enzyme? Kevin Ahern: So his question is, do you have to have a very specific molecule to get in the enzyme? And the answer is, yes, absolutely. That's why enzymes have very, very strong structure specificity, yes. So that is the sort of overview of them. Not surprisingly, one of the things that we see is that the binding of the substrate to the enzyme commonly occurs as a result of hydrogen bonds. Here we see some hydrogen bonds that are helping to hold this substrateóin this case, uracilóin the active site of an enzyme. That's not at all uncommon. What we will see next week is that, when we look at enzyme mechanisms, that some enzymes, as a result of their catalysis, transiently become linked covalently to their enzyme. I emphasize "transient" because if it becomes covalently linked and it doesn't come off, we just destroyed the enzyme. We'll talk about that next week. For right now, when we think of binding, the binding occurs essentially by hydrogen bonds. Hydrogen bonds are the most common bonds we see in nature. Now I need to just introduce a couple of concepts to you with respect to how enzymes work. How do enzymes do what they do? There are two common models that are used to describe mechanisms of enzymatic action, and these two models that are there, usually you've only heard of one of them. In basic biology classes one of the things that you're almost always taught is that the enzyme binds to the substrate very much like a lock fits a key. I see people shaking their heads before I even start to say that. Right? That there's a relationship. The key, not all keys will fit into an enzyme. I mean, not all keys will fit into a lock and not all substrates will fit into an enzyme. That relationship and that metaphor actually works very well to describe substrate binding. It does not, understand "not," do a very good job of describing how an enzyme accomplishes what it accomplishes. So the first modelóit's called the "lock and key model" it's also called the "Fischer model"óis shown here. There is your lock. There is your key. There is a perfect relationship between them, and it explains why enzymes only bind and act on specific molecules. Only certain keys will fit into the lock. So it's nice at explaining that. By the way, when you see this thing that says "ES complex" that's just the enzyme bound to the substrate before the reaction has occurred. So binding has to occur before the reaction occurs, and we call this thing the ES complex. Well, that's very nice about explaining specificity, but it doesn't do a very good job of telling us how it is that the enzyme catalyzes the reaction that it does. As a result of that, peopleóa man named Daniel Koshland, for exampleósaid that we're not thinking about this in the right way, and the reason that we're not thinking about it in the right way is this assumes a rigid enzyme. Enzymes really aren't rigid. You saw hemoglobin had very tiny shape changes, for example, that happened. It tells us that proteins, of which enzymes are included, proteins have flexibility, and that flexibility turns out to be very important for understanding how an enzyme works. So Daniel Koshland proposed this, and he proposed that, instead of a lock and key model for explaining enzymatic action, that an induced fit was a better descriptor. The induced fitóand this is not the best figure, but you get an idea hereónotice that here's the enzyme before it has bound to substrate. You'll see that there's something that resembles a, and there's something that resembles b, and there's something that resembles c. But it's not exactly what it ends up being over here. It's not exactly that. So what this tells us is, first of all, the enzyme is flexible. It has to be. But that the binding of the substrate actually induces the enzyme to change shape. It induces the enzyme to change shape. You think, "Well, how does that happen?" We could imagine that there's quite a few ways it might happen, even for something as simple as this. Imagine, if you will, that a fits a very well. When a binds to a over here, this causes a conformational change, not unlike the conformational change we saw with oxygen binding to hemoglobin. And now this starts getting to shape more like b, and changing of b makes it more like c, andóbang!óthe enzyme has bound to the substrate. So this induced fit is a fundamental difference from the Fischer lock and key model. The Fischer lock and key modelóand I'll explain the significance of that in a secondóbut the Fischer lock and key model says that enzymes bind to substrates, and then magic happens. It doesn't explain catalysis. On the other hand, the induced fit model tells us that the enzyme binds to the substrate, this causes a change in the enzyme, and this change in the enzyme may bring regions together that weren't together before, and this new structure favors catalysis. Very much like we saw in hemoglobin, where this new structure favored binding of additional oxygens. So the flexibility of the enzyme, and we'll see several examples of it this term, the flexibility of the enzyme is critical for how enzymes function. It turns out that when I talked the other day about the Kcat of enzymes that had a Kcat of a millionówe saw carbonic anhydrase had a Kcat of a millionóand I said there's no way that we'd get that with a chemical catalyst, we realize that the flexibility is what differs between an enzyme and a chemical catalyst, and that flexibility facilitates all of the properties of the enzyme. So the induced fit model does a very nice job of explaining how catalysis happens. The induced fit model, if we were to summarize it, says that not only does an enzyme change a substrate, but it says that a substrate also transiently changes the enzyme. That's a very, very key point in understanding how enzymes function. Not only does the enzyme change the substrate, but the substrate also transiently changes the enzyme. Yes, sir? Student: Could that explain how some enzymes can bind to an entire class of substrates... Kevin Ahern: Yes. Student: ...by catalyzing the activity at different rates for each one? Kevin Ahern: Yes, his question is a very good one. His question is, does that explain how some enzymes can bind different substrates and have different Kcats for those individual substrates? And the answer is, yes, it can be a factor, absolutely. It explains, as you could imagine, why some enzymes are more flexible, bind different substrates. They may have more flexibility of binding in that region. We'll... oh I'm sorry. Student: The change done to the enzyme would have to be less permanent, though, otherwise... Kevin Ahern: That's why I said "transient." Student: Oh, meaning... Kevin Ahern: It's transient, yeah. Okay? It's not a permanent change. If it's a permanent change, if we make a permanent change to an enzyme, I don't care what it is, the enzyme works one time. One time. So we don't want to make permanent changes because, again, by the definition of a catalyst, remember your definition of a catalyst from freshman chemistry is a catalyst speeds a reaction without itself being changed. If when I finish with this enzyme it has been changed, then I don't have a catalyst anymore. But the difference between an enzyme and a chemical catalyst is that transient change. That change can happen to an enzyme and actually enhance its ability to function, and that's why we see such an increase in activity. Question? Student: Yeah, just as a follow-up, if a enzyme generally is able to bind to a wider class of substrates... Kevin Ahern: Yep. Student: ... a larger, broader range, does that lack of specificity for a single target give it less catalytic activity? Kevin Ahern: So his question is, if an enzyme binds a bunch of substrates does that basically make it less functional or less effective as an enzyme? The answer is, as I'll talk about in a little bit, there's not a direct relationship between the binding of the affinity of the enzyme and the rate of the enzyme reaction. So there's not, no. Okay, you guys are thinking about this. I'm very pleased to see that. That's good. The induced fit model is a very interesting model as we think about it, and it really does explain how enzymes function differently from chemical catalysts. Well, at this point, I need to introduce and say a few words about the Michaelis-Menten model of explaining or studying enzymatic reactions. This is simply a way that we do our experiments. There are some reasons why we do experiments the way that we do to study enzymatic reactions. Let's imagine that I'm interested in studying carbonic anhydrase, and carbonic anhydrase catalyzes a reaction that goes very, very, very fast. Right? I'm interested in how fast this enzyme can make product. So I start with my enzyme. I start with my substrate. I let the reaction go and I start measuring how much product that there is. There's a problem with this, and the problem is that this becomes an issue when we think about equilibrium. Equilibrium happens when we have reactions going forwards and backwards at the same rate. Now, I'm interested in understanding how fast this enzyme is making product. I really don't want to see the backwards rate, right? Because that's going to make it look like my enzyme isn't going as fast as it possibly could go. So one of the considerations I have in doing my experiments is I want to measure what are called "initial velocities," Initial velocities. Initial velocities are done on a fairly short time scale, before the amount of product starts to accumulate. Once the amount of product starts to accumulate, I get way up here in this range, actually, up here in this range, once the product starts to accumulate, now the backwards reaction starts going and I'm going to see a less accurate measure of velocity. I'm really only interested in the conversion of substrate to product, not product back to substrate. So one of the considerations is that I should be working at initial velocity... fairly short time frames to do those measurements. There are a lot of considerations in the Michaelis-Menten and I'm not going to go through all of them, nor am I going to require you to do the derivation. There is some sophisticated math that we would have to do to derive Michaelis-Menten, and I'm not really interested in having you understand that. But there are some things that I want you to understand that arise from Michaelis-Menten. You've seen maximum velocity is one of those. Kcat derives from maximum velocity, and I'm going to illustrate another one to you in just a second. Now, what am I showing you on the screen? I haven't even told you this. Here's that V-versus-S that we did before. You'll notice that little zero right there. That zero means, initial velocity means measure that velocity very quickly. Don't let things accumulate. How would I do this experiment that you see on the screen right here? I think it's important for students to understand how we perform experiments in a lab. These points didn't just come out of the air, but, in fact, somebody had to do the work to do those. So how would I do that experiment? Well, when I do an experiment, I always want to have one variable, one thing that is changing and I keep everything else, as much as I can, constant, because I can't determine how the two variables are affecting an experiment. If I were to do this experiment, I want to generate my curve, I'm interested in velocity versus substrate concentration. Well, one of my variables is obviously going to be varying concentrations of substrate. Let's say I take 20 tubes. I put into those 20 tubes the same amount and the same concentration of buffer. I put into those 20 tubes the same amount and the same concentration of enzyme. I measure each reaction, ultimately, for the same amount of time, a fairly short period of time. But the difference is, in each tube I have a different concentration of substrate that I'm adding. Okay? Well, higher concentrations of substrate, way out here, are going to give higher velocities than low concentrations of substrate. Higher concentrations of substrate are going to give higher velocities. Yes? Student: That initial velocity, is that also going to be a max velocity? Is that the fastest it's going to be going at any point? Kevin Ahern: Well, maximum velocity is what I determine when I get the enzyme saturated with substrate out here. Initial velocity, every one of these is done at the same time point. So, no, not everyone is going to be maximum velocity, because that's way out up here. I'm trying to determine maximum velocity. So to determine maximum velocity, I have to do this experiment I've just told you. Clearly, if I'm way down here in terms of concentration, that is not going to be maximum velocity because, again, the analogy to the factory, where I've got this factory that doesn't have enough supplies to keep things going as fast as I want to keep the workers working. Okay, makes sense? Other questions there? I want to introduce, at this point, another parameter that we can learn by doing Michaelis-Menten kinetics. This parameter is an important one. It might seem at the surface like it's related to Vmax, but, in fact, it's not. This parameter that we're interested in, not only are we interested in how fast an enzyme works, but we're also interested in how much affinity the enzyme has for its substrate. How much affinity do you have for your significant other? Some of you have high affinity. Maybe, if you're not getting along real well, maybe you have low affinity. But affinity is a desire to, shall we say, grasp. [laughter] Why is that funny? I knew it would be, but why is that funny? It's always nice to think of real world examples for what are actually fairly abstract molecular concepts, and affinity is one of those that always generates giggles. So an enzyme's affinity for its substrate is very much like your affinity for your significant other. That's a real world example. Well, the measure of that affinity is called Km. It's called the "Michaelis constant," and you don't need to know that. Just Km is fine. Let's think about what Km is. Let's imagine I have an enzyme, and we'll come back to this graph in a second, so I'll explain the graph to you, what I'm telling you, in a second, let's imagine that I have an enzyme and it has a very high affinity for its substrate. That means when that enzyme is out there it's going to grab that substrate. Right? You see your significant other all the way across campus and you go racing over to grab the significant other. Right? Or maybe you have several significant others. I don't know if you have. [laughter] And that actually becomes important for the other consideration. Let's say I have an enzyme that doesn't have much affinity for its substrate. They don't really feel like running all the way across campus to go grasp the significant other. But, hey, there's one over here. Let me grab them. If I start flooding the enzyme, if I have an enzyme that has low affinity for its substrate, I have to flood it with significant others before it will start grasping. Okay? I have to flood it. Well, what does that mean? It means if I go to saturating amounts of substrate, way up here at Vmax, everything at saturating amounts is going to have substrate bound to it. So the amount of substrate it takes to get to Vmax doesn't really tell me anything, because everything that's saturated with gigantic amounts of substrate is going to have that. So if I want to measure the affinity, I don't want to be way up here in this region. I actually use something called the Vmax over 2. Now Vmax over 2, I don't actually have to have the saturating amount to be up here to get to that amount. Instead, what I have to have is enough substrate to get it to this point. If I do this for all enzymes, then I have a comparator for each one. That comparator turns out to be the Michaelis constant. Now, based on what I've just told you, does an enzyme that has high affinity for substrate have a high Km or a low Km? It has a low Km. It takes very little substrate to get to half maximum velocity. Something that has low affinity for substrate will have a high Km. It's important in science to be precise. One of the ways that students in this class learn this message the hard way is I'll ask them on an exam, "What is Km?" If I didn't tell you anything else, over half of you would say, "Km is Vmax over 2." Now, I will ask you, is that point the same as that point? No. Km is the substrate concentration that it takes to get an enzyme to Vmax over 2. They are not the same thing. That's like saying 60 miles per gallon is 30 miles per hour. It doesn't make any sense. It doesn't make any sense. So Km is a substrate concentration. Pound that into your heads. High Km, low affinity. Low Km, high affinity. Does that make sense? So Km is not the Vmax over 2. Now I know there'll probably be some questions at that point. Let me stop and ask if there are any questions. Was I that clear? Yes, question? Student: [unintelligible] Kevin Ahern: If an enzyme has multiple sites, multiple binding sites where it can catalyze a reaction, we study it in exactly the same way... exactly the same way. Good question, and I'll actually show you, when I talk about mechanism for that next week, that there are some interesting things that happen as a result of enzymes having multiple sites where we could actually change one site and that affects the whole enzyme. But let me save that for then. Wow, okay that's good. We're moving right along here. Everything I've shown you so far has been a very simple plot, and that simple plot has shown you velocity and how did I define velocity, again? Was it that big of a weekend, guys? Student: Yes. Student: Concentration of the product over time. Kevin Ahern: Concentration of product over time. So I plotted velocity versus substrate concentration. So everything I've shown you has been that, and you can see from this plot right here that, well, it's kind of hard to tell exactly where Vmax is, right? I mean, I have to kind of make a guess about where this is going to eventually run in up here. So because I have to make a guess in this, I have to make a guess in thisóbecause that's just half of itówhich means that I have to make, ultimately, a guess in this. I'd like to have a way of determining these values that are more precise and easier to do. So to do this, people have come up with other ways of plotting the same data... other ways of plotting the same data. In that example I gave you, I had 20 tubes. I had 20 concentrations and each one had a different velocity, which meant I had 20 different velocities that were there. Right? What if I took and I simply inverted the numerical value of each one of those? Instead of velocity, I take 1 over the velocity. If the velocity is 4 micromolar per second, then 1 over it would be 1/4 micromolar per second. That would correspond to a substrate concentration that might have been 2 micromolar, so that when I invert that, it becomes 1/2. I take and I invert every value and then plot the inverted values. If I do that, I create a different kind of a plot. It's the same information. I don't have to do the experiment any differently. I'm just inverting the values of these, and I create something called a Lineweaver-Burk plot. You can see from this that there's some real advantages to plotting the data in this way. I'll point out the advantages and then I'll point out a couple of other things to you. The advantages are that the place where the line intersects the y-axis is known as 1 over Vmax. I have a much more precise way of determining where Vmax is and I can see this thing is getting ready to jump, isn't it? Hehe, Alright. [class laughing] I feel like I accomplished something there, that I beat it! I beat the system here! Isn't that cool? So the y-intercept is 1 over Vmax. I now have a precise way of saying what that is. I take whatever this value is and I say, all right, if I want to know Vmax, then whatever this value is, I invert it. Bang, I've got Vmax. The x-intercept has a value of minus 1 over Km. Bingo! I take whatever this value is, I take minus 1 over it, and I get Km. Now, these give me very nice and simple ways of making these determinations. Don't worry about the slope. You can memorize that if you want, but I think if you know the intercepts you're in much better shape. The beauty of this is I get a straight line. Notice that all of the points that I did before are all in this quadrant. Why is that? Student: They're all positive. Kevin Ahern: They're all positive. Okay? I can't have a negative substrate concentration. I can only draw a theoretical line to that. So by drawing the line, after I align my points, to that, I get that value. So Lineweaver-Burk plots are very, very useful because these two intercepts, this one right here and this one right here, are consistent with the line that we see there, and these two intercepts give me very, very valuable information for me about an enzyme and how the enzyme works. We're just sailing through stuff. Other comments or questions? Student: You said that the Vmax was just a guess? So [unintelligible] on this plot? Kevin Ahern: Well, it's really not, because this is an extrapolation back from here. I can actually, if I do a least-fit square of this line, I get a precise value there. So, in essence, no, it's not. Yes? Student: So, in essence, this is more precise than guessing you would do... Kevin Ahern: Yeah, yeah. And this is very commonly done with data, we use a Lineweaver-Burk plot. The other plot doesn't have a name. It's called a V-versus-S. That's what it's called. Don't forget that that's concentration of S. So those little brackets mean concentration of S. In fact, that's what it means right here, 1 over the concentration of S. Brackets always refer to concentration. And, again, we're plotting 1 over the initial velocity. Student: The concentration for all of these is in molarity? Kevin Ahern: Concentration can be in whatever you want it to be, but it's generally molarity, yes. Okay, good. Now let's look at a variety of enzymes. We see that enzymes can have very different Km values, very, very different Km values. Here's the Km value for chymotrypsin in micromolar, again, micromolar is a measure of concentrationó5,000. So high Km means low affinity, so this guy has relatively low affinity. Whereas, here's lysozyme which has very high affinity for its substrate. Here's an arginine-tRNA synthetase, very high affinity for its substrate. Like the question that the gentleman in front asked earlier about what if an enzyme has multiple substrates that it can bind, it's not restricted to just binding one, there are different Km values that will be associated with those, different Km values. How could I explain that with an induced fit model? Well, we could imagine that the very first thing that makes contact could actually be a limiting factor, right? If that very first a that bound to a was more like an a-prime and it didn't exactly fit, it might have less likelihood of getting in there and making those other changes necessary for the enzyme to have the proper configuration for a reaction. So the induced fit model still fits multiple substrate interaction and how an enzyme works with that. Look at carbonic anhydrase here. It has a relatively high Km. But, boy! Once we flood that sucker with substrate, it takes off and goes. Now, carbonic anhydrase is an odd enzyme. I'm going to talk about it in just a minute, because it has a very interesting property associated with it that I'll tell you about. So we don't see a direction relationship between Km values and Kcat values. They're not the same. There's not a relationship, as such. We see fluctuation with those. Here are the turnover numbers. This turnover number 600,000, we will see variation in turnover numbers a little bit based upon different conditions that we might use. If I did it a different pH, I'd get a different Kcat value. But as long as we specify the conditions, we will get the same Kcat value from one batch of enzyme to the next. We see quite a wide range of velocities. Here's lysozyme. Lysozyme had a very low Km, but it doesn't have much turnover number. There's not a relationship between these two quantities. Multiple substrates. Well, if we talk about an enzyme that'll bind to multiple things, here's chymotrypsin. Chymotrypsin will act on different amino acids. Remember chymotrypsin is a protease, so it's recognizing and binding to specific amino acids in a protein and then cutting their peptide bond. Here are the different things that it will bind to and cut, and here is a measure of the Kcat over the Km. We see very wide range of these two. I'm going to talk about the significance of Kcat over Km in just a second, but we see that there's quite a fluctuation. The enzyme behaves differently with some substrates than others. Something like phenylalanine it really likes. Something like glycine it doesn't like so much. There's a millionfold difference between these two. Let's think about enzymes, in general. We've got parameters now for how fast an enzyme works, and we've got a parameter for how much grasping affinity the enzyme has for its substrate. If we were to define a perfect enzyme, what qualities would it have? In terms of Kcat, would it have a high Kcat or a low Kcat? High, right? We like efficiency. We like speed. So it would have a high Kcat. How about Km? Students: Low. Kevin Ahern: Low. So a perfect enzyme would have a high Kcat. It works very fast, but it doesn't take very much substrate to get to Vmax over 2. It likes its substrate. We can get this sucker going pretty readily. That's what they're plotting in this thing here. They're plotting the value of Kcat divided by Km. The most efficient enzymes will have a very high Kcat over Km, because the Kcat will be high and the Km will be low. Alright? The Kcat will be high and the Km will be low. I'm going to tell you something that's going to surprise you. There's very few things in nature that we describe as "perfect" but there's a certain category of enzymes that clearly are. Perfect enzymes. What is a perfect enzyme? Well, we talked about the ideal enzyme. Here's the ideal qualities. It's got a high Kcat and it's got a low Km value. But a perfect enzyme, when we study all the enzymes that are out there, we discover that there's a group of enzymes that get a very high Kcat over Km and they sort of hit a brick wall. They don't get that value much beyond a certain point. It doesn't go infinitely high. They get to a certain point and it gets no higher. And you think, "Why is that?" The reason that happens is that when an enzyme is perfect, two things occur. One, any changes to the enzyme will result in a less efficient enzyme. It'll have a lower Kcat over Km. So it has evolved to the point where it's going as rapidly as it can go. Well, what limits the rapidity of an enzyme? It turns out that what limits the rapidity of an enzyme is how fast a substrate can diffuse into the active site. We didn't think about this before. Think of carbonic anhydrase. If it's catalyzing a million molecules of product per second, that means a million molecules of substrate have got to bind in there per second. That's why it's really hard to think about how an enzyme works, compared to something in the macroscopic world. That's how fast that that substrate has to diffuse in there and get that reaction going. For perfect enzymesóthis is importantófor perfect enzymes, the limiting factor is the rate of diffusion of the substrate in water. If they could diffuse faster, that value would get higher. But the reason that they all get to that same point and have basically the same Kcat over Km value, is because water, things can't diffuse any faster in it than that. Student: Is that just for perfect enzymes or all enzymes? Kevin Ahern: That's for a perfect enzyme. These are what we call "diffusion controlled enzymes." You'll see their values do fluctuate a bit, but basically 10 to the 8th is about where they get stuck. The substrate can't diffuse any faster in water. And, yes, you could imagine some substrates will have different diffusion rates in water than others will, which is why we see a little bit of fluctuation, but basically they're all in that range. Now, these guys are perfect. I say that they're perfect because if we mutate anything in these enzymes we get something less efficient. The selection process of evolution is remarkable in terms of what it has produced. The common question I get at this point is, "How come all enzymes aren't perfect? "If we have an evolutionary process driving this, "why aren't all enzymes perfect?" And I answer that question very simply, "Why doesn't everybody have a Maserati "to drive to Fred Meyer? "And why don't they go there at 110 miles an hour?" There would be consequences, right? Enzymes that are capable of doing things a million per second are dangerous to a cell. Because if a cell makes too much of product and it can't be controlled, all of a sudden, "Oh my god, I made too much carbonic acid." That's a problem. In the case of carbonic acid, the cell kicks it out and it's fine. But there are other things that we could imagine. Let's say it's an enzyme that breaks down our glycogen. We'll talk about that later this term. It breaks down our glycogen. Our body needs sugar. It's nice to break down glycogen quickly. But I want to have it slow enough that I can control it so it doesn't overdo the breaking down of glycogen. Otherwise, I'm going to waste a lot of energy. In the case of the Maserati going to Fred Meyer, the consequences are you run over people. In the case of the cell, we're running over the needs of the cell. So why do we have any perfect enzymes? There are several reasons for this, as well. We'll talk about a couple of them later in the term, but one of the reasons that enzymes have evolved to being perfect is because the speed of the reaction actually prevents an unstable reaction from occurring. Some of the intermediates are unstable, so if we can make that reaction go blindingly fast, the likelihood we have things that we don't want get lower. There's other reasons, as well, but that's one of the reasons we have perfect enzymes. That's pretty cool! There's not a large number of them, but there's a decent number of enzymes that are perfect. Questions about that? I do have a song. We're maybe a little ahead for the song, but it's kind of fun and I think we've mostly covered this stuff, so let's do it. You'll see something that I'll talk about next time. The song is to the tune of an old song from the '60s called "Downtown." Anybody know the song? So with this song, when we get to the point with the "downtown" part, I want everybody to just jump up and throw your hands in the air and say the word "enzymes." Okay? Let's start. [Singing "Enzymes"] Lyrics: Reactions alone could starve your cells to the bone. Thank God we all produce... enzymes! Units arrange to make the chemicals change, because you always use... enzymes! Sometimes mechanisms run like they are at the races. Witness the Kcat of the carbonic anhydrases. How do they work? Inside of the active site it just grabs onto a substrate and squeezes it tight in an ENZYME! CAT-al-y-sis in an ENZYME! V versus S in an ENZYME! All of this working for you... enzyme, enzyme. Energy peaks are what an enzyme defeats in its catalysis. Enzymes! Transition state is what an enzyme does great, and you should all know this. Enzymes! Catalytic action won't run wildódon't get hysteric. Cells can throttle pathways with an enzyme allosteric. You know it's true. Kevin Ahern: Not yet, but you will. Lyrics: Well, when an effector fits, it will just rearrange all the subunits, inside an ENZYME! Flipping from R to T, ENZYME! Slow catalytically, ENZYME! No change in Delta G... enzyme, enzyme. You should relax when seeking out the Vmax though there are many steps. Enzymes! Lineweaver-Burk can save a scientist work, with just two intercepts. Enzymes! Plotting all the data from kinetic exploration, lets you match a line into a best-fitting equation. Here's what you do. Both axes are inverted then you can determine Vmax and establish Km for your ENZYMES! Sterically holding tight, ENZYMES! Substrates positioned right, ENZYMES! Inside the active site, enzymes, enzymes. Alright. [applause] Thank you. Student: All right, now you have to tell me what musical "Downtown" is from because it's... Kevin Ahern: What musical, I don't know. I don't think it's from a musical. Student: [unintelligible] Kevin Ahern: Is it? Student: [unintelligible] Kevin Ahern: I'm not sure it's originally from a musical, but... Student: Okay, I feel like there's a musical it's in and I can't... Kevin Ahern: Maybe, I don't know. Student: Hi, is there a way to make up a recitation? Kevin Ahern: No, no. Missing one recitation won't kill you. So hang in there. Student: I'm two hours away. Kevin Ahern: Oh, that's a bummer. Yeah, sorry about that. Student: Hi, I have a question. So, I saw the table [unintelligible] side chains. Does that come from the value of the Kcat over Km? Kevin Ahern: No, the enzyme shape does that. Student: Oh, really? Kevin Ahern: Yeah. Student: Okay. Kevin Ahern: Uh-huh. Excuse me, guys. Sorry. Squeeze in here. Student: When would be a good time to talk to you aboutó [END]
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Medical_Lectures
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24_Biochemistry_Gluconeogenesis_Lecture_for_Kevin_Aherns_BB_450550.txt
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[Ahern laughing] Student: Do it. Ahern: Do it. [laughing] I'm guessing if I gave everybody who came to class an A today then they'd never come to class again. That's just my hunch. Which would kind of be a self-defeating thing, right? So... Student: Not necessarily. Ahern: Maybe we would, right? What she should say is, "Ahern, you're a scientist, "let's do the experiment and fight out," right? Student: Exactly. Ahern: Well, we can't do that. Student: You can give us extra credit. Ahern: I could give you extra credit. There's a lot of things I could do. I could give you money. [class laughing] We could go have beer. We could have pizza. Student: How many of us would not get in trouble for you buying beer. Some of us are still under age, so... Student: Yeah, you could get in a lot of trouble. Ahern: No, actually the way I do that is I go to a place where they can serve people underage and you have to show an ID so it's not my responsibility. A lot of energy. I hope everybody's got a big Thanksgiving planned. Wild plans? Student: Family. Ahern: Oh. Family, huh? Like I said, if any of you are in town and would like to come over, give us a holler and I'll let you know where we're going to live and everything, but if you'd like to come over, we've got plenty of turkey and other things. And no, I won't get you drunk. But we'll have a good time. Today we're going to have a good time because we're going to be thinking of the making of glucose. I know that for many of you, that's been something you've dreamed of doing and you're going to get that dream today. Happy days are here again. Glucose synthesis is an interesting process. The phenomena of course is known as gluconeogenesis and it is a pathway that is very similar, very similar to glycolysis. Very similar. It's very similar to the reverse of glycolysis. However, there are important differences and specifically, there are two reactions. I'm sorry, specifically that there are 3 reactions. There are 3 reactions in gluconeogenesis, in glycolysis that are replaced by 4 reactions in gluconeogenesis. So gluconeogenesis has 11 steps, glycolysis had 10. One of the steps takes two steps to get around. So it's 2 step. If you learned glycolysis, gluconeogenesis for 8 of the steps, Let's get that right for 7 of the steps. I can't get my head right today. for 7 of the steps is identical to glycolysis except for in the reverse. Same enzymes, same intermediates going to the opposite direction. Three of the steps that are in glycolysis as I said are replaced by 4 steps. So let's take a quick look at that. Before I take a look at that, I'll show you something your book is distracted by and that is this process here, which is the making of glucose from glycerol. Why do we care about the making of glucose from glycerol. One of the reasons we care about the making of glucose from glycerol is glycerol is a byproduct of fat metabolism and so it turns out that the only portion of the fat molecule that can be converted directly into glucose is the glycerol. We don't convert fat into glucose for the most part, with the exception of the glycerol. I just show you this, I'm not going to go through and expect you to memorize these are anything, but just show you that glycerol is a 3-carbon molecule. It gets made in a couple of reactions into an intermediate in glycolysis, dihydroxyacetone phosphate. And of course, once it's dihydroxyacetone phosphate, we can now do the upwards pathway, going into making glucose via gluconeogenesis. And we see this, this is a phenomena you've seen before. We saw how other sugars entered glycolysis and gotten broken down by being converted into glycolysis intermediates. We saw, for example, fructose got converted into fructose-6-phosphate and then got metabolized as an intermediate in glycolysis. In this case, we see glycerol being converted into an intermediate in glycolysis or gluconeogenesis, it can actually go either way, and be made into either glucose or ultimately into pyruvate. Let's focus on gluconeogenesis since that's our main topic of the day. You'll notice in looking at the screen that we oriented very much like we oriented glycolysis except that we're going upwards in gluconeogenesis where as we were going downwards in glycolysis. So we start at the bottom and the place where we will start gluconeogenesis is actually pyruvate. But again, we remember that all these designations about where something starts and stops is really arbitrary. We could just as easily start it at lactate for some types of metabolism. We can start at amino acids as well, but we're going to start right here at pyruvate. So starting at pyruvate, and that's a good place to start because that's where we finished glycolysis, starting with pyruvate, cells can convert pyruvate into glucose. Well, not surprisingly, if pyruvate is a 3-carbon molecule and we want to make a 6-carbon glucose, we need to have two pyruvates to start everything. We're going to start with 2 of everything and eventually they're going to combine into 1 as we get higher up in the pathway. What we discover in gluconeogenesis is the first instance that we see of a phenomenon I call sequestration meaning we're sequestering something. All of glycolysis occurs in the cytoplasm of the cell. All of the enzymes of glycolysis are found in the cytoplasm of the cell. In the case of gluconeogenesis, we see 2 enzymes that are not found in the cytoplasm of the cell. These are sequestered into other organelles in the cell and I'll show you those. They actually end up being the first and the last enzyme in the pathway. All the other enzymes between the first and the last are all found in the cytoplasm. Let's look at what's happening in making glucose from pyruvate. If we recall in glycolysis, in going from PEP to pyruvate, I said that was the big bang. I said that was a reaction that was extraordinarily energetic. It had a large delta-G-zero prime. And as a consequence, that, you might imagine going in the reverse direction, would be an enormous energy barrier to overcome. And in fact, that's exactly what it is. It's because of this enormous energy barrier that cells can't go directly back from making pyruvate to PEP in one step. They have to do a two step around it. And the two steps around it are these two enzymes here. Pyruvate carboxylase and phosphoenolpyruvate carboxykinase, which you are more than welcome to abbreviate as PEPCK. Let's talk about the first one first, pyruvate carboxylase is an enzyme that is found in the mitochondrion of cells. It's not found in the cytoplasm. The very first reaction of gluconeogenesis occurs in the mitochondrion, not the cytoplasm. In this reaction, as you can see on the screen, carbon dioxide in the form of bicarbonate and ATP are used to convert pyruvate into oxaloacetate. You can see the structure here. Here's a 3-carbon, here's a 4-carbon over here. We've put an additional carboxyl group onto the end of pyruvate. The carboxyl group going right here. We can see the new carboxyl group on the right side. Here it was what pyruvate looked like over here. So what we did is we took this methyl group and we added a carboxyl group to it. That takes energy to put that on there. It makes a 4-carbon intermediate and that 4-carbon intermediate you're going to hear a lot about next term because oxaloacetate is one of those molecules that appears in so many metabolic pathways. It's a very, very important molecule. It's important in amino acid metabolism. It's important in the citric acid cycle. And it's also important as you can see here in the synthesis of glucose. This is an energy requiring reaction so if we started with 2 molecules of pyruvate, it's going to take 2 molecules of ATP and 2 molecules of bicarbonate to make 2 molecules of oxaloacetate. We haven't gotten to PEP yet because even with all that energy that we've put in, we've made a 4-carbon intermediate and we have to convert that 4-carbon intermediate into phosphoenolpyruvate or PEP. To do that, the oxaloacetate which was made in the mitochondrion has to be moved out of the mitochondrion and into the cytoplasm. Next term we'll talk about how molecules move across an organelle. But it turns out there are specific proteins that will shuttle a molecule across a membrane. There are specific proteins that will transport oxaloacetate out into the cytoplasm. When it's out in the cytoplasm, oxaloacetate can be acted upon by this second enzyme that's unique. By the way, all the unique enzymes are shown in red on here. By the unique enzyme phosphoenolpyruvate carboxykinase, or PEPCK. Notice that it also takes energy and that energy in this case comes from GTP. So GTP is just like ATP, a high energy source. GTP is used in some places in the cell for energy. The most common place we actually see GTP used is in the synthesis of proteins because all proteins are made using GTP, not ATP. We'll talk about that next term. If we have 2 molecules of oxaloacetate and we want to make 2 molecules of PEP, it takes us 2 molecules of GTP and this enzyme to accomplish that. In the process, a CO2 is released. Look what we did. We put a CO2 on the form of bicarbonate and we've released the CO2 up here, so no net gain of carbons have occurred. We've done a molecular rearrangement and in the process of doing that molecular rearrangement, we've also put a phosphate onto the molecule, creating that very high energy PEP molecule. As I said, this reaction occurs out in the cytoplasm. To go from pyruvate to PEP in terms of synthesizing glucose, we had to use 2 high energy phosphates for each molecule for a total of 4. So in order to go from here 2 pyruvates to 2 PEPs, we need 4 triphosphates. 2 ATPs and 2 GTPs. From an energy perspective, GTP is exactly equivalent to ATP. There's no difference. Going down, if you recall, when we went from PEP to pyruvate, we only got a total of 1 ATP for each one, or a total of 2 ATPs. So we can see that building molecules in anabolic pathways takes more energy than we get out in catabolic pathways. That's not surprising. We're thinking okay, well we're going. Yes? Student: So how does this prevent PEP from immediately switching back to pyruvate? Professor: She's reading my mind. My next point is, her question was how does the cell keep PEP from just going back to pyruvate? Well, that's a very, very important consideration because we know that if that molecule has the opportunity to go back to pyruvate in the presence of pyruvate kinase, it's going to do it. That's the big bang reaction, right? We see now why we have to regulate that pyruvate kinase, because if we want this thing to go upwards, we darn sure don't want to be turning this right back around and making pyruvate because we will have destroyed our purpose and we will have wasted energy and we would have gotten nowhere. This business of wasting energy and getting nowhere, where we're making something and breaking it down, going in sort of a circle is something that we'll talk about later. But it is a non-productive metabolic process that can occur in cells. We don't want that to happen. So for that reason, we want to turn off pyruvate kinase when we're turning on this process. Similarly, we want to turn these off when we are turning on the pyruvate kinase. Everybody follow? Student: Can you say that again? Ahern: Okay, well, to kind of go through that again, so basically we want to turn off the enzymes of gluconeogenesis when we have on the enzymes that's catalyzing those big reactions of glycolysis. In this case, pyruvate kinase. Conversely, we want to turn off pyruvate kinase, or turn on pyruvate kinase when we turn these guys off. So we want to have on one vs. the other. There's a name for what occurs if we put both of these on at the same time. Let's imagine we have a situation in a cell where these enzymes were active and so was pyruvate kinase. The cell would turn pyruvate into PEP, pyruvate kinase would turn it back into pyruvate and we would go around, and around, and around, and around. That phenomenon is known as a futile cycle. F-U-T-I-L-E cycle. It's futile because the cell is getting nothing out of it. It's burning 4 triphosphates going up and only getting 2 back on the way down. So each time it turns the cycle, it's losing 2 triphosphates. It's also producing one thing. What's the one thing it's producing? No, there's no net carbon dioxide because it goes in and it goes out. It's heat. Just like we talked about exercising, heat's generated. This process will generate heat and it's going to be totally wasted. Totally wasted. So we don't want to be running these two processes at the same time. For the moment, we will say yes, we've got the pyruvate kinase turned off, so PEP starts to accumulate. When PEP starts to accumulate, now the reverse reaction of glycolysis is favored, catalyzed enolase and we convert PEP into 2-phosphoglycerate. Next, we convert 2-phosphoglycerate into 3-phosphoglycerate. And next we convert 3-phosphoglycerate into 1,3-bisphosphoglycerate and we remember now that in going from here upwards, we have to use ATP again. So now we've got to put 2 more ATPs into the process. We get to 1,3-bisphosphoglycerate and we want to go back to glyceraldehyde 3-phosphate, now we have to reduce that molecule. We have to use electrons from NADH to convert the 1,3-bisphosphoglycerate into glycoaldehyde 3-phosphate. That involves loss of a phosphate as well. Now we've got a two molecules of glycolaldehyde 3-phosphate. Our triosephosphate isomerase converts one of them to DHAP, leaves the other one alone, they combine together to make fructose 1,6-bisphosphate. We're climbing the ladder. You see in each case all that we're doing is we're reversing every blue enzyme reaction. We're just reversing. We're going upwards instead of downwards. How do we do that? We do it by increasing the concentration of these from the bottom filling upwards. When we get to fructose 1,6-bisphosphate, we have another consideration. The other consideration is that if you recall during the discussion of the glycolysis pathway, I said that PFK catalyzed a reaction that released a lot of energy. Why did it release a lot of energy? I said if we did a reaction with just phosphate, it wasn't very favorable. But we used something to make this reaction favorable. What was it? ATP. This reaction became energetically favorable in the glycolysis direction going down because we put ATP energy in. One thing that we could do is we could say okay, well let's reverse that reaction and we'll remake that ATP. That would be tough for us to do because A, the reaction is very favorable energetically going down. That doesn't make sense to try to do so instead we do a different reaction. So instead of trying to remake that ATP, we use a different reaction, and we use consequently a different enzyme. The enzyme that we use to catalyze the conversion of fructose 1,6-bisphosphate into fructose 6-phosphate is this enzyme known as fructose 1,6-bisphosphatase. Now you see these names start sounding like the intermediates. I'm going to help you on this one. We're going to call this guy FBPase-1. FBPase-1. Alright, that name will sound different than fructose 1,6-bisphosphatase. We're going to call the enzyme by a different name and you'll see why later why I want to call that enzyme FBPase-1. And now what we're doing is instead of remaking ATP by a reversal of the reaction, we're simply clipping off a phosphate. It turns out that's energetically favorable to clip off a phosphate. Why? Remember phosphate bonds a higher energy and if we just simply clip it off, we make that upwards reaction become favorable. It's a very cool trick that the cell is doing to make fructose 6-phosphate from fructose 1,6-bisphosphate. Questions about that? Yes, sir? Student: Is there a time when the body chooses to run futile cycles to make heat? Ahern: Very good question. His question is are there times the body runs futile cycles to make heat. As a matter of fact it turns out there are. Not this reaction, but other reactions I'll talk about next term. And one's a very important consideration in something in our body called brown fat. It is a way of helping to up the temperature. It's not this reaction, but another reaction that's done in a futile sense. Very good question, yeah. Other questions? We're getting near the end. We're at fructose 6-phosphate. We need to convert that back to glucose 6-phosphate. That's simply a reversal of the reaction of glycolysis. Again, we use the phosphoglucose isomerase to make that isomerasation and we're at glucose 6-phosphate. At glucose 6-phosphate, we have exactly the same problem that we had with converting fructose 1,6-bisphosphate into fructose 6-phosphate. If we try to simply reverse the glycolysis reaction, we would have to make ATP. That would be an energetically unfavorable reaction. It wouldn't make much sense for us to do. Instead, cells use a different enzyme to catalyze a different reaction. The reaction that they catalyze is again parallel to the one catalyzed by FBPase-1 and that is we simply clip the phosphate off of this guy to make 3-glucose. This last enzyme is found only in the endoplasmic reticulum of cells. It's only found in the endoplasmic reticulum of cells. Now if we look at this, what we see is, here's the glucose 6-phosphatase. This is what it looks like in the membrane of the endoplasmic reticulum. This glucose 6-phosphatase, you can see, is embedded in the membrane of the endoplasmic reticulum and in order for a glucose 6-phosphate to be converted, it must be moved first into the endoplasmic reticulum and here is another one of those proteins that does the transport of specific molecules. In this case, it's moving glucose 6-phosphate into the endoplasmic reticulum. There it interacts with the enzyme, gets its phosphate clipped off, and then both of them are kicked out into the cytoplasm. So as a result of that, cells now have made a functional glucose molecule starting with 2 pyruvates and they're left with one glucose. In the process of making this glucose, they have required six triphosphates, four ATPs, and two GTPs. They've also required two NADHs. And obviously two pyruvates to start the process. It takes more energy to make a glucose than we get out of glucose when we burn it. That's why we have to eat. If we relied only on our energy from that which we made and then broke down and made and then broken down, we would run out of energy. We have to eat to make up that deficit of energy. Connie? Student: So you need 2 NADHs, 4 GTPs, 2 ATPs, and 2 pyruvates? Ahern: No, you need 2 NADHs, 2 GTPs, 4 ATPs, and 2 pyruvates. So there's only 2 GTPs, that's the reaction of PEPCK. We have made that. I want to tell you a little something about gluconeogenesis that's important. That is gluconeogenesis is not found in every cell of our body. In fact, it's fairly carefully sequestered again as it were. Not now where in the cell, but actually where in the body. So the primary organs that we have in our body that make glucose are our liver, part of our kidney. That's the 2 primary places that we make glucose. So muscle cells for example do not make glucose. Muscle cells are really good at burning glucose. They're not designed to make glucose. That means that when we're running and we're exercising very heavily, we have to have a way of getting that glucose that's made in the kidney and more importantly in the liver into our muscles. That's where our blood stream is very important. That's why our heart starts beating faster when we start exercising more heavily is to carry more nutrients in the form of oxygen, in the form of glucose, and in the form of carrying away carbon dioxide. So all those things are important when we're exercising. That's one of the reason our blood flow increases as a result of that. Glucose turns out to be a wonderful compound for this purpose because glucose is very soluble in water. It can move in the bloodstream very easily because it's an aqueous environment. The liver dumps it into the bloodstream and poof, it's off to its target tissues in seconds. It gets there very, very quickly. So glucose is very, very useful for that. As we will see next term, fat is not so good for that quick energy because A, fat is not water soluble, B, fat has to be packaged up into lipoprotein complexes that have to made to be able to be soluble in a water aqueous environment. That's a broad view of gluconeogenesis. Now gluconeogenesis as I said, if you know glycolysis, you basically know gluconeogenesis because gluconeogenesis uses 7 of the enzymes that are the reversal of those in glycolysis and yes, you should know the 4 enzymes of gluconeogenesis that are different from those other enzymes in glycolysis. I'm not asking you to know additional structures though. I'm not asking you to know additional structures. Student: Does glucose 6 - phosphatase have an acronym? Ahern: Does glucose 6 - phosphatase have an acronym? If you want to call it G6Pase, you may. The other thing I noticed I didn't mention here and I'll mention it very briefly is the enzyme pyruvate carboxylase. That was the first enzyme that I talked about. That was found in the mitochondrion. We'll see next term why that's kind of an important thing. It catalyzes the reaction that you see on the screen. There's the oxaloacetate molecule, there's the ATP, the carbon dioxide, actually in the form of bicarbonate, carbon dioxide, that's all the same thing. The enzyme is one that uses a coenzyme. I haven't really talked about coenzymes yet and I want to say just a brief thing about them. Coenzymes are molecules that enzymes use to help catalyze a reaction. They're a non-amino acid that an enzyme uses to help it catalyze a reaction. That's what a coenzyme is. The coenzyme that pyruvate carboxylase uses is one you'll see commonly for reactions that involve an addition of a carbon dioxide to something. The coenzyme it uses it known as biotin. And biotin is really great at grabbing onto carbon dioxide and getting it to the enzyme to do something with. That's what biotin does. Whenever you see the name carboxylase in an enzyme name, it tells you A, that it's putting carbon dioxide onto something, and as a consequence of that, it almost always uses biotin to help it do that. It turns out that the carbon dioxide is carried at this end of the molecule out here. The lysine is the place where it attaches to the protein. So the protein has a lysine side chain, the biotin gets attached and out here there's a carbon dioxide that this biotin will carry to the active site to the enzyme so the enzyme can use that carbon dioxide. PEPCK, just a brief word about that, there's the reaction that it catalyzes. PEPCK is one of those enzymes, although your book shows some allosteric effectors, it's really not very regulated allosterically. A much more important regulation of this enzyme is control of where it's synthesized. So for example, my muscle cells are not going to make PEPCK because they're not going to go through gluconeogenesis. There's no reason for them to have and use that enzyme. My liver cells on the other hand, which do go through the reactions of gluconeogenesis, uh oh. [laughing] My liver cells, which do have the reactions of gluconeogenesis will in fact make PEPCK. What I just introduced for you, you probably didn't realize, was a third mechanism of controlling enzyme. We talked about 3 earlier in the term. One was allosteric control. 2nd was covalent modification. Now the third is whether or not an enzyme is made. If the enzyme is not made, that's the ultimate control. So PEPCK is regulated primarily by whether or not a cell makes it. That of course involves control of transcription and/or translation, and we'll talk about that next term. Questions about that? You guys look like you're asleep today. Why is everybody smirking? Yes, sir. Student: So when you eat like a meal that's really full of fat and you feel super lethargic, is it just your body trying to put all the energy into breaking it down? Ahern: So when you eat a meal that's super high in fat and you feel how did you say? Student: Lethargic. Ahern: Lethargic. Is that because your body is putting all its attention into breaking down that fat? Partly. One of the things that happens with eating a meal of anything, even if it's not a big meal of fat, is that your digestive system diverts a lot of blood supply to it to help carry things away. So instead of more blood being available to your muscles and brain and so on and so forth, your digestive system is kind of taking over. As a consequence, there's less oxygen for you to think and there's less oxygen for your muscles and so forth to do things. So it's a natural response of your body with that lower supply that you're just not going to feel like going out and doing stuff. Student: Tomorrow, when we all eat ungodly amounts of food... Ahern: When we eat I believe you said ungodly amounts tomorrow, is that going to be a factor? It is going to be a factor. One of the things some people say is a factor with Thanksgiving, if you eat a lot of turkey, the claim is that turkey is full of tryptophan and tryptophan can be converted into molecules that help you to sleep. That's a little controversial so whether that is true or not I don't know. But I'm willing to take the risk. [class laughing] Other questions? Would you guys like to sing a song? Alright, let's sing a song. I've got two songs. Maybe we should sing, let's sing the short one first. To do this one, I have to get you in the right frame of mind, okay? The right frame of mind goes as follows. There was a song that was written back in the 1930s. The 1930s was the Depression. And everybody was very upset, kind of like you guys will be after the lecture is over. Oh damn, I don't get to hear anymore biochemistry. They're very upset, they're very depressed. They needed something to build them up, right? They needed something to make them feel better. America's songwriters created some really amazing songs at that time. I'm going to get you started on one of them. You'll see why I get you started on this song in just a second. The song is called "Happy Days Are Here Again." Does anybody know this song? We're going to start to sing this song together. Lyrics: Happy days are here again The skies above are clear again. Let us sing a song of cheer again. Happy days are here again. Aren't you happier now? Happy days are here again. The skies above are clear again. Let us sing a song of cheer again. Happy days are here again. One more time! Happy days are here again. The skies above are clear again. Let us sing a song of cheer again. Happy days are here again. Next verse. Crappy days are here again. The sky above's not clear again And the sun has disappeared again. Crappy days are here again. Rain is falling from the sky. I wish I knew the reason why. Guess I'll have to wait until July For the weather to be dry. I do not mean to harangue. Since rain provides yin and yang. Because the flowers every one Love moisture followed by the sun. Let's stay happy til the rain is done. In Corvallis, Oregon. Ahern: Now don't you feel better? [class laughs] I know you do. Pretty much, yeah. This time of year it is kind of relevant i think. I have another one but we'll save it until I tell you about a couple other things. I'm actually going to go easy on you guys today because you came here on a tough day. Thanksgiving's tomorrow and everybody else is leaving down and why didn't he give a pop quiz so that we got extra credit? You know, and why didn't he give us those As that he talked about and so forth? I'm not going to go off the deep end, I'm going to save that for Monday. Going off the deep end is something I can do. And I figure why should we do that now? Let's do it Monday. I'm going to tell you about a cycle that you'll find very interesting and very easy to understand. It's called the Cori cycle. I'm going to skip this and come back and talk about this on Monday. But the Cori cycle is what I want to finish with and then we'll sing one more song. So the Cori cycle is an important cycle that was discovered by a husband and wife team named Cori. It turns out to be of critical importance in our body. The Cori cycle is a way for our body to handle very diverse sets of exercise situations. So let's imagine, forget the screen for a moment, let's just imagine that I am out on my morning jog which I haven't be able to do all week because it's been raining and I've been sick. But I'm out for my morning jog, which I'll be out tomorrow morning. Anybody who wants to run, come with me. And out running, and I will take off from my house and go for a ways. My body will fairly quickly recognize that my muscles really want glucose because my muscles burn glucose to get pyruvate, ultimately get ATP and all kinds of things from that. My muscles don't have a tremendous amount of glucose in them so my liver wakes up and starts producing glucose. That epinephrine hormone that we talked about, one of the things it does is it stimulates the release of glucose by the liver. So my liver takes that glucose and it dumps it into the bloodstream because the liver doesn't need the glucose. The muscle cells need the glucose. The glucose travels to my muscle cells, it gets to the muscle cells, the muscle cells go thank you very much. Student: So it's doing more of the releasing previously stored glucose, not running gluconeogenesis... Ahern: Her question is "is it releasing previously stored "glucose or doing gluconeogenesis," the answer is it's doing both. So the liver releases the glucose, the muscle cells grab it. And I keep running and running and I'm an old guy so I have a pretty good heart, but my heart probably isn't delivering as much blood and as much oxygen as fast as my muscles can use it. That's especially true the longer that I run. So the longer I run, the less oxygen my muscle cells are going to have. My muscle cells are going to take this glucose and as long as they've got oxygen and they can make pyruvate and acetyl-CoA, they're happy. But what happens when they start running out of oxygen? Well, we remember that the only thing that they can do to keep glycolysis going at that point is convert pyruvate into lactate. And they do that. Lactate, as I said in class at the time I mentioned it, is a biological dead end. It doesn't go to anything else. All we can do is convert it back to pyruvate. And to do that, we need oxygen. The muscle cell doesn't have oxygen, lactate's sitting there, it's not doing the muscle cell any good. Moreover, lactate is an acid, so it's starting to drop the pH of the muscle cell and that's a problem. The muscle cell says hell with that, and it dumps lactate into the bloodstream. The bloodstream goes right back to the liver which has plenty of oxygen because the liver is close to your lungs. It converts that lactate to pyruvate and then boingo! It does gluconeogenesis. What we see is a cycle that's occurring in the body and it's known as the Cori cycle. The liver is making glucose, dumping it in the bloodstream. The muscles are using the glucose, making ultimately lactate when they run out of oxygen. Lactate's going back into the blood stream and back up into the liver. It's a beautiful system and it works amazingly well. Make sense? One last thing. This cycle is needed because, again, muscle cells can't make their own glucose. They're depending on the liver. It makes sense for the liver to do it because the liver has plenty of oxygen. The muscle cells don't have plenty of oxygen. Yes, sir? Student: Eventually, the liver will run low on oxygen. The oxygen demand will eventually out stretch your lung and hearts' ability to put out, right? That's like hitting the wall in a marathon run or something? Ahern: So his question is, is hitting the wall in marathon running the equivalent of the liver's basically losing the ability to maintain oxygen and so forth and people argue about what hitting the wall in marathon running actually means. A lot of thinking is it actually is resulting from the depletion of your stores of glycogen which liver is storing. So in addition to making glucose by gluconeogenesis, the liver can also release glucose as she was referring to up here, by breaking down glycogen. It's thought that hitting the wall occurs when you really don't have hardly any glycogen left. Again, that's argued. Other questions about that? Connie: Speaking of hitting the wall, I heard that after you hit the wall, you start burning fat. What is that about? Ahern: She says she heard that after you hit the wall, you start burning fat. Yes, you will be burning fat. You'll be burning fat to some extent as you're running as well. It's just that it's not as readily available source of energy for quick things at that time. But yeah, if you didn't have some back up energy source at that point, you'd be in pretty deep trouble. Burning up fat's important. Turns out burning up fat is very important for your heart. Your heart uses fat as a big energy source. Though I don't talk about it much here, another thing that your heart can actually use as an energy source is lactate. The heart can pull lactate out, convert it to pyruvate, but then instead of making glucose, it will convert pyruvate to acetyl-CoA again because the heart has plenty of oxygen and acetyl-CoA is a source of ATP. Lactate can be used by the heart as a way of keeping the heart going. That's an important thing to do, too. Other comments, questions? Student: Is it generally? Ahern: Generally, that's considered an important thing. Depends on how well you like your relative, I guess. Did you have a question? Student: If the heart muscle has developed the ability to use lactate, why haven't regular muscles developed that ability? Ahern: Why haven't regular muscles developed that ability? Well, they would, but remember the regular muscles are away from the lungs. So they run out of oxygen. They could, if they have oxygen, but they don't have that. If they had oxygen, they wouldn't be making lactate in the first place. They're only making lactate because they have to when they're out of oxygen. It's the oxygen that's determining that. It's not any limitation that's there of theirs. Make sense? How about a last song? The last song I always get out of breath. That's why I didn't sing it after the last one. because the last one is a long song. It's my favorite song I've ever written. I hope you guys like it. It's to the tune of "Supercalifragilisticexpialidocious." [class laughing] You know what it is. Here we go. It's called gluconeogenesis. Lyrics: When cells have lots of ATP and NADH too They strive to store this energy as sugar, yes they do. Inside the mitochondria they start with pyruvate. Carboxylating it to make oxaloacetate Oh gluconeogenesis is so exhilarating Memorizing it can really be exasperating Liver cells require it so there's no need for debating Gluconeogenesis is so exhilarating Glucose, glucose come to be glucose, glucose come to be Oxaloacetate has got to turn to PEP Employing energy that comes from making GTP From there it goes to make a couple phosphoglycerates Exploiting ee-nolase and mutase catalytic traits Oh gluconeogenesis is liver's specialty Producing sugar for the body most admirably Six ATPs per glucose is the needed energy Gluconeogenesis is liver's specialty Oh glucose, glucose, joy to me Glucose, glucose, joy to me. Converting phosphoglycerate to 1,3BPG equires phosphate that includes ATP energy Reduction with electrons gives us all NAD And G3P's isomerized to make DHAP Oh gluconeogenesis is anabolic bliss Reversing seven mechanisms of glycolysis To do well on the final students have to learn all this Gluconeogenesis is anabolic bliss Oh glucose, glucose factory Galactus, glucose facotry The aldolase reaction puts together pieces so A fructose molecule is made with two phosphates in tow And one of these gets cleaved off by a fructose phosphatase Unless F2,6BP's acting blocking pathways Oh glucogenesis a pathway to revere That makes a ton of glucose when it kicks into high gear A cell's a masterminding metabolic engineer Glucogenesis a pathway to revere Oh glucose, glucose jubilee Glucose, glucose jubilee From F6P to G6P that is the final phase The enzyme catalyzing it is an isomerase Then G6P drops phosphate and a glucose it becomes Inside a tiny endoplasmic-al reticulums Oh glucogenesis is not so very hard I know that on our final we will not be caught off guard Because our professor lets us use a filled out index card Gluconeogenesis is not so very hard. Yeah! Thank you. Happy Turkey Day.
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Medical_Lectures
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Hemostasis_Lesson_2_Platelet_Activation_and_Aggregation.txt
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this is the second lecture in this series on hemostasis and will be part one of a description of normal physiology which will focus on primary hemostasis the learning objectives of this video will be first to describe normal endothelial function as relevant to hemostasis and the endothelium's reaction in response to vascular injury next to describe the formation and structure of platelets and last most of the video will focus on the list of triggers and consequences of platelet activation I'm showing this overview of humas again just to orient you in that I'll be focusing this video on this part including the role of collagen Von Willa brand factor and fibrin on platelet activation and their aggregation into a platelet plug what do the endothelial cells which normally lme blood vessels do in the absence of vessel injury in order to prevent inappropriate thrombosis first they secrete prostacycline which for a reason that will be explained in a few minutes is usually abbreviated pgi2 prostacyclin inhibits platelet activation and aggregation they they also secrete nitric oxide which locally vasod dilates and also inhibits platelet activation and aggregation they express a molecule called heprin sulfate which activates an enzyme called antithrombin which inactivates several members of the coagulation Cascade they express thrombomodulin which changes the enzyme thrombin's Affinity away from activation of proclotting factors and towards activation of anti-coagulant factors and last endothelial cells normally Express a protein called tissue Factor pathway inhibitor which inhibits the tissue Factor 7A 10A complex heprin anti-thrombin thrombomodulin thrombin and the tissue Factor 7A 10A complex will all be discussed in the next video in addition to these normal functions of the endothelium which prevent spontaneous thrombosis the normal laminar flow of blood through the vessels typically results in a layer of cell-free plasma immediately adjacent to the endothelium preventing platelets from making physical contact with endothelial cells which could have the potential to trigger activation the earliest response to vascular injury is vasospasm which essentially occurs instantaneously vasospasm is shortlived however and most relevant in minor injuries of smaller vessels while the vasospasm of endothelial cells following injury is important platelets have a much more important and complex role I'll first discuss where platelets come from and what their structure is like platelets come from megacar oyes megacar oyes are large cells produced in the bone marrow largely under stimulation from the hormone thrombo poetin here's a path slide from a bone marrow biopsy in which two Mega caros sites are seen near the middle over 1,000 platelets are formed from the cytoplasmic fragments of a mature megaros site oneir of the platelets in the body are sequestered in the spleen and this splenic platelet pool freely exchanges with the platelets in circulation normal platelet lifespan is about 10 days in this peripheral blood smear the many lavender circles with Central clearing are the biconcave discs of red blood cells the two larger cells with a multi-lobed dark purple nuclei are white blood cells and the tiny purple dots that are spread around among the other cells are the platelets you can appreciate that they are much smaller than the other cells since they are cytoplasmic remnants platelets lack a nucleus and have relatively few traditional organel such as mitochondria and ribosomes however their cytoskeleton and membrane structures are highly complex the details of the platelet cytoskeleton are beyond the scope of this course but there are other structures of which you should be aware for example platelets contain unique cytoplasmic structures called Alpha and dense granules these granules contain various compounds involved in platelet adhesion activation and aggregation as well as the coagulation Cascade platelets also have many different types of membrane glycoprotein receptors of which there are debatably four particularly important ones there is the gp1b 59 complex which binds to the multimeric protein Von willbrand Factor after it has been immobilized on exposed collagen gp1 a2a also called integrin Alpha 2 beta 1 binds to exposed collagen directly as does gp6 and a receptor called GP 2b3a also called integrin Alpha 2B beta 3 binds to free fibrinogen and Von willbrand Factor if you're wondering about the confusing nomenclatures for these receptors the GP system which stands for glycoprotein designates numbers and letters to proteins according to their electrophoretic Mobility on polyacryamide gels the integrin nomenclature is based on the protein structure unfortunately neither of these are remotely helpful in understanding or remembering their functions the normal response of platelets to vascular injury is dependent upon something called Von willbrand Factor this is a large circulating glycoprotein that is produced by platelets and endothelial cells and which has key roles in both platelet adhesion and aggregation as well as in coagulation Bond willbrand Factor exists as a heterogenous mixture of multimers of various sizes which are linked by dulfi Bridges smaller multimers are constitutively secreted by endothelial cells and megaros sites while larger multimers are stored in endothelial cells within structures called wo pade bodies and in platelets within structures called the alpha granul the larger multimers have greater activity than the smaller ones bonda brand Factor also plays a role in the coagulation Cascade in which it binds to circulating factor 8 greatly increasing its halflife the Von willbrand Factor protein contains many different binding sites including ones for collagen gp1 B gp2 b3a factor 8 and heprin the largest multimers are broken down by an enzyme called Adams 13 an acronym which stands for a disintegrin and Metallo proteinase with a thrombospondin type 1 Motif member 13 that must be one of the most ridiculous enzyme names in the body the enzyme is important however because reduced activity can lead to a life-threatening condition called thrombotic thrombocytopenic perpa or TTP which patients develop microscopic blood clots throughout the body TTP will be discussed in a later video interestingly patients with type O blood have lower levels of V willbrand Factor compared to other blood types which may be related to the observations that patients with type O blood may be slightly more prone to bleeding complications and patients with non O blood groups may have slightly higher rates of heart disease and Venus thromboembolism as platelets travel in the circulation they exist in an inactivated State inactivated platelets are non-sticky so to speak and do not actively secrete compounds into the blood that promote hemostasis so what are the triggers for Activation in which platelets transition from a relatively quiet anti-thrombotic state to one that is pro thrombotic there are four primary compounds that trigger platelet activation Each of which uses one or more specific receptors the trigger for the first wave of platelets to be activated is subendothelial collagen which is exposed to the blood following vascular injury ADP is another trigger which is actively secreted out of activated platelets in an autocrine and paracrine fashion this means that it acts only very locally in which an activated platelet that secreted ADP can have its activation further enhanced by ADP binding to itself or ADP can activate inactivated platelets in the immediate vicinity thromboxane a 2 is a compound that diffuses out of activated platelets and can also act in either an autocrine or paracrine fashion finally thrombin an enzyme which itself is activated from the coagulation Cascade can activate platelets as well so just in the last several minutes I've mentioned a lot of different receptors proteins and other factors and you may be starting to lose track of what each of these things is responsible for to help synthesize all this information into a cohesive picture of how the individual components of plet activation fit together let me walk you through a brief animation so here's a blood vessel wall with red blood cells flowing past the endothelium beneath the endothelium is the subendothelial Matrix and a layer of collagen fibers now be aware this is not to scale now let's suppose that the endothelium is damaged somehow exposing the blood to the subendothelial collagen although there's a lot of over in the chronology of the steps of platelet activation conceptually you might consider platelet adhesion as being the first step Bond willbrand factor that is either circulating in the blood or which is released by endothelial cells binds to the collagen local turbulence in the flow of blood brings inactivated platelets close enough to the site of injury that the gp1 b59 receptor complex binds to the immobilized Bon willbrand Factor other platelets will rely on the gp6 or gp1 a2a receptors to bind directly to the collagen and any of these can activate the platelets activated platelets begin secreting the contents of their Alpha granul which include fibrinogen more Von Willer bin factor and a member of the coagulation Cascade called Factor 5 they also secrete the contents of the dense granules which most importantly includes ADP activated platelets will also start to generate thromboxane A2 which diffuses directly through the cell membrane at this point aggregation starts to occur this is dependent upon platelet activation triggering a confirmational change in the GP 2b3a receptor from an inactive to active State now fibrinogen and Bon willbrand Factor can function as Bridges between the GP 2b3a receptors on neighboring activated platelets as platelets start to aggregate more prematic compounds are secreted activating more platelets leading to more aggregation and before long an enormous bunch of platelets are stuck together in what is referred to as the platelet plug in addition to the aformentioned consequences of activation two more are critically important first platelets change shape when activated in response to an activation triggered increase in intracellular calcium rearrangements of the internal cytoskeleton causes their normal discoid shape to transform into a highly irregular morphology with numerous projections of cytoplasm sticking out in all directions here's an electron micrograph of the three major types of blood cells on the left is the biconcave disc of a red blood cell on the right is the round irregularly surfaced white blood cell and in the middle is the highly irregular form of an activated platelet this shape change increases the platelet surface area and increases their ability to physically interlock with and adhere to neighboring platelets one final important consequence of plet activation is the surface expression of a phospholipid called phospho tial Serene the concentration of negative charges from the phospho tial Searing on the surface of the outer plasma membrane supports a assmbly of clotting Factor complexes to be discussed in the next video because the mechanism and consequence of platelet activation is difficult to remember I'm going to run through it one more time using a different diagram with some additional details about the platelet receptors involved once again after vascular injury subendothelial collagen becomes exposed the gp6 and gp1 a2a receptors bind to this directly and the gp1 b59 receptor complex binds the collagen via Von willbrand Factor the process by which platelets initially become stuck to exposed collagen is platelet adhesion the active platelet adhesion to collagen can itself lead to platelet activation platelet activation can also be triggered by thrombin which is formed from the coagulation Cascade and which binds to the par one receptor it can be triggered by thromboxane A2 which binds to the thromboxane A2 receptor and it can be triggered by ADP which binds to the P2 y1 and P2 y12 receptors each of these receptors is important because they are potential drug targets some of which are used by drugs already on the market and some of which are being investigated consequences of platelet activation include secretion of granules which release ADP more Von willbrand factor and fibrinogen pl activation also leads to their shape change and it leads to activation of the GP 2b3a receptor activation of this receptor allows it to bind fibrinogen and Von willbrand factor which aided by the shape change results in platlet aggregation the consequence of significant platelet aggregation is the formation of the platelet plug the last topic I'll discuss in this video is the synthesis and regulation of thromboxane a 2 which is important in understanding the mechanism of action of aspirin thromboxane A2 is an example of an eosino these are an important and diverse group of 20 carbon polyunsaturated fatty acids other important members in this group include arachadonic acid and prostacyclin the synthesis of eosino is very complicated and unless you are concurrently enrolled in a graduate level biochemistry course you should not bother memorizing this chart however there are a couple of specific details important enough to remember the first is that prostacyclin synthes is more active in the endothelium and the consequences of its product prostacyclin is that platelets are inhibited and there is baso dilation conversely thromboxane synthes is more active in platelets and the actions of thromboxane A2 are to activate platelets that we' as we've already mentioned and also to cause basoc constriction so there is physiologic antagonism in the body between those two enzymes normally prostacyclin predominates until platelets become activated at which time a greater proportion of prostag gland in H2 gets converted to thromboxane the final detail from this chart to remember is the actions of and distinctions between a cyc oxygenase or Cox one and Cox 2 enzymes while both of of these enzymes work to convert arachadonic acid into prostag gland in H2 via a multi-step pathway the expression of these enzymes differs Cox one is constitutively expressed in most tissues while Cox 2 displays an inducible expression in response to inflammation this gives rise to the category of medications called Cox 2 Inhibitors which are thought to be more selective to areas of inflammation and thus carry fewer side effects the world's most common anti hemostatic medication is of course aspirin which is a cox1 inhibitor aspirin will be discussed more in the fifth video on antiplatelet medications that concludes part one of the normal physiology of hemostasis part two will focus on the coagulation Cascade in fibrinolysis
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Medical_Lectures
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21_Biochemistry_Glycolysis_Lecture_for_Kevin_Aherns_BB_450550.txt
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Kevin Ahern: I hope you had a good weekend? I hope. I had a nice weekend. I went to the coast and got away, which is why I didn't get the video posted until Sunday. I know there were several of you who were very anxious to see that, so I'm sorry it took a few minutes. There was actually a problem on the OSU site on Friday so I couldn't get the thing done on Friday afternoon, which is what I prefer to do. That's why it waited until yesterday. I have not yet officially announced a time for the BB450 review session for Exam 2. I am planning on that being tomorrow night at 5:00 p.m. I will videotape, as before, and I will announce that for sure when I get a room secured. I haven't done that yet, so I need to do that. But I'll send an email out to the classroom listserv. I'll also post the information for it on the class web page. Material for Exam 2 will go through whatever I cover today. Today is the end of material for Exam 2. Clear as mud? We're just about done talking about metabolic control, and this last thing I'm going to talk about is sort of an add-on in the chapter. It's kind of like your authors of the textbook weren't quite sure where to put this one, so they stuck it here under metabolic control. This refers to the reaction types that enzymes catalyze and it's actually a fairly interesting set of information. The reaction types are shown down here and this categorization of the reactions that enzymes catalyze is an attempt to systematize the classification of enzymes, making them systematic. There are, it turns out, six separate categories of reactions that are catalyzed by enzymes. When we take all of the reactions that are catalyzed by enzymes then we can place them into six groups based on the chemical reaction being catalyzed. They're shown on the screen. I'll give you examples or show you some examples of these, but the six different types are oxidation-reduction, ligation requiring ATP cleavage, isomerization, group transfer, hydrolytic, and addition or removal of functional groups. Let me show you some examples of these types of reactions. Oxidation-reduction reactions I talked about last time, and they involve loss of electrons by one substance and gain of electrons by another substance. In this case, the top case, we see succinate, which we will study next term in learning about the citric acid cycle. We see succinate losing electrons to FAD, meaning succinate is becoming oxidized. FAD is becoming reduced, and the products of those reactions are fumarate plus FADH2. Whenever you see an electron carrier in a reaction, you can pretty much assume it's a redox reaction, because the electron carrier is there to carry electrons. Here's another reaction we'll talk about next term, malate going to oxaloacetate. Malate has the electrons that it loses. It gives electrons to NAD to make NADH, and oxaloacetate is a product. So oxidation-reduction reactions are very easy to spot, largely because of the involvement of the electron carriers. Ligation reactions are also fairly easy to understand. Ligation reactions generally involve the joining together of two different molecules, the joining together of two different molecules. We can see, in this case, pyruvate is being joined to carbon dioxide. We'll talk about this reaction in about a week or so. In this reaction, carbon dioxide is joined with to three-carbon pyruvate to make a four-carbon molecule oxaloacetate. So joining two substances together is what is involved in ligation reactions. Isomerization reactions are also very easy to recognize. We're changing the configuration of a molecule. We're not causing oxidation or reduction or breakage or ligation or any of that. We're simply rearranging it. In this case, citrate, we can see here, is being converted to isocitrate, and basically this hydroxyl group is moving from here over to here, swapping places with a hydrogen. So a rearrangement reaction is an isomerization. You'll also notice that the reaction types that I give here are slightly different than the ones that were in the table, it's jumping around, slightly different than the ones they gave in the table. I would prefer that you use my categorizations than the one that's in that table. Group transfer may not sound very intuitive by its name, but it basically involves the movement of a portion of one molecule to another molecule. Notice that I said a "portion" of one molecule to another molecule. A ligation joins two molecules. A group transfer transfers a portion of one to another. In this reaction, which we'll talk about later today, glucose is gaining a phosphate from ATP. That gives it glucose 6-phosphate and leaves behind ADP. This is a prime example of a group transfer type reaction, the phosphate being the group that's being transferred. Hydrolytic reactions, as their name implies, you've already seen a bunch of these, are reactions that involve cleavage of a molecule using water. The proteases, for example, that we talked about, used water to break peptide bonds and they are an example of hydrolytic reactions. You see a prime one on the screen right there. The last one is probably the hardest to understand, and, to be honest with you, I don't place much emphasis on it, but I'll just give you a notion of what it is. The last group is called lyases, and lyases are enzymes that basically catalyze the splitting of a molecule. That's a little bit of a simplification of what they do, but you can see in this reaction from glycolysis that fructose 1,6-bisphosphate is being split in half, and so lyases cause the breakage of bonds to split molecules into two. That's one way of thinking of them, and, for our purposes, that's all we really need to pay attention to. I'm not going to expect you to memorize the names of these enzymes, but I do think that you should know the categories. The six categories. I'm not going to say, "Give me an example "of an enzyme that is a lyase," for example. I think that's just busy work. But understanding the six categories is useful. Now, the six categories actually, as I said, are an attempt to systematize the naming of enzymes. This gave rise to what was called the "EC number. " The EC number, "EC" stands for "Enzyme Commission," and they were the group that came up with these categories and they are involved in classifying enzymes into each of the six different categories. So when we see, for example, an enzyme that is an oxidoreductase, it would all fit into a category like this and it would have a specific EC number. So the EC number, I don't remember the numbers off the top of my head, but I think this is Category 1, all oxidoreductases would have, in their numbering scheme that corresponds to them, the very first digit corresponding to the category that they're in. So, in this case, an oxidoreductase is Category 1. All of the enzymes in the EC commission that are oxidoreductase would have as their first number, number 1. They would have a 1, point something, point something, point something, point something. Each point would designate a little bit more specificity about the type of reaction that the enzyme catalyzes. But, six general groups, and that first digit of the EC number is what is critical for identifying these six groups. Yes, sir? Student: Do the EC numbers correspond to the order you gave those to us in? Kevin Ahern: I don't know if they correspond to the order, to be honest with you. I'm not worried about you memorizing those numbers, but know that there are six. That's basically what I want to say. I'll just show you this last thing as a trivia item. It's not something I expect you to memorize or anything, but it shows the sort of central importance of the molecule ADP in a variety of places, as it appears in biochemistry. ADP is, of course, a part of ATP. ADP is shown in blue, red and yellow. That same ADP is found in NADH. It's found in NADPH, which is not shown on here. It's found in FAD, and it's also found in coenzyme A. I don't know why it's not red over here but it's also found in coenzyme A. You don't need to know that. I'm just showing you that for trivia purposes, but these molecules play very, very important roles inside of cells, and ADP is a significant component of every one of them. So much for metabolic controls. We're going to turn our attention now, for the first time, to the pathways of metabolism. "Metabolism" I'm going to define for you as a collection of reactions that are found in cells. Metabolism we can think of as the chemical reactions or the biochemical reactions going on inside of cells. As I showed you on the road map the other day, these molecules or these reactions are actually linked into highway-like pathways, that one molecule leads to another, leads to another, leads to another. As such, that means that all of metabolism in a given cell is linked. There's no escaping that. They're all linked. Now, I want to emphasize to you that a given pathway, we're going to be talking about glycolysis, a given pathway is a man-made invention. It's a man-made invention. "But it's there, it's on the table." Well, that's right. But where I call it glycolysis really depends upon where I define the starting point and the ending point. If we look at that big road map, it's kind of like saying, "Where is the road to Portland?" The road to Portland could start in Ashland, if we're thinking about southern Oregon. It could start in Salem. It could start in Seattle. So where we define a given roadway is really a man-made invention, and so glycolysis is like that. For our purposes, glycolysis will be a pathway that starts with glucose and ends at pyruvate, starts at glucose and ends at pyruvate. You'll see, as we get going further into that, that that is a bit of an arbitrary thing. I also want you to keep in mind that when we talk about metabolism and when we talk about these pathways, that the pathways are interconnected. They're interconnected. I could take the road to Portland by going, for example, over to I-5, riding it up to Salem and then cutting over to, say, 99, and then going back up. There's a lot of ways I could take that pathway to Portland, and the metabolic pathways are the same way. Now, the significance for us, of the interconnectedness of pathways is the interconnectedness of the molecules in those pathways. We can't say that “This molecule is in glycolysis,” because, A, glycolysis is a man-made invention, and B, that molecule may be part of what we define as glycolysis, but now if I think of the pathways going out instead of up and down, that molecule might be tied to other things, as well. So there's a very, very strong interconnectedness with that. You'll see that especially next term when I start talking about the citric acid cycle and fatty acid oxidation. Well, let's take an overview of the pathway of glycolysis. The pathway of glycolysis starts with glucose. Glucose is the most common and most abundant sugar on the face of the Earth. It has six carbons. The pathway of glycolysis causes it to be broken into two pieces that are identical, known as pyruvate. So one six-carbon piece gives rise to two three-carbon pieces. Then, at pyruvate, we see that pyruvate has three different "fates," as we describe them, meaning pyruvate can be converted into three different things, depending upon what the cell needs, what the cell has, and the type of cell in which the reaction is occurring... what the cell needs, what the cell has, and the type of reaction in which this reaction is occurring. Let's just very briefly go through these. I'll probably remind you of them later when I get to pyruvate, talking about the pathway. Pyruvate, most commonly, when we're talking about aerobic metabolism, goes to acetyl-CoA. It's not even shown on here, but acetyl-CoA is an intermediate. Acetyl-CoA plus oxygen ultimately leads to carbon dioxide plus water. So the first and most common direction is to go where oxygen is available and it goes to acetyl-CoA. We'll see how that acetyl-CoA gets used next term. But this pathway, this very first and most important pathway, requires oxygen. Well, what if cells don't have oxygen? If cells don't have oxygen—and yes, that does occur in our bodies, when we're exercising heavily our muscles can't get oxygen fast enough—they have to have a backup way of generating energy, as we shall see. In our cells, when they're lacking oxygen, pyruvate gets converted to lactate. Notice, again, lactate is not the same as lactose. Lactose is a sugar. Lactate is this guy over here. Well, what if it's not my cells? What if it is a bacterial cell or a yeast cell and it runs out of oxygen? That, of course, that pathway goes from pyruvate and leads us to ethanol. It's the foundation of brewing, micro brewing and all these great things that happen with that. For any of you who've ever made your own beer and so forth, you know that you mix all the stuff up and then you cap it off so that there's no oxygen available, and that lack of oxygen is what leads to the production of ethanol, and we'll see later why that's the case. So, three different fates for pyruvate. No oxygen leads to either ethanol, in the case of bacteria and yeast, or lactate, in the case of us. In any organism that is aerobic when there's oxygen available it goes to acetyl-CoA. Questions on that? I'm hearing none. Let's dive into glycolysis. This figure is a fairly good overview of the process of glycolysis. It has all the players on it. You can see glucose, glucose 6 - phosphate, blah, blah, blah. The first question I get is, "Well, what of this do we have to know?" Well, I'm going to spell that out for you fairly explicitly right here. First of all, you need to know the names of the ten intermediates. Second of all, you need to know the names of anything that you could easily learn. What does that mean? Well, I've told you previously that you need to know the structure of glucose, and glucose is right there. Duh, right? If you know the structure of glucose, then learning the structure of glucose 6-phosphate means you simply need to know where to put the phosphate onto there. Duh-uh. You also had to know the structure of fructose, and fructose, if you put a phosphate on it, you have fructose 6-phosphate right there, so that's an easy one. Fructose 1,6-bisphosphate, fructose, you put a phosphate on positions 1 and 6, and you've got that guy there. So those four, given the fact that you already are supposed to know glucose and fructose, should be no-brainers for you. I will mention other ones, depending upon how far I get through this pathway today in the lecture, that you'll be responsible for, and I'll save those to see how far along that I get. This overview of the pathway, we can also see in blue, over here, the names of the enzymes. The enzyme names you are responsible for, yes. These are very important enzymes in the body, and the enzyme names actually tell you something about what's going on in the reaction. Here's hexokinase, "hexo - " meaning "six," "kinase" meaning "puts a phosphate onto." This guy puts a phosphate onto six-carbon sugars and, guess what, glucose is a six-carbon sugar, so hexokinase puts a phosphate onto glucose. So the ten names of the intermediates, the ten names of the enzymes, and the molecules that it's easy to learn the structure of, and I'll point out two more for you. Let's look at this process. We see that the process is divided on your screen into two main phases, and those two main phases are actually different than what the last version of your book did. If you have the 6th edition of the book, you will see it splits it into three phases. I was never very fond of the three phases, so I'm glad they went to this two-phase model. The two phases are known as "energy investment" and "energy realization" or "energy generation," you can call it whatever you want. It means that in the first phase of glycolysis we have to put energy into the molecules, and in the second phase of glycolysis we get more energy out than we put in. Now I want to emphasize, that glycolysis is a source of ATP in the cell. It is not usually a very large one. Only when cells are hurting does glycolysis become a very large one. When are cells hurting? One of the times might be if they're low on oxygen. Then that ATP that they can get from glycolysis turns out to be very, very important. I'll also give you some numbers here that we will talk about again probably more next term than this term, but this gives you an idea of the importance or the relative importance of glycolysis in the scheme of things. If I start with glucose and I go down to pyruvate, what you will discover is I produce a net gain of two ATPs... two ATPs. That's not a lot, but I also generate some other things along the way that are very useful. I produce, first of all, some NADH. I produce two of those, and I also produce two pyruvates. Now, if I take these NADHs and I take those pyruvates, and I oxidize them all the way down—this happens in the citric acid cycle and in the electron transport system if I oxidize them all the way down and I count the number of ATPs that I get, depending on who's doing the counting, you get about 38 ATPs. You get 38 ATPs, and you think, "Wow, this is just basically getting the process started," and that's correct. But you only get those 38 ATPs under one condition and that's if there's plenty of oxygen. If there's not plenty of oxygen, you're stuck with two ATPs. So it's important to be able to get as much energy out of glucose as possible if you want to be efficient. If you don't want to be efficient, that's fine. It turns out that's good. Did you know that? Not being efficient is good? Any speculation on when you might not want to be very efficient? Student: When you're running a marathon? Kevin Ahern: Well, when you're running a marathon, you'd kind of like to be efficient, I think. Yeah. Student: Maybe when you're asleep? Kevin Ahern: Maybe when you're asleep? Well, not really. I can think of a better condition. It's a condition I found myself in earlier this term. Student: Well, I was going to say when you're stressed, you don't want to be overloaded, but... Kevin Ahern: No, It's not when you're stressed. Student: When you're on a diet? Kevin Ahern: When you're on a diet! Absolutely! You want to be the least efficient when you're on a diet because the more it takes to burn to make ATP, the more stuff you're going to burn up. I'll tell you later about how people have this notion about going aerobic is the best thing that you can do and I'm going to try to convince you that going anaerobic is the best thing that you could do if you're trying to lose weight. We'll talk about that in times to come. Let's look at the energy investment phase. The energy investment phase we can roughly think of as about the first five or six reactions of glycolysis... It sort of depends on how we count them. In this phase, we have to use energy from ATP to get the process started. So here's a catabolic process, and we'll see that in several ways. glycolysis is an unusual catabolic process, here's a catabolic process that is sort of breaking the rules and requiring us to put energy in before we can get energy out. Most catabolic processes don't do that. There's two different places, as you can see on the screen here, where ATP is used. In this first phase, no ATP is generated. The ATP is generated only in the second phase, as we shall see. Well, let's take a look at the reactions. Here's the first reaction. Glucose combines with ATP to produce glucose 6-phosphate and ADP. I showed you this reaction earlier. It's catalyzed by the enzyme known as hexokinase. By the way, I'm also going to tell you, in a few cases, the Delta G zero primes for reactions, and I think you should know the generalities, not the absolute numbers. This guy has a fairly negative Delta G zero prime. If I tell you about the Delta G zero prime for a reaction being fairly negative, fairly positive, you should know those. This is one of them. This reaction has a Delta G zero prime that's fairly negative, meaning that, if we start this reaction with all concentrations of everything equal, that it will go strongly in the forward direction. What if we try to do this reaction with phosphate instead of ATP? The enzyme will actually do it. If we measure the Delta G zero prime for the reaction using phosphate instead of ATP, what we discover is that the Delta G zero prime for the reaction is very positive. Well, this is a prime example of where coupling the hydrolysis of ATP to an energetically unfavorable reaction converts it into a favorable reaction... a prime example of that. The same reaction with phosphate instead of ATP has a Delta G zero prime that's very positive, meaning not very favorable in the forward direction. It's going to be much more favorable in the reverse direction. However, when I link it to ATP and the hydrolysis of ATP yields energy that drives this reaction forwards. This reaction is an interesting reaction. It's one of three reactions in glycolysis that is one of the regulated reactions, meaning the enzyme itself is regulated. This one turns out to have an unusual regulation and I'm going to spend very little time talking about that. I won't talk about regulation until at least Wednesday. But this is one of three. Glycolysis is also unusual in having not one regulatory step, but three steps that are controlled. Student: [unintelligible] Kevin Ahern: What's that? Student: [unintelligible] Kevin Ahern: In the pathway. The second step of—oh, by the way, the induced fit, I've showed you guys that before, but you may recall when I talked about glycolysis before and I said that, remember how the substrate changes the enzyme upon binding it? And the example I actually gave you was this enzyme. Hexokinase starts out, it's got two molecules that it's got to bind and it's got to crunch together to transfer that phosphate from one to the other. We can think of these as rather like the jaws, the teeth of the jaws, right here, where up above I could have, let's say, ATP, and down below I could have glucose. The binding of both of these causes the jaws to close so that the phosphate of the ATP is brought into close proximity of glucose, and the phosphate is able to jump. When that phosphate jumps, the jaws open and the molecules come back out. This is a really good example of an enzyme that has an induced fit. It changes its shape as it binds to its substrate. You can see that here, a little bit. They're only showing you the glucose. They're not showing you the ATP. But you can see that the addition of glucose, in this case, is converting the unbound enzyme to the bound enzyme, shown in red. You can literally see those jaws closing down. Reaction two of glycolysis is a reaction catalyzed by phosphoglucose isomerase. It simply involves the conversion of the glucose 6-phosphate to fructose 6-phosphate. We see it goes through a linear intermediate in order to do this, and all that's happening here is an aldehyde group, right here, is being converted to a ketone group. So the position of the double-bonded oxygen is moving in the molecule. That is an isomerization. Student: Could you say the name of the enzyme one more time, please? Kevin Ahern: Phosphoglucose isomerase or glucose phosphate isomerase. You see it listed both ways. Reaction number three is one of the most important reactions of glycolysis. This reaction, first of all, is catalyzed by the enzyme known as PFK, phosphofructokinase, and, yes, you're more than welcome to call it "PFK." Later in the term you'll see that we refer to this as PFK1 and you can call it either PFK1 or PFK at this point, because we haven't encountered PFK1 or 2 yet. But this is PFK. Why is this reaction so important? First of all, this reaction, this enzyme, PFK, is the most important regulatory enzyme in the glycolysis pathway. It's the most important regulatory enzyme in the glycolysis pathway, and, as we will see, there are several things that can allosterically affect this enzyme, several things that allosterically affect this enzyme. You'll notice also that we're adding a second ATP in this reaction. So we're converting fructose 6-phosphate to fructose 1,6-bisphosphate, and, by the way, you may use these abbreviations. There's only one abbreviation I'll ask you not to use, and I'll show you that one later. But all the abbreviations that they use here are certainly acceptable in this class, with the exception of the one I'll give you later. ATP is required, and, just as we saw before, this reaction, with the hydrolysis of ATP, the Delta G zero prime of this reaction is very negative. The hydrolysis of ATP helps to drive this reaction forward. If we try to do this reaction with phosphate instead of ATP, we discover it doesn't go forward very well, at all; in fact, it goes very strongly backwards. So the hydrolysis of ATP, again, is making this reaction much more favorable. So at this point, we've built a molecule that has two phosphates into it and, now, in the next step, we're ready to break it apart. We can look at this next step as being a lyase-type reaction because what we see is that this guy is going to get split right in the middle, and this enzyme is the only enzyme in the entire pathway whose name doesn't immediately tell you what it does. It's called “aldolase.” All the other enzyme names in the pathway tell you what the enzyme does. So what's happening here? In this reaction, fructose 1,6-bisphosphate is being split in half. One half becomes dihydroxyacetone phosphate, or DHAP. The other half becomes glyceraldehyde 3-phosphate, or what your book calls “GAP,” and I don't like that. Let's call it “G3P.” G3P, okay? So instead of calling it “GAP,” you're going to call it “G3P.” This reaction is interesting. This reaction is interesting because the Delta G zero prime for this reaction is very positive. It's very positive. Now, if I told you the Delta G zero prime for a reaction is very positive, would that tell you if the reaction goes forwards or backwards? No. It wouldn't tell you anything, because the Delta G would tell you. Right? Delta G zero prime will influence that, but Delta G will not tell you, right? Same as with the other reactions. Delta G zero prime, even though it's very negative, it doesn't determine the direction. Only if I tell you that all the concentrations of things I start with are equal, then it tells me the direction of a reaction, right? Well, how do I have a reaction that has a Delta G zero prime that's very positive, yet the reaction still gets to go forward? What would it take to do that? Student: You need a covalent intermediate? Kevin Ahern: A covalent intermediate. No. Student: A high concentration of the reactant? Kevin Ahern: A high concentration of the reactant is one thing. And what else would help this reaction to go forward? Student: [unintelligible] Kevin Ahern: Well, enzymes don't change Delta G. Student: The Delta Gs in the first two parts of the reaction? Kevin Ahern: Well, that's a good point. So the Delta Gs in the first two parts of the reaction do actually help, and they help by increasing the concentration of reactants, which is related to what Connie said. Student: A decrease in concentration of the products? Kevin Ahern: A decrease in concentration of the products. Remember, these reactions, we see that one's connected to the next, connected to the next, connected to the next. So if I have something sucking away the products of this reaction, those two can work together to overcome this energy barrier, because that's going to change the value of that log term. Increasing concentration of reactant and decreasing concentration of product. Now, as we will see, the cell has a really cool trick for accomplishing both of these things. Student: [unintelligible] Kevin Ahern: I'm sorry? Student: [unintelligible] Kevin Ahern: Oh, yep. That's not the cool trick it uses, either. Blast you! So we'll come back. We'll talk more about the aldolase reaction in a bit. We turn our attention now to the last of the energy investment phase. The last of this energy investment phase involves the conversion of dihydroxyacetone phosphate into glyceraldehyde 3-phosphate. Notice this is a reversible reaction, so it can go either way. But in the direction of glycolysis it moves to the right. In the direction of synthesizing glucose it moves to the left. We'll talk about the synthesis of glucose next week. Student: [unintelligible] Kevin Ahern: The dihydroxyacetone phosphate was one of the two products of the previous reaction. Student: That's the DHAP. Kevin Ahern: That's the DHAP. So in this reaction, we're converting DHAP into G3P. Now we have two G3Ps. This simplifies things for keeping track of stuff, because now every reaction is just duplicated. We have two copies of everything that's reacting, from this point forward, and they're two copies of the same thing. This reaction is catalyzed by the enzyme known as triose phosphate isomerase, and it is a prime example of something we talked about earlier in the term. Triose phosphate isomerase is a perfect enzyme. It's a perfect enzyme, meaning that it has a very high kcat over Km value. It's limited primarily only by the rate with which the substrate diffuses to the enzyme. The reason that this enzyme is perfect, it appears, is because there's actually an unstable intermediate that is produced in the mechanism of the enzyme. The faster the reaction goes, the less time that unstable intermediate is around to cause problems. By making the enzyme perfect, we make the reaction go so fast the unstable intermediate doesn't have a chance to fall apart. Perfect enzymes commonly have that strategy. Well, the upshot of this is, at this point, we now have two molecules of glyceraldehyde 3-phosphate and we are ready to move to the oxidation phase of glycolysis... the oxidation phase. By the way, there's the unstable intermediate right there. You don't need to know that, but just to tell you I wasn't lying, you know it's there. In the energy generation phase, what we see is that there are two places that ATPs are produced. But remember that, since every molecule is duplicated, the ATPs themselves are also duplicated. So we're going to produce, in this second phase, a total of four ATPs. That's how we get our net gain of two. Let's start the process off with a bang, and starting that process off with a bang is this mouthful of an enzyme, glyceraldehyde 3-phosphate dehydrogenase. By the way, whenever you see the word "dehydrogenase" involved in an enzyme name, it's always a redox enzyme. That, of course, is reaffirmed by the fact there's an NAD plus that is present here. This reaction is actually two reactions that are occurring on this molecule. Remember that we're starting with two of these G3Ps. Everything we've got two of, from now on, so we've got two G3Ps, we have two NADs, we produce two of these, and we produce two of these, and we produce two of these. What's happening in this molecule? The first thing that happens in this molecule is an oxidation. An oxidation is occurring. The NAD is giving us a clue to that, and the structure of this aldehyde is the clue. Notice we have an aldehyde to start with. Over here, we have an ester. That means this guy had to have gotten oxidized to an acid first. So the oxidation happens and then, after the oxidation happens, phosphate is put onto the acid, making the ester. So oxidation to produce an acid, then addition of the phosphate to the acid to produce the ester. The oxidation converts NAD into NADH, as you can see here. This reaction is interesting. If we compare it, for example, to the hexokinase reaction or the PFK reaction, those reactions were putting a phosphate onto something, right? And I told you that putting a phosphate onto something required hydrolysis of ATP. That's why they used ATP. We're putting a phosphate onto something and we're not using ATP. What does that tell you about this reaction? This reaction is energetically favorable, by the way. What does it tell you? It says we have to have an energy source, right? Previously we had an energy source from ATP. Now we have to have some other energy source. What do you suppose the energy source is? Student: The 1,3-BPG has a higher [unintelligible]? Kevin Ahern: Well, this molecule has a higher energy, sure, but we have to make this molecule. Student: [unintelligible] Kevin Ahern: What's that? Student: Is it the NAD-plus? Kevin Ahern: No, it's not the NAD-plus. It's the oxidation reaction. Remember that oxidation produces energy. So this oxidation that's occurring in this process is giving sufficient energy to add that phosphate. In the previous reactions we saw the energy came from ATP. Here the energy is coming from oxidation. So putting together an oxidation, in this case, with a phosphate, gives us a high energy intermediate, and 1,3-bisphosphoglycerate, this guy's loaded with energy. 1,3-bisphosphoglycerate, or 1,3-BPG, is one of those molecules on that table I showed you last time that had a higher energy than ATP did. It's got a higher energy than ATP does. This guy is full of energy. That negative charge and that negative charge really repel each other. They really don't want to be together. This is not the next reaction. This is the mechanism of that reaction. There's the oxidation. There's the phosphate coming on, so you can see the sort of two steps to this process. In the next reaction, we generate our first ATPs. This enzyme is known as phosphoglycerate kinase. If you want to call it PG kinase, that's fine, too. In this reaction, a phosphate from 1,3-BPG is being transferred onto ADP to make ATP, and that leaves us behind with 3-phosphoglycerate. Now, in order for this reaction to go forwards efficiently and, yes, it does—in order for this reaction to go forwards efficiently, this guy has got to have a lot of energy,and it does. 1,3-BPG has more energy than ATP does, so therefore it can transfer the phosphate onto ADP and make ATP very efficiently. This type of a reaction is a type of a reaction that we haven't seen previously. We're actually making ATP using this step. It turns out there are three ways of making ATP. One is what you see on the screen. It's what known as "substrate-level phosphorylation"... substrate-level phosphorylation. In this mechanism, a high energy molecule is transferring a phosphate to ADP to make ATP. That's only one of three ways that cells make ATP. The second way that cells make ATP is what's called "oxidative phosphorylation." We'll talk about that next term when we talk about the electron transport system. Oxidative phosphorylation in animals is, by far, the most abundant form of making ATP, the most abundant way of making ATP. The third type, or the third mechanism for making ATP is what's called "photophosphorylation" and that's what plants use in photosynthesis. Now this is a relatively minor way of making ATP. Substrate-level phosphorylation doesn't contribute very much to the overall ATP pool. Questions? We're getting close. We're getting close. The last three reactions you see are all bundled together. We're going to make it through. Actually, we might not make it through. Maybe we won't. Yay! I'll just take up the time with something else that you'll be responsible for, so... But I'll spend a little bit of time on this first reaction. This first reaction is kind of cool. I don't know why your book bundles all three of them, as if they're not important, because this one turns out to be a really important reaction. Look at what's happening in this reaction: 3-phosphoglycerate is being converted to 2-phosphoglycerate. It looks like it's a simple isomerization reaction, and overall, it is, but you'll notice the enzyme's class is not described as in isomerase. It's described as a mutase. What the heck is a mutase? Well, the mutase tells us something about the mechanism that this uses. So if I had an isomerase, what would an isomerase do? Well, it would grab that phosphate on position number 3, and it would move it to position number 2, and we would have the same end product. Well, how does a mutase work? A mutase works by making an interesting intermediate. It makes an intermediate by putting an additional phosphate on. So at one place I have two phosphates on, and then it removes this phosphate and leaves the other one behind. Well, maybe you can see where this is headed. The intermediate in this process of a mutase is known as "2,3-BPG." 2,3-BPG is an intermediate. It's a byproduct and it's a stable byproduct of this reaction. 2,3-BPG, you remember, I said was a byproduct of metabolism. It binds to hemoglobin and favors the release of oxygen. The more glycolysis I have going on, the more likely some of that 2,3-BPG is going to escape from the enzyme, not make this thing, and now go out and affect hemoglobin. The mechanism of this enzyme is actually telling our body, "Here's where a lot of glycolysis is going on." It's a really cool thing. 2,3-BPG can be released from the enzyme, without making this, in some cases, and at a low efficiency, let's say 5% efficiency, 2,3-BPG is released and this is not made. The more I have glycolysis going on, the more 2,3-BPG is made, and, of course, this is a flag that this is a cell that's needing energy very quickly, very rapidly. So 2,3-BPG is produced in that way. Questions about that? You look like you've been struck dumb. "Oh, my God! I have seen the light!" Right? "I have seen it, I know." And just so that you won't feel like I'm rushing you through it, why don't we just call it right there for the exam? How's that? Oh, one question over here, yeah? Student: [unintelligible] Kevin Ahern: Yep, this equilibrium is fairly, Delta G zero prime is fairly close to zero. See you guys on Wednesday. Exam material stops right here. Captioning provided by Disability Access Services at Oregon State University [END]
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Medical_Lectures
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13_Biochemistry_Catalytic_Mechanisms_II_Lecture_for_Kevin_Aherns_BB_450550.txt
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Ahern: So the exams are not graded. The TAs have exams of their own and I have told the TAs that, fingers crossed, I would like to have exams back sometime on Friday. That may not be until afternoon, I don't know at this point. And I'm not even sure if that's possible. But the aim is to get things back by Friday. When exams are available, what I will do is I will send an email out to the class announcing that they're available and announcing where to pick them up, okay? So they won't be available here, for example. They won't be available in my office. But I will announce where to pick them up. You will need your ID to pick up the exam. So keep that in mind and the minute, I can assure you the minute they are ready to go, I will put an email out to the class and get them to you. I don't like to have exams out too long before you get a chance to look at them. So I always like to ask, just sort of informally with the class, what did you think of the exam? Student: Good exam. Ahern: Good exam? I'll comment one thing, I have never had an exam where I had fewer questions. There were maybe 10 questions I got on the exam and that was for a class of this side, really unusual. So I hope that's a good sign. But you don't have to like my exam, I don't hate you if you don't like my exam. Student: Why do you videotape your exams? Ahern: Why do I videotape my exams? I find it decreases the looking around factor by a factor of about 100. [student laughs] It does. Student: I figured, but. Professor Ahern: Yup. I don't mean to be mean, but there's just too many eyes here and not enough of our eyes. And what I have found is that really works very well. That's why I videotape those. I find most students are honest. I've only had a handful of situations where in this class, where I've had dishonesty as an issue. And the only students I've ever had with serious dishonesty issues actually were in smaller classes interestingly enough. But it has happened here and I do have taped evidence when it does happen. So that's why I do it. Student: Was the zwitterion question a trick question? Ahern: Was the zwitterion question a trick question? No. How many zwitterions, how many amino acids can form zwitterion? None. They all can. That's not trick. You guys should know that, right? Is the answer "None" a trick? I'm sorry, I don't think so, folks. There are no amino acids, everything, so think about it. What is a zwitterion? A molecule that has a net charge of zero? Doesn't every amino acid have a PI? and what's PI? It's the pH at which the charge is zero. I know I will get some answers on that that don't say none and so I hope that's very few, but that's something that you should know. Yes, Shannon? Student: What did you have in mind for the last question? The one about the Kcat. Ahern: What did I have in mind for the one about the Kcat? Oh, yeah, yeah, yeah, okay. So the question, and by the way, I will post the key outside my door after we give the exams back. I don't post those now because students get all anxious until they see their exam. So I don't post the key until the exams are back. When the exams go back, there will be a key outside my door. But answer your question about Kcat. How can you have two apparent Kcats? and the answer is it depends on how you calculate the concentration of the enzyme. If you take Vmax and divide it by the total concentration of enzyme, you will see a reduced Kcat compared to the uninhibited enzyme. Why? Because much of that concentration of enzyme is not active, it's inhibited. However, if you take the inhibited out of it and you take Vmax and divide it by that modified concentration of enzyme, you will see exactly the same Kcat as if you have an uninhibited enzyme because you're only measuring the velocity of the uninhibited enzyme when you do that. So that's the answer to that. That was the thinker question and I like you to think, I like you to think about the principles that you've hopefully learned in the class. How about length? Was length a factor or not? Student: Yeah. Ahern: How many thought it was too long? Oh, that's relatively small, okay. I really did try, this is the one exam where time is sometimes a factor, and so I was going to have 3 of the longer answer questions and I decided not to do that and I made them shorter. So I had 2. There's trade offs in all these as I've told other classes. So one of the things that happens is the fewer the questions I have, the more points each question is worth. And so people are always worried about that. So I have to try to strike a balance. And this balance, I think works fairly well. I'm glad to hear there weren't too many at least who indicated time problems. Yes, sir? Student: Have you ever offered alternate formats like putting out a 5 part exam and saying choose the 4 you want to attack? Ahern: Have I ever done alternate formats where I say okay, pick the ones you want to do? I have done that. They cause logistical problems for grading and, to be honest, I'm not sure that's the best way to test. I don't rule them out, but in my opinion, the, when I'm giving an exam, I'm sampling. And so I'm sampling your knowledge and while that sampling can be large and I can do what you recommend there. My concern is, well, what I always see is there's a bias. Everybody decides to do this question over here. And it seems to me that when I have that bias, it suggests that all of the questions aren't equal in difficulty and so that kind of makes me think well maybe that's not the way to go. But I have considered it and I have done it on some occasions. And I won't rule it out. I always reserve that as an option. But I've shied away from it for that reason. Anybody hate the exam? You can say it. As I said, I don't...okay, there you go. Do you want to say anything? Student: 20 point questions suck. Professor: 20 point questions suck, okay. Student: If you don't know it, that's like 2 letter grades gone. Professor: Yeah, that's why I like having more questions rather than fewer. Next term, the format of the exam changes in 451. Most people like 451 better because the maximum number of points on a question I think is 3. Student: There we go. Professor: We have a lot of shorter answer questions. But we're working problems, we have to work through problems and that's why. If you're going to spend a fair amount of time on something, it should be worth more points. I mean I can make a 3 point Henderson Hasselbalch question, but if you spend 20 minutes of your time working on it, that wasn't a good investment of your time, right? Other comments or questions? Student: You said this is the exam was the one where time is an issue. Ahern: This is the one exam where time is sometimes an issue. Student: Okay, so the second midterm we'll have a little extra time? Ahern: So the second midterm will be exactly the same format. The final will be exactly the same format. So all the exams are the same format, but what I think happens on the second and third exam is that you've gotten used to Henderson HasselBalch format. The working of the problems, which usually is the time limitation, isn't as much of a factor because you've seen how to do these sorts of problems before. And so I rarely, I won't say never because you, in a class of this size, I could give an exam where I ask you to sign their name and people would say they didn't have enough time. But literally, I rarely have much of an issue with the other two exams. And usually I'll tell you. When I ask that question to this class, I would say I've seen as many as 2/3 of the hands go, 'I didn't have enough time.' So I was very careful to try to make sure I didn't ask you too many things. But it meant I had to have 20 point problems because we have to have a 100 point exam. Yeah, Elizabeth? Student: About the extra credit question...[inaudible] Ahern: The extra credit question, okay. I'll answer that and then I'll go onto other things here. So the extra credit question for hemoglobin, alright. So most people got, the book sort of eluded to the fact that in both cases, you had shape changes. That you're changing from R to T or T to R and so fourth, and that's true. But the root of that change is the binding of a molecule. And that's the answer. Because in the case of hemoglobin, you're binding a molecule exactly in the same manner that the enzyme is binding a substrate. And so the parallels of those with respect to concentration and so forth, that's really the reason you see those two curves being essentially the same. Everybody's all [Ahern makes groaning noises]. Alright, I hope everybody got how I start my lecture. Right? Student: No. Ahern: No!? Student: [Inaudible] Ahern: I said if you said something about starting the exam, we would give credit. Or starting the class, we would give credit, yeah? Student: I said, "How's everybody doing Today?" That's always the next thing you say. Ahern: That's the second thing I say. So the first thing I say is "Okay folks, let's get started." I don't live consciously, I just blurt it out. [class laughing] Alright, well let's turn our attention. Thank you for your feedback. That's not a lot of feedback, but I do appreciate feedback and I'm always happy to listen to what you have to say about exam formats. And I do take suggestions. The suggestion about having other possible choices is one, as I said, I've done and I won't rule out. But other thoughts or feedback, I'm open to 'em, very much appreciate that. Okay, last time I talked about, in some detail, mechanisms by which this, these S1 proteases worked. And I hope you've had a chance to look through those mechanisms and sort of lay things out. And I know it probably wasn't the first thing you did when you went home after the lecture on Friday, but I think it will pay you to go through and analyze that. Because you're going to see when you do that is by understanding that mechanism by which S1 proteases work, you will see similarities in mechanisms that other enzymes use to catalyze reactions. And I'm going to talk about some of those today. But before I do that, I want to talk a little bit more about S1 proteases because so far all I've told you about them is that they are a class of enzymes. And they're a class of enzymes that are called S1, are called serine proteases, I'm saying S1 proteases, they're serine proteases. They're called serine proteases because they all use serine in the active site. So they all have the catalytic triad. But there are some related proteases that have things like that that I want to spend a few minutes talking about today. One of the things that we think about with the proteases, the serine proteases, is the fact that as I said, they all have serine, they all have histidine, and they all have aspartic acid as a catalytic triad at the active site. So one of the questions people have asked is, "Well, what is the relative importance of each of these "in the catalysis of these enzymes?" And so using genetic techniques today, it's very easy to alter the genetic code for any of these proteases and change which amino acid is presence at any given place. Doing that, researchers have changed, for example, a serine residue of 221, which is the serine, gives it its name, to an alanine. Or changing the histidine position 64 to an alanine. Or changing aspartic acid at position 32 to an alanine. Or changing all 3 to an alanine. And when they do that, and they compare the activity, so this is the log of Kcat. So Kcat of course is a good measure of velocity and the wild type enzyme has an activity up here. I know this is a log scale. So this isn't like these are half, this is 1, 2, 3, 4, 5, 6, 7 orders of magnitude, meaning that the wild type is 10 million times more active than the enzyme that has its serine changed to an alanine, okay? So obviously that serine is a very important residue. You might wonder how in the world it even happens if it doesn't have a serine, and I won't go into that, but this tells us that serine is very, very critical for the catalytic acid. One 10 millionth as active when that serine is changed to an alanine. What this graph also tells us is that histidine is also very critical. And that's not surprising because, as you saw in that catalytic, excuse me, in that catalytic mechanism, histidine had to pull that protein, protein, proton off of the serine to make the alkoxide ion. If you don't have something that can pull that proton off, then it's a much tougher go. Yes? Student: Is the 221, is that what amino acid number is? Ahern: That's the number of the amino acid in each case. So without something that can pull that proton off, the enzyme is just as dead in the water as if it didn't have a proton that could be pulled off in the first place. So these are important. When we look at removing the aspartic acid, we still see about 5 or 4 or 5 orders of magnitude lower. But this tells us that that aspartic acid residue is not as important as the other two. The other 2 are much more important. And that's not totally surprising because the aspartic acid was mainly crowding that histidine. That tells us it's not absolutely essential for its catalysis although it does play some importance in that. Yes, Neil? Student: [inaudible] Ahern: You're saying would this be equivalent to what would happen if I put DIPF there, is that your question? Student: No, [inaudible] Ahern: Would this one respond to DIPF? Student: [inaudible] Ahern: This one? Would it respond to DIPF? You mean if I made the mutation? Student: Yeah. Ahern: Well if I made the mutation, I wouldn't have serine in there anymore, right? So it would not be possible to treat with DIPF. I can only treat something that has serine and expect DIPF to have an effect. Yes, sir? Student: Are there studies done on [inaudible] such as this where you can replace the serine, histidine, or aspartic acid with something other than alanine and perhaps increase the function? Ahern: Okay, so it's actually a very good question. His question is what if I mean if I go from serine to alanine, at least they look similar, but they're not chemically similar. What if I change the serine to a threonine for example? Threonine also has an hydroxyl group. The question is A, would I see activity? and B, might it even be better? and the answer is it's all possible, of course, but my production would be, in this case, that it would not be, because we're thinking about very precise orientations inside of that active site and threonine is going to have a slightly different configuration. It's possible it will be better but my suspicion is it will probably be somewhere in here. And his question is also good because when we think about the process that gave rise to this proteases in the first place, they were mutation and selection. Mutation and selection. We think of mutation as mostly being detrimental because most mutations give rise to things that aren't functional, kind of like what these individual ones are here. But some mutations actually give rise to more functional enzyme. That's how enzymes evolved evolutionarily in the first place and so that tells us something that it's possible, it could happen. But histidine to alanine here, that's a pretty big change. What if I put something like let's say a tyrosine here that might have a lot of electrons and ability to perhaps influence that proton. That might have an intermediate effect or some other effect. Make sense? Yes, sir? Student: Why is it just as ineffective when you replace as three as when you take off one them? Ahern: Very good question. Why is this one the same as this? Why would that be? It says that essentially once we disable these, we're pretty much dead in the water. I mean, one ten millionth is there. And that the contributing, these aren't additive effects but in fact this is any one of these mutations, well I guess they don't have that. Any one of these mutations alone is enough to knock the enzyme out essentially. You might say well why do you have any activity at all? And that's also a good question but remember we have a very little activity and we have the rest of the structure of the enzyme intact. We have the binding site, we have an environment in there that maybe, in addition to the specific amino acid we see in an environment there that's favoring to some extent these activation and breaking of peptide bonds. We're focusing on the nucleophilic attack and the other components to breaking peptide bonds that are still present in the structure of the enzyme. Does that answer your question? So that's an interesting observation. What I want to do now is show you some other proteases. So some related proteases. Subtilisin is a protease, it's an S1 protease, but there are other proteases that aren't S1 that behave very much like S1, well like serine proteases, alright? I keep saying S1, it's in my head. One of these classes of proteases is known as cysteine proteases. And the cysteine proteases, a good example is papain, it comes from papaya fruit. When we look at this active site, we see something interesting. We don't see a catalytic triad. We see, in this case, a catalytic diad. And the diad consists of a cysteine residue adjacent to a histidine. If I were to ask you to predict what you think happens in this catalytic mechanism, what do you suppose is going to happen? How will this enzyme function? I will give you a hint and tell you it also performs a nucleophilic attack very much like S, like the serine proteases perform a nucleophilic attack. Yes? Student: The nitrogen on histidine withdraws the electron from the sulfhydryl group on cysteine, which is then nucleophilic and attacks the substrate. Ahern: Basically yes, except you said electron. It's actually removing the proton. The histidine is removing the proton from the sulfhydryl cysteine, that makes a sulfur left behind with extra electrons. Those extra electrons are nucleophilic, they attack the peptide bond and very much like we saw, and I'll show you a mechanism in a second, very much like what we saw with the serine proteases, the cysteine proteases work the same way. So let's take a look at that mechanism and that mechanism is here. It's flipped because now the histidine is on the left instead of being on the right. But we see the histidine, there's exactly that same geometry that we saw with the serine proteases. There's that hydrogen out there for the taking. The binding of the proper substrate to the active site changes the orientation between the histidine and the side chain of the cysteine, making it favorable for those electrons in the histidine ring to pull off that proton from the cysteine side chain. That leaves behind a negatively charged sulfur ion. It's very reactive, it attacks the peptide bond and exactly all the other things we saw in the serine proteases happen. We form a covalent intermediate just like we saw before. We form, we see a fast step, we see a slow step, just like we saw before. So mechanistically, this class of proteases is essentially identical to that of the serine protease, at least for our level of understanding. Yes, Shannon? Student: When there's an X, does that mean collagen? Ahern: When there is an X, that means the rest of the molecule. Remember these are peptides that we're cutting here, right? Another class of proteases that is slightly different but interesting are called the aspartyl proteases. An example which is are renin. And the aspartyl proteases, at first glance, look somewhat different than the serine proteases but I hope to show you in the mechanisms some similarities that I think you will agree with. But those similarities aren't all the way through like we see with the cysteine proteases. How does this work? First of all we have, as you can see on the screen, in the active site we have 2 aspartic acid residues that are very close to each other. And between them, forming a bridge is a water molecule. One of the protons of the water molecule is attracted to the negative charge that's on the side chain of the aspartic acid and it doesn't matter for our purposes here which one. And when we look at the mechanism, we see this. Here we have one of the aspartic acids on the left, the other aspartic acids on the right. Here's the polypeptide chain that's up here. And here's the water being held in place. Now what's happening here, if we think about what happened with the cysteine proteases. [Ahern coughs] Excuse me. The cysteine proteases and the serine proteases, we had to extract a proton to make a reactive molecule. In the case of the serine proteases, we had an alkoxide ion. In the case of the cystein proteases, we had a negatively charged sulfur ion. What happens with the aspartyl proteases is very similar. We extract a proton using the electrons of the aspartic acid residue, creating a reactive hydroxide ion. It's got an extra electron, it's negatively charged and just like we saw in the other two cases, this acts as a nucleophile. The nucleophile attacks the peptide bond just as we saw before. The peptide bond falls apart and we've broken the polypeptide chain. Now there is one part of this mechanism I just described to you that is different from the other two classes of proteases. Does anybody recognize one thing that's going to be different here? Student: It flips which residue starts with the water between each one as it switches. Ahern: He says it flips between which residue starts the water and the answer is no. That's not the difference. Student: [inaudible] Ahern: What's that? Student: Is it still the same covalent bond being formed? Ahern: What covalent bond are we talking about? You're on the right track, that's why I'm asking you. Student: The intermediate. Ahern: She's right. If we think about the, for example, the serine proteases, we made an alkoxide ion, right? We had that oxygen atom that had an extra electron. It attacked the carbonyl group that caused one part to fall off and what happened to the oxygen? It became covalently bound to the other change and that chain is therefore stuck at the enzyme. That oxygen was attached to the rest of the enzyme, right? Do you see this water being attached to the rest of the enzyme? No. So we're not going to see a fast step and a slow step with this. Once we break the bond, we break the bond. There's nothing holding it to the enzyme such that the other chain has to be released. That's what made the slow step in the serine proteases. Everybody understand that? In the case of the serine and the cysteine proteases, the nucleophile was physically attached to the enzyme. The oxygen was the side chain of a serine, the sulfur was the side chain of a cysteine. Those are covalently attached to the enzyme. So when they become covalently attached to half of the polypeptide chain, that half the polypeptide chain has to get released. Here, water is not attached to anything. When water attacks that carbonyl group, the bond breaks, it's free. One step, bam. Everybody got that? Questions? Student: So it's all just fast? Is it all very fast? Ahern: I won't say it's fast or faster because remember we have to still get that proton off. I'm not trying to compare speeds, I'm just saying we won't have two steps, a fast step and a slow step. We won't have that with an aspartyl protease. Yes sir? Student: Does this reset itself by [inaudible] stealing the hydrogen off...[inaudible] Ahern: yeah, yeah. Student: Is that how it resets itself? [inaudible] Ahern: Yeah your question is a very good one. How does the enzyme reset itself. And if we think about the water that was added, one of the protons gets grabbed here. And the hydroxyl does the attack, but to break a peptide bond, we have to add both an OH and an H. And so that H can either come from here, that H can come from here. And then the enzyme to reset itself is going have to get back to its original state. So if it comes from here, then it's a simple matter. We've just gone back to the place where we were at. Let's move on to the 3rd class of proteases. The 3rd class. These are called metalloproteases. Now we're getting further and further away from serine proteases but I hope you see a common theme. An example of metalloproteases is thermolysin. These enzymes derive their name by virtue of the fact that they use a metal ion as their way of holding on to a water molecule. We'll see this theme come up also. A metal ion to hold onto a water molecule. We saw the aspartyl proteases that the water was in place and the water had to get positioned right there. The water played an important role in that catalytic process. The metalloproteases have a means of holding water there. That actually could make the enzyme more efficient because in the aspartyl proteases, that water could be bouncing around. That's why I didn't say they were faster. In the case of the metalloproteases, we've got something that's going to hold onto the water. The most common ion that's used is zinc. And I'm going to show you that mechanism here. In the case of metalloproteases, we're looking at the active site. And we remember that if we're going to break that peptide bond, we have to have a nucleophile. And that nucleophile and coming from water, we're going to make a reactive hydroxyl group just as we did for the aspartyl protease. But the way in which we make it is slightly different. First, water is bound by this zinc ion. The zinc, you'll notice, is positively charged. The positive charge of the zinc is attracted to the negative, or the relatively negative charge, partial charge, on the oxygen of the water. That interaction helps to hold water at the right place. And then, and this is going to vary from one enzyme to another, there is a side chain that will help to remove a proton from water. That might be a histidine, that might be another like lysine, or arginine, or something else capable of pulling a proton. That's listed on there as a B for base. So some other side chain is going to help in the removal of that proton from the water and again, once we've removed the proton, we've got a hydroxide behind that is very electron rich, attacks the peptide bond, and bang, we're off and running. Is this going to have a fast step and a slow step or just a fast step? Student: Two step. Fast and slow. Ahern: Fast and slow? What does it take to have a fast and slow? What's the slow step in the serine protease? Releasing the polypeptide chain from the enzyme, right? Do you see the polypeptide chain getting attached to the enzyme anywhere here? No, this only has one, this doesn't have a fast and a slow. This does not have a fast and a slow because there's no covalent intermediate with the enzyme. The water, again, is the attacker. The hydroxide in this case is the attacker. Student: Is the water bounded to the zinc? [inaudible] Ahern: The water is attracted to the zinc. Student: Okay, so it's not a covalent bond? Ahern: It's not a covalent bond of any sort, no. These are partial ionic interactions, not unlike a hydrogen bond, except for it's not a hydrogen that's involved. It's an oxygen. Hopefully what you see in that is a common theme. First of all, every single protease that I've shown you so far has a nucleophile. Every one. That nucleophile is what attacks the carbonyl group and it's the attack on that carbonyl group that results in the breakage of the peptide bond. That common theme goes through all of them. The only ways in which these enzymes which I've described to you so far different is the means by which they generate the nucleophile and what the nucleophile itself is. That's the only ways in which they differ. The basic mechanisms are the same. We created a nucleophile, the nucleophile attacks the carbonyl group, the peptide bond breaks, and the pieces go their way. Well this business of creating nucleophiles is not unique to proteases. There are other enzymes that use nucleophiles and generation of nucleophiles in their catalytic mechanisms. One of these is an enzyme we've been talking about some already, that's the carbonic anhydrase and that's right here. You guys look very tired. Ahern: Would you like a joke? Students: Yeah. Ahern: Would you like to stretch before the joke? Okay, so let's stretch. So this my, this is my magic genie joke, alright? My magic genie joke. This guy's walking along, down the street, kind of bummed. He kicks a bottle and all of a sudden realizes, "Whoa! What's that?" He grabs it, you know. Wipes the dust off of it. Of course in the process polishes this thing. And out pops this magic genie. And the magic genie says, "oh master, thank you, I will grant you 3 wishes." and the guy says, "oh, this is really great' he says, 'first of all, what do I want? "I guess I want a billion dollars "so that I can be a very rich man." and poof, a certificate appears in his hands and it says he has a billion dollars in a Swiss bank account. This is really good, right? Second wish he says, "I'd really like to be a very powerful guy." and poof, a certificate appears in his hands and he's the president of Apple, or Microsoft, it doesn't really matter. This is not a computer joke. Depends on where you lean, I suppose. "This is really awesome, "I'm a really powerful guy. "Third wish, I've got money, I've got power. "I want every woman to love me." and poof, he turns into a box of chocolates. [class laughing] And you thought it was going to be a dirty joke, didn't you. [class laughing] That's a dumb joke, I know. But it always gets a big laugh. I hate telling joke about sex or the other, but... Carbonic anhydrase, so you've heard about carbonic anhydrase already. I hope I've connived you it's a pretty amazing enzyme. carbonic anhydrase, you recall, was the enzyme I said when we first talking about enzymes that had a Kcat of a million per second. It can catalyze the conversion of a million molecules of substrate into product per second, per enzyme. Remarkable thing. Here's the reaction that it catalyzes. We see carbon dioxide plus water going to carbonic acid. Carbonic acid can then ionize and form bicarbonate. And that can happen with incredible rapidity under the right conditions. Under the right conditions. Now what does that mean? Well, there's something very odd that happens with this enzyme. Most enzymes have a fairly narrow pH range where they work that's ideal. And most of those ranges are set pretty much to correspond to the physiological environment in which we find them in the body. Most of our body tissues are at a pH of 7 to 7.4. Blood, for example, is about 7.4. We look at the activity of carbonic anhydrase, and by the way, you're seen Kcat changing here. Kcat will be constant for a given set of concentrations. If we change the concentrations, Kcat itself will change. We look at the Kcat of this enzyme, and we discover something very odd. We see, not surpassingly at a low pH, it's very low. We get to about pH 7 and we're looking maybe half a million per second. But this guy tops out at about pH 9, where we get up to a million per second. Why is this enzyme odd in this respect? Why is its Kcat so high at such a high pH that it really doesn't encounter in the body. Well the reason that this is the case is it gives us a clue in to a very important consideration in how the enzyme works. And so that's what I want to show you next. It's going to relate to the things I've been talking about. During the catalytic process inside of this enzyme, we see something happens that happened very much like what was happening the metalloproteases. That is we have a zinc ion, and then zinc ion functions to hold water so that we can create a nucleophile. It functions to hold water so we can create a nucleophile. The rate limiting step in the catalytic action of this enzyme like that of the proteases that you saw before, is the rate of formation of the nucleophile. The rate of formation of the nucleophile. The faster the nucleophile can form or the easier the nucleophile can form, the faster the enzyme is going to be. If we go to pH 9, we could imagine that it's going to be a lot easier to remove protons off of water than if we're at pH 7. We have more basic conditions, water's going to give up its protons more readily, and so at pH 9, we see that this enzyme is far more active than it is at pH 7, indicating that a very, very important step is the removal of that proton. That proton can come off many ways, but if the buffer is helping that proton to come off, it does even better. Does that make sense? I see a lot of nods, okay. My next question is do you suppose if we kept raising the pH, it would get faster and faster and faster? I see nos. They say "it's a trick question. "I know Ahern.", right? The answer being no, why? Within H of the enzyme. So at pH 9, the enzyme is still holding its shape enough that it's actually able to continue catalysis. When we get above that, we're going to see the ending drop off precipitously. So that's a very interesting step. This actually shows the mechanism and again, it's nothing like you haven't already seen. We see the water being bound. We see something pulling off that proton, whether it is just ionization of the water itself, which will happen in water to a limited extent, or it's like a side chain of a histidine, or a lysine, or an arginine that is pulling that off. Once we've got the nucleophile created, the nucleophile attacks the carbon dioxide as it defuses into the active site and that actually creates this, in this case, bicarbonate ion that is released. So it's the formation the nucleophile just as we've been seeing all along that is the critical step in this process. Would you product that aspartyl proteases might work better if we raise the pH on them? You can stick your neck out, I won't chop it off. I hear a no. Do I hear any yesses? What do aspartyl proteases have to do, folks? What about the metalloproteases? They both use water. How about metalloproteases? Do they work better at a higher pH? Probably would. They probably work a little better at a higher pH. What's going to determine if they work better at a higher pH or not? The stability enzyme structure. That's going to be the only limitation. If the enzyme structure is stable at pH 10, the enzyme will be way better at 10 than it is at 7. Because again it's easier to make that nucleophile at pH 10 than it is at pH 7. Yes, sir? Student: So that being in consideration, is there any movement, like the biotech industry to create man-made enzymes by taking a natural enzyme, sticking in a bunch of cysteines and crosslinking it back and forth like crazy and making it stable at a higher pH? Ahern: You should work in the biotech industry, sir. The answer is yes. There is a lot of interest in manipulating enzymes to infact increase their activity and increase their efficiency. In the case of carbonic anhydrase, though, remember what's limiting them is actually defusing it into the active site. It's probably not going to get much better than that million even if we were to improve that. But for a metalloproteases, it may very well be useful because now we can stabilize the enzyme with disulfite bonds, we can use it at a higher pH, that might very well be a strategy. You bet. We're slithering along through this. Let's take a, spend the last 5 or 6 minutes talking about restriction enzymes. We'll see some similarities we saw before. So you really hope, I hope you're starting to see the themes now. Restriction enzymes are enzymes that many of you have used if you've ever worked in a lab that does DNA work. And restriction enzymes catalyze the breakage of DNA double strains and specific nucleotide sequences. Restriction enzymes, they're also called restriction endonucleases, I'll take either one, that's fine. They catalyze the breakage of DNA double strands at specific nucleotide sequences. Now that's what they do. That's not how they do it. How do they do it? Well I will tell you that they hydrolyze them. They use water to break the bond. And that should give you some hint about the mechanism that they use and you can start seeing the wheels turn, thinking 'I'll wager that they're going "to make an activated water molecule in some way, "like taking a proton off, "making a nucleophile, "nucleophile's going to attack, "and that attack is going to result "in breakage of a bond." and you would be exactly correct. You would be exactly correct. Please turn your phone off, whoever has it on. Now let's take a little bit closer look at this in terms of how they operate. Actually that's not a very good figure. One of the things we see in restriction enzymes is that they all require magnesium for their action. They all require magnesium and magnesium, like zinc, is a divalent KCat ion and one of the things that magnesium helps to do is it does help to position the water so that it can lose a proton and make an attack on, in this case, a phosphodiester bond. We're not breaking peptide bonds, obviously. We're breaking phosphodiester bonds because those are the bonds between the adjacent nucleotides in a DNA molecule. Now I'll show you something that's really cool and interesting. Actually, I have to explain it to you because it doesn't show it very well. Let's take a look at this enzyme. There are many different restriction enzymes. This one is called ecoR-five. EcoRV. EcoRV recognizes the structure, the sequence GATATC and it cuts right in the middle of it. Now if you look at this carefully, you'll see that on the top strain, we go GATG, or ATC, on the bottom strain, we go to GATATC. These are what are called symmetric sequences. And these symmetric sequences are very common features of sites that restriction endonucleases recognize. It means that we can cut right in the middle of this guy and we've cut both of them. We just cut this guy right in the middle. Not all of them work in the middle, but this particular guy works in the middle. Now I want to explain to you just physically how a restriction enzyme works and then I'll show you very briefly a little bit of mechanism. A restriction enzyme is a protein. That protein grabs a hold of DNA. So many proteins will grab hold of DNA. DNA is a negatively charged molecule, we would expect that to grab a hold of DNA, perhaps positive or neutral, we certainly wouldn't expect the protein to be negatively charged because it wouldn't interact very well with DNA. The enzyme grabs a hold the DNA molecule and what it does is it literally slides down the DNA molecule. Now most of the time it's going down that trip down the DNA molecule, it does not encounter the sequence GATATC. In fact that sequence will occur randomly, only about once every 4,000 residues. So a lot of the time it's just floating along here, sliding itself down the DNA molecule and then all of a sudden in the binding side of the enzyme, GATATC is there. It's found the right site. What happened in the serine proteases or in any of the proteases or in any of the enzymes when the proper substrate bound? What happened physically inside those enzymes? Shape change, right? We saw a shape change that happened in those. Those very tiny shape changes caused the enzyme to have all of its properties. In the case of the serine proteases, that shape change resulted in the creation of the alkoxide ion. In the case of the restriction endonucleases, that shape change is more dramatic. What it does is it actually causes a bend to occur in the DNA. So we think of the DNA molecules of being straight and linear, but when the enzyme is bound to that proper site, the enzyme goes "oh, whoa!" and it bends. The DNA molecule is physically bent at that point. It's physically bent. Now that bending turns out to be critical for the catalytic action. Now, we can see that bending happening right here and when we analyze the structure that's present at that bending, what we see first of all is that there's a nice little water molecule that gets positioned right here, held in place by a magnesium right here. And that only happens when the bend occurs. Without the bend, there's no pocket for that holding to occur. So when the proper substrate has bound, the water's in place, the magnesium is in place, the proton can get removed and the nucleophile can be created. Okay, so we've said a lot today about nucleophiles. Once that nucleophile is created, it attacks the phosphodiester bond, the bond gets broken just like we saw with the peptide bond and everybody's happy. I will very briefly go over that next time and I will see you on Friday. [END]
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Medical_Lectures
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How_to_Create_a_Differential_Diagnosis_Part_3_of_3.txt
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hello this is part three of a guide to clinical reasoning or how to create an accurate differential diagnosis from a patient's presentation to remind you here is our practical five-step bedside approach in this part I will demonstrate how to use the approach with an intern level case as I did in part two I present this patient to you the same way that an intern might present the patient to his or her attending on rounds or to their colleagues during a morning report or teaching conference again I'll keep a running list of the key features as I go for practice consider writing down the key features while listening to the presentation but not watching it I also would consider stopping the video before the discussion of the problem representation framework and application of key features to first see what you can come up with on your own the chief complaint is a 58 year old man presenting with recurrent chest discomfort over the past two weeks mr. Jones is a 58 year old smoker with a history of hypertension who is in his usual state of health until two weeks prior to admission at which time he experienced in the episode of chest discomfort the pain came on over the course of several minutes while he was moving some boxes in his garage it is located in the center of his chest and he qualitatively describes it as a dull aching and heaviness he stopped moving the boxes and the pain gradually subsided over the course of five minutes while he was resting he also was a little lightheaded during the episode he denies any radiation of the pain or other associated symptoms including shortness of breath nausea vomiting diaphoresis or palpitations since that initial episode he has had approximately five more similar episodes each lasting about 10 minutes in total and each associated with varying degrees of lightheadedness some episodes occurred with some form of exertion such as walking up a flight of stairs but at least two began at rest including one on the morning of admission that lasted longer about 20 minutes his past medical history is notable for hypertension and obesity his past surgical history includes a cholecystectomy 10 years ago and a right knee replacement one month ago medications are lisinopril 20 milligrams daily the top roll all 25 milligrams B ID aspirin 81 milligrams daily and simvastatin 20 milligrams qhs for his social history he lives with his wife in San Jose and has two children all of their health is excellent he is a retired mechanical engineer he continues to smoke one pack per day as he has for the past 40 years but does not drink alcohol his family history is notable for a father who passed away at 76 from pneumonia his mother is alive at 83 with diabetes on physical exam he is a moderately obese man in no apparent physical distress but appears slightly anxious his temperature is ninety seven point nine Fahrenheit heart rate 86 blood pressure 152 over 92 respiratory rates 20 and oxygen Sat 94% on room air his BMI is 35 H E and T exam is normal cephalic with normal oral pharynx neck is supple with no breweries no thyroid megali or nodules and no lymph adenopathy his chest has normal excursion is clear to percussion in auscultation bilaterally his cardiac exam shows a regular rate and rhythm s1 and s2 are normal there is an s4 hurt at the apex but no s3 there is a 2 out of 6 uniform systolic murmur heard best at the left lower sternal border and without radiation is jbp is not visible presumably secondary to his habitus there is no chest wall tenderness abdominal exam is non distended there is no tenderness regarding spleen is non palpable liver is non palpable his musculoskeletal exam shows full range of motion at all joints and no bony abnormalities he has no edema in his lower extremities on neural exam his mental status is fully intact cranial nerves 2 through 12 are intact and one was not specifically tested patellar ankle and biceps reflexes are two plus bilaterally motor strength five out of five and all major muscle groups gait was normal on labs his CBC complete metabolic panel troponin CK and dua are all normal a urine tox screen was also negative a chest x-ray was also normal his EKG shows normal sinus rhythm with a QRS axis of negative 45 degrees high voltage and T wave flattening in leads 1 AVL v5 and v6 there is no pathologic q waves or ST segment deviation so I just completed steps 1 & 2 the next step in our five-step approach is creating the problem representation remember the problem representation is a one to two-sentence summary using precise medical terminology of the most relevant key features of the patient's history exam and diagnostic tests these are the key features that I feel are the most important in this case for the problem representation for this patient we would start with mr. Jones say 58 year old man since one of the goals the problem representation is to be concise if it's possible to group together or summarize related parts of the history you should sow hypertension obesity and act of smoking become multiple cardiovascular risk factors the surgery may become relevant so let's include that as well when listing the key features of the chief complaint I've already started using some semantic qualifiers so if I continue that trend we would say that he is presenting with acute episodic inconsistently exertional chest discomfort finally the highly relevant diagnostic data includes the unremarkable vitals which we can summarize as he is hemodynamically stable which to a much lesser degree also includes the fact that there were no signs on exam or labs of shock he is mildly hypoxic and we'll end by summarizing the normal cardiac enzymes lack of Q waves and lack of ST segment deviation as without evidence of acute infarction so altogether mr. Jones is a 58 year old man with multiple cardiovascular risk factors and recent surgery presenting with acute episodic inconsistently exertional chest discomfort he is hemodynamically stable mildly hypoxic and without evidence of acute infarction okay now let's go back to our five-step method we're now at step four adopt the framework what framework should we use here I'm going to display our problem representation again since the framework we choose is essentially driven by this one of the major reasons for creating a problem representation is to help with framework selection the most common framework for chest pain is one based on the organs or anatomic structures in the chest and upper abdomen for example structures in the chest include the heart lungs a order esophagus and chest wall here are some specific diagnoses related to those structures that can cause chest pain and in the abdomen we have the stomach the small and large bowel and some relevant diagnoses this is conceptually similar to the framework used for abdominal pain in the last case although I said that this is a common framework to use for chest pain and I think it would be similar to the one most missions would use it's not the one I personally would choose for this specific problem representation I would not classify using this framework as a mistake per se I just happen to think there's a better option let me read that first sentence of the problem representation again mr. Jones is a 58 year old man with multiple cardiovascular risk factors presenting with acute episodic inconsistently exertional chest discomfort this description suggests that mr. Jones doesn't have undifferentiated chest pain but rather he may have something quite a bit more specific unstable angina that is chest discomfort that is specifically caused by lack of adequate blood flow to part of the heart which is either new or getting worse with time since this problem representation is more specific than chest pain the best framework to use should be more specific as well therefore I would use this one instead which focuses on the mechanisms of angina rather than anatomic structures and organs this will reinforce an improved understanding of the patient's underlying pathophysiology we have angina caused by decreased oxygen supply engine the caused by increased oxygen demand and then angina mimics also while you may notice many diagnoses have shown up in both frameworks by using the most appropriate framework additional potentially relevant diagnoses have been suggested that you might otherwise not have considered two additional notes about this framework first the general category of angina due to increased oxygen demand is frequently referred to as demand ischemia second the diagnosis at the top of the decreased oxygen supply category is listed as unstable plaque from CA d or coronary artery disease that's sort of equivalent to common usage of the term unstable angina the problem is unstable angina is technically a symptom or a symptom syndrome and not necessarily a path of physiologic mechanism despite its frequent usage as such therefore while I'll be using the term unstable plaque moving forward many people would refer to this specific diagnosis as unstable angina or alternatively as acute coronary syndrome some of the terminology is definitely a bit confusing returning to the overview of the five-step approach we're now at the last step applying the key features from step two to the framework from Step four in order to generate a differential diagnosis so here's our framework on the left and our key features on the right there are two reasons why I consider this case to be an intern level case and not a student level one the first is because the optimal framework is one that is not based on anatomy but rather on physiology which is less commonly employed and requires a little bit more thought the second is because the provisional diagnosis seems so obvious that it requires conscious effort to simultaneously consider other plausible diagnosis without falling into the trap of including the complete list of every possible cause of chest pain to generate the focus differential the best place to start is with the diagnosis mentioned in the framework which is overall the most common cause of the chief complaint as characterized by the framework so in this case the framework categorizes mr. Jones's chief complaint as unstable angina and overall the most common etiology of unstable angina at least in a 58 year old man with multiple risk factors is an unstable plaque from coronary artery disease so we'll start with that diagnosis and try to link the most relevant key features to it the fact that the patient is a late middle-aged male raises the probability of unstable plaque many of the individual elements of his description of the chest discomfort slightly increase the chance of unstable plaque when looked at individually but when considered together collectively the chest pain story so to speak is almost textbook for unstable angina from an unstable coronary artery plaque likewise when each cardiovascular risk actor is analyzed in isolation each one slightly increases the probability that this is from CID but with a dramatic cumulative effect the physical exam is surprisingly unhelpful in diagnosing unstable Angela from CID the fact that the troponin and CK are normal actually have minimal to no impact on the probability of CID here since they are only elevated in infarction and not in the scheme iya however the EKGs lack of any evidence of present or past ischemic changes very mildly argues against CID in no way does that mean a patient can't have a normal EKG in either CID or even during an acute MI that certainly happens it's just that a normal EKG is more commonly seen in most other causes of chest discomfort than in CID so from the fact that we've already identified that unstable plaque is the most common cause of unstable angina in this age and gender with this collection of medical problems and from the relatively consistent chest pain story it's safe to conclude that unstable plaque from CID will probably end up as the most likely explanation for the patient's presentation and thus the provisional diagnosis for additional connections between key features and diagnoses that my brain might make at this point let me just run through each diagnosis as it's listed on the left the next diagnosis on the list is coronary vasospasm which is also known as variant angina and occasionally by the eponym of Prince metals angina coronary vasospasm is neither particularly common nor particularly rare among the patients key features there are two which are most relevant first he is an active smoker which is a risk factor for vasospasm and thus this argues in favor of the diagnosis second as vasospasm is typically not related to exercise the patient's observation that the episodes are occasionally exertional in nature mildly argues against this aside from its non relationship with exercise vasospasm otherwise symptomatically mimics engine uh from cid quite well and there are no typical exam a routine test abnormality Sene therefore the other key features are not particularly relevant for this specific diagnosis anomalous coronary Anatomy and coronary thrombosis Ammar both quite rare entities that I would not entertain in the absence of some elements of the presentation that specifically suggested them how about tachyarrhythmia a tachyarrhythmia such as paroxysmal atrial fibrillation could transiently increase the heart rate too dramatically elevated levels increasing oxygen demand beyond what the coronaries can supply leading to symptoms indistinguishable from angina the biggest argument in favor of the tachyrhythmia is its relatively high prevalence compared to most other items in the framework additional key features that argue for this diagnosis include the lightheadedness with some episodes lightheadedness is not particularly common with Angela from CID but can be caused by low cardiac output in the setting of severe tachycardia also the EKG has features of LVH which can lead to or is indicative of increased left ventricular pressures which then would eventually lead to left atrial enlargement predisposing the patient to arrhythmias such as a fib though that's a little bit of a stretch a key feature that mildly argues against an arrhythmia is the lack of associated palpitations but certainly there are people with paroxysmal afib and other tachyarrhythmias who lack palpitations but they are probably in the slight minority will pick up the pace a little for the remaining diagnoses a or text enosis is argued for by the presence of an s4 and the LVH on EKG both of which can be the consequence of a s however a s severe enough to cause angina almost always has physical findings of the diagnosis particularly a classic systolic harsh crescendo decrescendo murmur at the upper sternal borders radiating to the carotid although this patient has a systolic murmur its other characteristics are not consistent with a s that's not to say it's impossible to be caused by a s just unlikely for the most part the arguments for and against s are the same as for hyper the cardiomyopathy including a murmur that is not characteristic an additional consideration is that hypertrophic cardiomyopathy typically presents younger than this patient also overall hypertrophic cardiomyopathy is significantly less common than a s pheochromocytomas can definitely cause chest discomfort from excessive oxygen demand during episodes of extreme hypertension however in addition to being quite rare pheochromocytoma is almost always present with other symptoms during the episodes such as headache palpitations and tremor their absence combined with the rarity of the diagnosis and significantly more likely alternatives rules out Pheo surreptitious stimulant use is essentially ruled out by a negative urine tox screen there are certainly stimulants that aren't routinely tested for but nothing in the history is suggestive of this pulmonary hypertension is an interesting consideration to historical key features argue for this his obesity and his smoking which can lead to OSA and our obesity hyperventilation syndrome and COPD respectively all of which can cause pulmonary hypertension primarily through the mechanism of hypoxic vasoconstriction - exam features also argue for it his hypoxia and the systolic murmur the description of which is most consistent with tricuspid regurgitation which is always seen in severe pulmonary hypertension there are two key features to argue against it first the EKG the majority of patients with pulmonary hypertension severe enough to cause angina type symptoms have EKG findings of right ventricular hypertrophy failure or strain this patient has none of those also the patient has no shortness of breath although pulmonary hypertension is overall a relatively common and underappreciated diagnosis it usually presents with either shortness of breath or other findings of right heart failure which this patient lacks still as the only plausible etiology that clearly explains the hypoxia and the murmur it will probably make the cut when it comes to the differential now we move on to the Angela mimics which includes several don't miss diagnosis for aortic dissection although smoking is a risk factor pain from dissection is usually described as sharp or tearing and not as a heaviness it is rarely if ever episodic in nature and given that it is rapidly fatal if not immediately treated it seems highly unlikely the patient could be having symptoms for two weeks and not yet have died also although at lesser consideration a normal chest x-ray argues mildly against the diagnosis as a or dissection usually causes mediastinal widening overall although it's a don't miss diagnosis I think we can rule out a or Tek dissection on the basis of the highly inconsistent history and by the term ruled out I mean the diagnosis is so unlikely that despite its severity if it's missed it no longer merits further testing or further consideration another don't miss diagnosis is PE he has to PE risk factors the more obvious is the recent surgery the less obvious is his smoking another key feature in favor of PE is the hypoxia hypoxia does not impact the probability of a PE among patients presenting to the ER with shortness of breath as a chief complaint as many other causes of shortness of breath are associated with hypoxia however it most likely increases the probability of a PE among patients presenting to the ER with chest pain since the only other etiology of chest pain clearly associated with hypoxia is pulmonary hypertension does anything argue against PE first though to a lesser extent than pulmonary hypertension PE s usually present with some degree of shortness of breath pain from PE s also is usually neither episodic exertional nor mid sternal while none of these elements or key features in isolation argue very strongly against PE taken as a collective of features would say that the overall description of the discomfort is highly uncharacteristic of PE and although I did not previously consider it a key feature the lack of physical findings of a DVT also argue against the PE although the patient is not tachycardic as we would usually see with a PE he is also on a beta blocker so his normal heart rate cannot be counted against the diagnosis overall although this is less clear than it was with the aortic dissection I think the arguments against PE are strong enough that I probably would consider it ruled out at this point based on history and lack of DVT evidence anxiety is relatively common and of course it's suggested by the fact that the patient subjectively feels anxious however the exertional nature is not typical for this diagnosis and the chest discomfort description overall is not particularly consistent with anxiety though largely because it is so consistent with several other previously discussed diagnoses so although I don't think I'd consider anxiety completely rolled out it's definitely relatively low on the differential finally chest wall pain sometimes given the name costochondritis when it's localized to the costochondritis theresa jest this diagnosis and although I haven't previously listed it as a key feature if you remember from the exam there is no chest wall tenderness which argue significantly against the diagnosis so now we've applied the key features to the framework to adjust the likelihood of the various diagnosis away from their overall prevalence in the population it may have felt like quite the exhaustive and thorough process and in some aspects it was such as the time it took me to discuss my thought process the good news is that when you aren't discussing this out loud but rather thinking through these diagnoses and making those links to specific key features in your head this process is much much faster and only increases with speed and accuracy with experience and deliberate practice from this process we can construct our differential diagnosis as mentioned a few minutes ago the combination of his highly consistent symptom description his multiple risk factors and the overall high prevalence of CI D an unstable plaque from CA D usually referred to as unstable angina is clearly the most likely diagnosis next the combination of its prevalence in the general population the associated lightheadedness and the suggestion from the EKG that the patient may be prone to high intracardiac pressures and/or left atrial enlargement suggest a tachyrhythmia leading to paroxysmal demand ischemia as the second most likely diagnosis from this point forward the exact order and overall number of the remaining diagnoses is open for debate if we consider that the differential diagnosis should include all diagnoses whose likelihood or severity is high enough that they warrant additional testing and may also include one or more other diagnoses which are common but which you may not be specifically testing for at this time I would round out this patient's differential with coronary vasospasm pulmonary hypertension and anxiety while most differentials also include one or more so called don't miss diagnosis the only such don't miss diagnosis for this patient is the a for mentioned unstable plaque or unstable angina and unlike the case from Part two there are no unusual feature diagnoses in this differential I want to be clear that this is not the only right differential diagnosis nor even necessarily the best or most accurate it's just the one that's generated by the combination of my personal knowledge and my personal experience so that concludes part 3 of 3 of this video series on the clinical reasoning I hope you found it interesting and useful if you found these videos helpful please remember to give them a thumbs up and if you have any questions please post them as video comments I tried to personally answer all questions in a timely manner you
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Medical_Lectures
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27_Biochemistry_Glycogen_Metabolism_III_Metabolic_Melodies_Lecture_for_Kevin_Aherns_BB_450550.txt
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okay folks let's get started how's everybody doing pretty good Santa Claus has come to town and you know Santa does with naughty kids Hees them finals he gives them finals he gives them very evil finals is what he does okay so look out for Santa Claus he's really a really a bad guy uh let's see let's think about a couple of things in terms of announcements and we have a couple surprises today one of which is standing in front of you with all this on and there's more surprises as well um let's see first of all uh this is not a surprise it's been announced on the web page but I'll remind you that we have a review session tonight at 6:30 in ALS 401 so I will videotape that and I will uh get it posted uh maybe later this evening but most likely sometime tomorrow okay um and that's there for those of you who had regrades on exams um they are available for pickup in the BB office and you can pick them up there so that's um available to you and last but not least we have a final exam I have heard and the final exam is in this room on Monday at 9:30 a.m. so uh get here in plenty of time remember to position yourselves with seating as I said before so we SE sit in theice odd-numbered seats number one right here number one right here and number one on that row over there same thing is true if you sit up above okay um and uh you will get a full hour and 50 minutes to take uh the exam um and uh what was it going to say um it's got 150 points okay so I've written the exam it has 150 points it has um the first section which is the short answer has 75 points that's half of the exam the second section which is the problem solving section has 36 points and the last section which are the longer answer has 39 points now it hasn't been duped yet so there may be some changes and one of those changes I'm kind of got to decide today is will we have an extra credit question or not I don't know so maybe you guys can help convince me today that we should have an extra credit question I see well nodding yes doesn't do it it's got to be no that's okay I'm not I'm not waiting for Applause or anything here I I think you know what I like to to have when it comes to extra credit right so we'll we'll see how that plays out there might be some music today I don't know so that's that's that's possible okay um let's see so that's basically it uh the the format as I said is the same as before the point changes are different there are three questions in section two and there are three questions in section three okay and there are 25 questions in section one that tells you anything and um people always say is it hard is it easy do I make them harder Etc and the answer is I I can never tell I never would have guessed the last exam that the average would be 76.5 so I was very pleased so I'm probably the last person you want to ask if an exam is hard or easy I always try to write them in the same way I always do okay and I think this looks like a like an average exam in that respect if you did not get a note card on uh Wednesday when I passed them out uh I don't have them here you'll have to come to my office to get them from me I will remind you that with the exam you have to turn in a note card that you got from me okay you have to turn in a note card that you got from me even if you don't use it or you don't plan to use it you need to turn that note card in with your exam with your name on so if you don't have one make sure that you get one to turn in and know I won't bring them to the exam all right so you want yeah how do you know well if I told you how I knew they were mine then that would kind of give it away wouldn't it yeah so I know which ones are mine all right so make sure that you bring an a note card that I gave you to the exam if you don't or use a different card or something you will lose points and again I need to make sure that you're using these cards and you're not passing them on to your roommates because that's that's important um for them to have the benefit of filling out their card s next term or next year whatever okay we don't have an awful lot more to do that's very good so we'll have uh in fact if I finish early which I uh might I don't know we will U I'll take a few questions relative to the review session and we will have a nice surprise at the very end so that's uh all in in store today okay now um I've had several questions from I've had several questions from people about this rather confusing regulation of glycogen phosphor and so I just want to take a couple minutes and sort of summarize that for you okay so glycogen phosphor is U an enzyme that's regulated in several ways all right so when we talk about coent modification it exists in two forms it exists in the glycogen phosphor a form which is the form that has the phosphate on that people describe as the more active and it has the form without the phosphate known as the glycogen phosphor B that people describe as less active Okay um the reason people describe glycogen phosphor ASE a as the more active form is that usually in the cell when it's present it's present in the r State because only when glucose is present will it get converted to the t-state if glucose is not present or glucose is is present in very very low quantities then there will be no gluc glycogen phosphor as a in the T State R state of course being the much more active form glycogen phosphor B on the other hand okay is much more likely to be found in the T State right to convert glycogen phosphor a b into the r State requires what am okay it requires am and that isn't again very common inside of cells only when cells are really really low in energy are they going to have much in the form of am on the other hand ATP and glucose 6 phosphate convert it into the Tate quite readily and those are usually fairly abundant okay glucose 6 phosphate okay now so those things all come together now again I'm not going to ask you to rank them or do complex scenarios with these that's not the point of telling you all that information but it is important that you understand all those different types of Regulation it's very important that you understand those different types of Regulation and what's there so the phosphorilation def phosphorilation of course involve a kise or phosphatase right the r&t involve allosteric affectors okay all right okay so that's uh where we start now um there's a couple things about regulation that we haven't talked about and you say my God it's already you know murderous right well there's a nice organizing scheme to it the organizing scheme that we've seen is that in general phosphorilation of proteins is favoring the breakdown of glycogen and the stopping of synthesis of glycogen in general okay we phosphorate glycogen phosphor first all we we phosphorate phosphor chinise what does it do well it turns around makes it active and it phosphates glycogen phosphor as B to a what does glycogen phosphor as a do it breaks down glycogen protein kise a can also phosphorate protein kyese a can also phosph correlate glycogen synthes that converts it into the inactive form the synthes B because now it can't make glycogen that makes sense breakdown and making we don't want having going on at the same time def phosphorilation on the other hand favors okay in general again these are General things in general def phosphorilation favors synthesis of glycogen and inhibition of glycogen breakdown now lay out the schemes yourself and see that and it will make sense I think to you all right well what we haven't talked about hardly at all is that Def phosphorilation and Def phosphorilation has to itself be regulated just like everything else has to be regulated because if we don't regulate it then everything's going to be def phosphorated all the time and cells aren't going to have energy like they need okay so we need to think about that Def phosphorilation and that's the wrong slide okay here's what we're after okay now what you see on the screen is a um a scheme that shows the regulation of phosphoprotein phosphatase that's the pp1 okay pp1 what we're looking at here m stands for muscle so what we're looking at is a scheme that exists in muscle all right well what does this tell us it tells us first of all that pp1 is bound to another protein and the unfortunate name is this protein is called G subm in this case and we'll call it g subm that's not a g protein we've used the term G protein before to refer to proteins that bind to guanosine nucleotides this is not a g protein it's just holding on to phosphoprotein phosphatase phosph protein phosphatase okay now what happens well this phosphotase has to be regulated okay when this phosphatase is in linked to GM as you see it here it is the most active okay it is the most active so we're starting out over here we're a situation where we haven't been doing anything and then all of a sudden we get our epinephrine so we're going to follow from left to right what happens when epinephrine gets uh Bound by the cell surface receptor all right so we've got an active pp1 and we're going to ultimately convert it over here on the other side to an inactive form this is the most active form all right well and again we're looking in muscle what happens is when we have epinephrine being synthesized we see that this protein GM gets phosphorilated it's phosphorilated by there's our friend protein kyes a again phosphorilation of GM causes it to let go of pp1 now that actually makes pp1 less active it's most active when it's bound to GM it is less active when it's been released it's still active but it's not as active as it was over here okay what else happens well look what protein kyes a does protein kyes a phosphorites the inhibitor this is an inhibitor of the enzyme over here no inhibition over here the inhibitor gets phosphorilated and that makes it a perfect binding molecule for the phosphatase in this state over here it's blocking the active sight and the phosphatase is completely inhibited so why is this significant well let's think about this when I said we do phosphorilation we want to favor the breakdown of glycogen right if we want to favor the breakdown of glycogen we're putting phosphates onto things we sure as heck don't want phosphoprotein phosphatase one on taking phosphates off of things right because if this is active we put a phosphate onto glycogen phosphor a this guy is going to turn around and take it right off so this one system protein kisee a by going out in phosph forting all these proteins is favoring completely the breakdown of glycogen it's stopping the reversal of that process that is the removal of the phosphates from those proteins now I know that's a little confusing okay so I'm going to stop and take questions relevant to that or give you a moment to digest it perhaps nobody has any questions yeah the break of so this is consistent with the breakdown of glycogen and it's stopping that is the reversal of that breakdown those breakdown enzymes okay those breakdown enzymes are favored by phosphorilation the synthesis enzymes are inhibited by phosphorilation as long as those phosphates are sitting on those enzymes we're going to be breaking down glycogen and this keeps that reversal from happening that is the taking the phosphates off now when insulin comes along I don't have a scheme to show you insulin is going to stimulate a different phosphatase it's going to take this phosphate off and everything's going to go back over to the left when everything goes back over to the left you can see what's going to happen we're going to start taking phosphates off of everything when we take phosphates off of everything glycogen synthesis will be favored glycogen breakdown will be inhibited okay we're almost there there's one other surprising thing that shows up yeah Connie insulin from the and the or just insulin doesn't do anything except stimulate processes in the cell insulin is binding to the to a receptor outside the cell insulin doesn't come in so insulin is stimulating a process that will ultimately result in the removal of this phosphate but not that one this one is well yeah I'm sorry I'm sorry yeah sorry I didn't say that but this this phosphate is well sorry maybe I misunderstood your question yeah okay so that's pretty cool pretty darn cool now there's one last really interesting and odd thing about glycogen metabolism and then we'll actually be done with the regulation all right and that's right here this was a really big surprise somebody takes out and they're really interested in studying glycogen phosphor ASE a they're interested in studying U uh glycogen synthes and they put them in the same tube I've got purified pure glycogen phosphor I've got Pure glycogen synthes I put them together in a tube and I want to see what happens I add glucose and something very odd happens I see that by the addition of glucose the breakdown of glycogen phosphor I'm the conversion of glycogen phosphor ASE A to B occurs and the break the conversion of glycogen synthes B to a occurs what does it take to have that happen what does it take to convert this guy to this guy or this guy to this guy what's happening there the first person that answers that gets a free metabolic Melodies calendar for 2012 are they different active sites in the same nope what' you say def phosphorilation come by and get your a calendar okay this Santa Claus is here okay all right so def phosphorilation is happening but I only had two enzymes here how do I get DEF phosphorilation I have a free metabolic Melody CD for the person who answers that what's that one of them does whates one def phosphates the other answer is no if anybody read ahead their notes they know yeah read the question say the question again okay how does that def phosphorilation happen nobody read ahead in the notes oh man okay I'm gonna have to tell you you guys are G gonna miss a valuable you you could sell this on eBay CD you have you have does um does it synthesize glycogen us the does it synthesize glycogen using the phosphorus no it doesn't okay all right boy that's I figured somebody would jump up at that one it's actually on the next slide okay it turns out glycogen phosphor a in the liver carries around with it look what it's holding on to it's actually holding on to this G in the liver that has the phosphoprotein phosphatase all right when this guy is in the r state State this is blocked nothing can get in there and the phosphatase is inhibited when this is in the r State that's what happens what does adding glucose to glycogen phosphor a do it converts it to the t-state and when it converts it to the t-state look what happens it lets go of this guy so when it lets go with this guy this guy is just like the one with the GM it's completely active and what does it start doing it starts taking those phosphates off now glycogen phosphor a gets converted to glycogen phosphor ASE B glycogen synthes b gets converted to glycogen synthes a all by Def phosphorilation that's pretty darn cool all right now why is that important why is that important that this thing carries this thing around with it any thoughts about that should I put a c out for that there's a very important reason why this guy carries this around your book actually talks about it nobody's read the book this is quiets a Class B all term all right so why why are cells breaking down glycogen why are they doing thisy they need energy and why do they what's their typical needs for energy quick quick running from running from the grizzly bear right or the teacher that has is wearing a hat that's right either way okay I want this quick which means I want to break things down quick and what did we learn earlier in the term about enzymes with respect to speed they work very fast yes okay that's good what else did we learn what did I say about driving your Maserati to Fred Meyer you got to control them right if you turn something on and it's really powerful and it's going to break things down really quickly don't you want to be able to turn it off really quick carrying this carrying around its own inhibitor is the most efficient way that it can turn itself off really quick now notice this happens in the liver this is happening in the liver the liver is producing what what's one of the things that the liver produces that only one other tissue produces glucose glucose okay so let's think what the what the liver is doing the liver is sitting there this guy is dumping epinephrine it's making I'm sorry is is yeah is um I've got to get it right now he's he's U making epinephrine so two things are happening one is glucose one phosphat is being produced by the breakdown of glycogen the other thing that's happening is gluconeogenesis is being stimulated both of those converge in the synthesis of glucose one's a break down one's a synthesis process but they're both producing the same thing which is glucose what's the liver doing with that glucose where's where's the glucose going exporting it's exporting it's dumping out of the liver into the bloodstream and what's happening when it gets in the bloodstream tissues that need it it goes to tisues that need it and when I stop exercising my tissues don't need it what happens blood glucose levels start going up right do I want my blood glucose levels to go up no so when my blood glucose levels go up what starts happen happening to my liver's ability to push out glucose I don't want to push out anymore right so now the liver stops pushing out glucose and what happens to the concentration of glucose in the liver cell up and that's what we see right here this stops the liver from making too much glucose and it stops it very very quickly glucose is a poison we're avoiding the poison with this very very Rapid Control that we've got here it's a very very important Point okay everybody understand what I've just told you so being able to carry around its own off switch allows this guy to merily go along its way cutting through that glycogen like butter but once this starts to accumulate it literally gets turned off by the very thing that it's carrying that phosphatase the phosphatase in turn activates glycogen synthes and what's that going to do with the glucose it's going to make it into glycogen ultimately right and what's that going to happen to the concentration of glucose in the cell it's going to fall we've just reduced the amount of poison okay make sense clear as mud yes sir clear as mud you knows how you get the end of the term and your voice starts going up mine does I don't know I'm not sure what that means but that's what happens to my voice okay I've got one last thing to show you and maybe one more chance for somebody to win a SE a very valuable CD I will autograph these for you you can sell them on eBay I you know you don't know how much you might make from your knowledge here okay so knowledge is power knowledge may be money so you might meet some exciting people from having these things too you never hey come over to my house we can listen right or come Watch Me Turn the page of the calendar over that's what you're going to say right so okay um glycogen storage disease all right so we think about enzymes and we think about metabolism and we think about reactions we always think about what happens when things don't work what happens when we have problems well it turns out that there are a variety of what are called glycogen storage diseases and these diseases arise as a result of efficiency of certain enzymes either in that pathway or related to that pathway okay so um there are several of these voner disease uh this one lacks a glucose 6 phosphatase and affects liver and kidney and what happens to the glycogen well it's got an increased amount of glycogen but it has a normal structure well that sort of makes sense because this doesn't build glycogen is involved in doing other things what's this guy involved in doing anybody remember nobody's going to say until I say how about a CD right glucogenesis that's right okay so glucogenesis can affect glycogen metabolism all right so we may see some problems arising from that here's Pomp's disease it's lacking alpha1 14 glucosidase okay and my this is not going here um the one I want to talk about the one I think is the most interesting is actually mcardle's disease gunite wow uh it's it's a deficiency of glycogen phosphor now you think about this and you think how can a person be deficient in glycogen phosphor and be alive right well it turns out that this uh disease is um makes it's possible for a person to be alive by virtue of the fact that it only affects muscle cells that is only the muscle cells are lacking the glycogen phosphor all right lacking glycogen phosphor what happens when we look at the glyc there it's got moderately increased amount not surprising because this is what breaks glycogen down it's got a relatively normal structure meaning we've got branching enzyme and all that other stuff that's there okay but what's surprising is it has fairly limited effects limited ability to perform strenuous exercise because of painful muscle cramps otherwise patient is normal and welldeveloped a person lacking glycogen phosphor in their muscle loses ability to do strenuous exercise but that's the main thing if we look at what happens with the thing here's here's a a plot of the concentration of ADP that's what we get from the breakdown of ATP okay um for a person who has uh mardal disease all right and uh this person is doing this this is a person who doesn't have mardal disease this person has mardal disease look at the levels of ADP you rest your levels are low you do some exercise your ADP levels stay fairly constant the person with McArdle disease sees ADP levels go high and then it falls meaning that the cells are catching up and making ATP now for that last chance at a CD my question to you is what's making this possible it's something we learn during the term something we learned a very important process I'll give you a hint a very important process we learned during the term that allows these people who have this disease to lead a fairly normal life any guesses yes is that going to be fermentation in lactate is that going to be fermentation resulting in lactate it could be slightly but no that's not the right answer the it's the Cory cycle right what does the Corey cycle do glutose the liver has a normal enzyme right and so when the muscles start running out of energy what happens oh wow we need some glucose they can't get glucose from breaking down of glycogen but the liver sure as he can do that and the Cory cycle kicks in okay so in essence it's a a backwards thing to the lactate that you talked about but it's the fact that the phosphor in the liver is perfectly normal that makes sense you guys are looking like you're ready for the surprise or something okay yes sir is this something that's easily diagnosed is this something that's easily diagnosed I think it' be fairly easily di I I'm not a physician so I don't know I I would think it'd be fairly easy to diagnose because it would manifest itself pretty readily in the fact that you're really just you know as as a kid you're going to have problem with exercise and you're just worn out with this so yeah I think it would be fairly easy to to detect that other questions I need to look in the back of the room and see if my surprises are ready yet or not are the surprises ready are the surprises ready oh okay are you ready for a surprise okay at this point you can put your pens down and your pencils down because we're not going to do anything more that's going to be on the exam it'll be fun and if you really don't want to hear some bad singing you can leave also so I I will tell you that there's going to be some singing well this is a little bit different singing than you've heard before okay so um you guys know I like the Beatles I write a lot of stuff to Beatles music and you know the Beatles were like this but I really think they should have been like this you know I mean if this were an ideal world that's what it would have looked like you know they may not have sold as many of billions of records as they did but anyway so as I was putting this together I sat in I thought about you know the Beatles and so forth and then I thought you know that's really odd you know there's all these things out there that relate to music that involve the letter B okay so the Beatles I mean they were wow what awesome things and you guys aren't old enough but what succeeded The Beatles was this group called The Beggs anybody hear the bgs oh you heard the BS okay Night Fever Night Fever right okay so anyway and what succeeded the bgs was a really important group known as right the back street boy what a group huh and of course what succeeded them we know of course was um yeah low moment in music I think but if you thought you've seen low moments in music You Ain't Seen Nothing Yet let me introduce the next bees that are going to turn the music world upside down through the bio comical choir please come down I have some people to help me sing today so they're going to drown me out yes Applause is appropriate yes thank you thank you let me introduce our participants including one person in the class starting with Santa on the end this is Bonnie this is Heather this is Ki Shannon andir Andrea Valeria and Linda so uh thank them all for uh thank you all for coming to help me with this I thought about you know introducing them you know we think about flash mobs I thought maybe we could just call these guys the world's first flush mob you know but that might be a little mean so I didn't do that so okay so we're going to do some of Britney's favorite hits we're oops we're going to do it again and you guys know the rule about singing loud right okay so we have four songs for you today that we're going to sing yeah and are you going to have sing with us they're going to sing with you yes this this group has questions you you guys don't ask me questions my group up here ask me question question so this is this is good now the first song that we think about is the fact that how many people in here are really sick and tired of that Sunshine there we go okay these are the real oregonians in the room okay so we're sick and tired of that Sunshine we don't want that Sunshine anymore please join us in singing our first big hit known as Let It Rain oh the Oregon weather's Dy the is mostly cloudy you can't stop it if you complain so let it rain let it rain let it rain it doesn't show signs of slowing and it's RAR right for snowing and driving some folks insane let it rain let it rain let it rain when it finally turns out dry will be put away our reer it will probably be but I'll surely miss the Reindeer cuz the sound of the Falling Rain P down the makes music inside my brain so let it rain it rain let it rain okay I couldn't quite hear you guys Heather is going to pick the key for Heather will pick the key okay I've been told I'm not picking the key on the next one so okay we have someone here who's much more musically talented than I in fact I should put the microphone with her perhaps we do that who I'm destroying all my equipment here all right somebody put oh I'm not I don't want you don't want that okay all right maybe I'll just hold it close to you all right wrong one here anyway all right the next song um we learned a lot during the term about hemoglobin you heard a lot about what happens in the blood and how all those things work it's appropriate that we talk about the bloody things okay this is an old song that many of you may not know and some people here may not generation I grew up with this ad it's an old Coca-Cola ad it goes Heather S put some oy [Applause] [Music] ring Yank on his the globe and shapes will change a bit what a sight to see the way they toy cooperatively and as I exit from the lungs to swim in the blood stream metabolizing they allpress their needs to to them I give upy change from R to [Music] while the [Applause] protons but that's not all the tricks I know more [Music] G when [Music] C the protons andat from L2 that's the way it is when are at go all right two down okay the next one may sound a little familiar to you and with good reason so I I this one I'm expect to hear a lot of voices on it's about sering proteases all right okay okay hea Heather started go for SES work almost identically Amino a TR tight changing their structure when they S1 then there are elect shs at the AC as theaction next [Applause] theide elak without ACH so one piece is bound to it the get set free has to act next toag where it started waiting for a pepti chain that it can itself to go and start all again okay now I don't know about you guys but I don't I can't really hear that much out out here you know do you notice and I don't know I didn't tell but they got on the final exam if they want to have an extra credit question that one of the rules was I had to really be able to hear you know like loud like louder than you've sung before okay you do know you will know the last one it's a it's a very easy Christmas tune it's to the tune of winter wonderland okay it's a song I wrote about this class we sang it last year and I can tell you the group that sang it last year they were Magnificent the bar is very high very high okay are we going to Belt it out I can't hear you all right all right let's go with it it's called BB Wonderland Heather take us no we're all starting all right ready one Anna two mil Hall dirty and they gety he walks to and not louder [Music] started MP3's got added to my iPod some sometimes were and exams when the Cur turned out I don't think it's so my scores are too low sliding by finally there's examination on December 5m I'll have my car pack with information so I don't have to memorize it and I'll feel like aart with my party just one to go and then ho I'll be all right I think you got it thank you thank you thank you see you on Monday not soon thanks guys [Music] [Applause] oh yes thank you thank you [Music] sh if if you're too crowded for the final and I don't get a chance to say it I hope you have a great holiday and I will see you next time thank you indeed you too study hard okay it goes how you doing uh they're back in my office come back and see me my office there now uh I'm going there in a minute yeah I got to take everything down but [Music] yeah e e e
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Medical_Lectures
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How_Bacteria_Cause_Disease.txt
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good evening everybody and welcome my name is Carol Fox and I'm the co-director of UCSF mini medical school which is brought to you by the Osher Lifelong Learning Institute here at UCSF MiniMed is always a popular UCSF program and we're delighted to see so many of you here tonight a lot of you I know are people who have been to our other mini med courses so welcome again we designed mini med to take you through a consolidated version of the core curriculum for second-year medical students and the course is taught by the same distinguished faculty here at UCSF who teach our medical students this year's mini med is structured around the medical students a three course on infection inflammation and immunity these biological processes are the core of a large number of the world's most pressing health problems and the course will cover key topics in microbiology immunology infectious diseases and international health we're privileged to have as our opening night speaker dr. Warren Levinson he's a professor of microbiology and immunology here at UCSF he lectured at our first mini med program in 1999 and so we're delighted that he's consented to come back and do a return lecture for us it's a different lecture but we're glad to have them he joined our faculty here at UCSF in 1967 and he's received many of our distinguished teaching awards here on the campus including long including outstanding faculty major contribution in teaching and inspirational teacher Awards so I think we're in for a treat tonight he's been the leader of our microbiology and infectious disease course for the medical students since 1981 and he was also instrumental in helping us put together this course we had long conversations on the telephone about what should be included in the course and who our speakers should be it's a matter of fact our speaker for next week I understand is hiding here in the audience in the back dr. Levinson is also an author and the ninth edition of his textbook which is right here medical bike-rack Medical Microbiology and immunology for any of you who want to purchase it I was published earlier this year I want to be put to sleep I'm in a commercial but it is one of the best-selling medical textbooks of its kind and it's been in publication now for a number of years so since it was just published we know that we're getting the most up-to-date current information that there is to be had on this topic in his non-academic life I always like to know what people do in their off time so we always ask I think in the BIOS that were provided to you it says that he was on the City Council of Mill Valley for a number of years and was also the mayor of Mill Valley for those of you who might be from Marin and I also learned today that he and his wife are ballroom dancers and that they just got back from Argentina where they did the Argentine tango and have been doing since so if that's probably a topic for another lecture right but I always like doing these introductions because I learned a lot of fascinating things about our faculty that you don't hear otherwise and please join me in welcoming dr. Warren Levinson who will tell us about how bacteria cause disease thank you thank you very much for that kind introduction Carol and welcome to the first in this series of six lectures on infection inflammation and immunity so as Carol said I am the leader of this course for the second-year students and in fact they just had their final exam yesterday we used to say that they all did well okay so one of the tasks of the first speaker in a series is to set the stage for the remaining lectures and so for the first portion of this talk I'm going to discuss describe to you the world of microbes these all of the organisms that cause human infectious diseases and I'll show you how they differ and compare and contrast them will then move on to infectious diseases themselves and how physicians think about them and describe many aspects of their the pathogenesis the way in which disease is caused and and then move on specifically in the last section to how bacteria cause disease and you'll see the various ways in which that happens so let's address the first thought and to to emphasize to you the importance of infectious diseases so here are the causes of death worldwide it's a little bit out of date but the data hasn't changed very much this is from World Health Organization and you see at the top of the list is infectious diseases vectors and parasitic diseases but but basically infectious diseases worldwide more than cardiovascular diseases and cancer put together that's not true in the United States and fexif disease is much less important but worldwide we are talking about the most common cause of death just to emphasize the importance of these infectious diseases so now let's talk about the cast of characters as I've termed it so starting from the most complex the largest group we have the worms and I'll show you a few of those and I see some faces scrunching up at that and then we moved lesser complex smaller the protozoa again I'll show you examples and discuss each of these then the fungi and then bacteria we're getting much smaller and much simpler and lastly the virus is the smallest of all of them and I'll show you images of all of these as we go along so here are the microbes that cause infectious diseases we have the animal world we have the worms technically called the helminths so they are eukaryotic meaning that they have a true nucleus the DNA the chromosomes are inside a nuclear membrane there's a cytoplasm cell membrane there are sort of classic eukaryotic cells and as worms they are multicellular many cells and so the organism that you know is the tapeworm and I'll show you a picture of that in a moment there's also the round worm the asterisks also very important and widespread these hundreds of millions of people throughout the world are infected by these organisms so here's the tapeworm so now you see why it's called the tapeworm looks like in fact a long tape the one interesting comment is that this little head is how it attaches to your intestinal tract and then these multiple feet long worm and here's for example this is one of the segments here's another segment and you can see the outline there so the hundreds of segments of these what are technically called pro gluttons now make up the body of this tapeworm and here's the round worm so we're now this is colonoscopy we have a scope in looking and here's the person's intestinal tract and what is that is that a round a law that has now living in this person's colon yes it is and and when you treat someone with these round worm infestations and paralyze the worms and then they expel these worms these are the worms from one person so when you read in the paper about the worm burden you see where that term comes from the try to imagine trying to function trying to do a day's work or be productive with this worm burden in your intestinal tract now a few of these worms up just parenthetically don't cause symptoms but when they are of this magnitude than a testicle obstruction or something similar may result okay so now we're moving again down the scale both in size and complexity here are the protozoa which are eukaryotic true nucleus but now our single cells in contrast to the worms which are multicellular these are now single cells and they have the capacity to move and the organism you know is Giardia you're very familiar with that term you might not know is Plasmodium but it's the cause of malaria and I'll show you images of those so here's Giardia single cell here are the flagella that allow it to move and here these are the suckers by which it attaches to your intestinal tract mucosa thereby disturbing the intestinal motility and causing the watery diarrhea so characteristic of Giardia infections oh here's Plasmodium the cause of malaria and here it is in the red blood cells all of these structures are in fact the organism the Plasmodium that caused malaria this is a so-called signet-ring stage for obvious purposes here's the little gem and here's the ring portion called a signet ring stage very characteristic yeah infection now so again single cells a protozoan parasite moving now to the other group of the fungi they have a true nucleus they are single cells but now they are non motile they don't move they don't have flagella and the organizing GLE cells that reproduce now by budding in the yeast category and molds that are in that reproduced by mitosis I'll show you examples of each of these the organism that you know is Canada which causes thrush main use or familiar with that term but the organism because it's such a beautiful yeast form that reproduces by budding is Cryptococcus very important cause of meningitis particularly in patients where HIV infected have who have a low cd4 count now so here's the body of the yeast and here is the little bud so this is unequal division it's not mitosis where one cell makes two identical daughter cells this these yeast reproduce by producing a little bud and so that's characteristic of the yeast forms the mold forms on the other hand are long filaments of cells technically called the hyphy that reproduced by mitosis where one cell makes two and the organism that you know because it's the green mold on the orange that sits in the back of the refrigerator for a month or two you've all experienced that is Aspergillus very important cause of human disease now here it is under the microscope and these long hyphy long strands each one of these is a single cell divided by these cross walls or septa as the technical term let me just parenthetically say in what I think is apt really remarkable is that some fungi have the capability of switching back and forth between these two modes absolutely remarkable that it can as a function of temperature be in the mold form in the soil and in the yeast form in the body and organisms that do that have you heard of valley fever coccidioidomycosis can change its shape in this fashion as well as histoplasmosis those of you who come from the Ohio Mississippi valleys you know that histoplasmosis is very common there that organism also can change its morphology change its shape change its appearance bold form in the soil and a yeast form in the bodies absolutely remarkable capability of these organisms okay so now moving yet further down in size and complexity we have the bacteria and again we're going to focus on that a little bit later but these now are prokaryotic organisms so these now don't have a true nucleus they don't have a nuclear membrane the chromosome lies free in the cytoplasm much simpler organism yet you'll see how complex and sophisticated they are when we address them specifically and they reproduce by binary fission and the P I'll show you a few here's streptococcus you're all familiar with strep throat or the flesh-eating streptococcus that ate my arm when you're at the checkout counter okay so this is so this is the streptococcus so here these are our red blood cells here's one here's another and these small blue spheres are these streptococcus repto means chains so you can see that these are in Chains here's our quite a long in some short chains very characteristic appearance of these streptococci so bacteria again much smaller than our human cells and lastly we in our description of the organisms that cause human disease are the viruses so they are non cellular there's no cytoplasm no nucleus they they are non cellular and grow only within cells they are what we term obligate intracellular parasites that is to say it is obligatory that they enter cells intracellular within cells in order to make more of themselves in order to replicate in order to grow so one virus enters our cell and a thousand or hundred thousand progeny viruses are produced a much different mode of replication than binary fission or mitosis where one cell makes two or even budding where one cell makes two but of different sizes this is now a completely different mechanism a mode of growth of replication so what again viruses non-cellular grow only within cells because they do not have the ability to make their own energy and they do not have the ability to make their own proteins they depend on our cells to make the virus proteins so they again are obligate intracellular parasites okay because viruses are so fascinating I just want to spend a moment on them normally we think of viruses is something very small and have some rather hazy idea and think that they're quite simple but in fact they are remarkably complicated and interesting let me just give you a little bit of a feeling for that so here's a coli which you're all familiar with it's been in the news these are the spinach outbreak so here's here's E coli it's and here's the small one of the smallest bacteria call chlamydia cause of sexually transmitted disease very important and here are the viruses so heat on this side are the DNA viruses viruses that have DNA as the genetic material on the right hand side are the viruses that have RNA as their genetic material and let me make one point and that is this is the viruses are the only place in the biological world where RNA is the genetic material all other organisms we the the worms the protozoa all the other ones that I've mentioned so far use DNA is the genetic material here's a whole group of viruses and I'll give you some names in a moment that have that use RNA as a genetic which are completely new and different idea so here we are as far as size is concerned here's the chlamydia one of the smallest bacteria one of the smallest free-living organisms and here are the PAC's viruses so-called smallpox virus is the one that you're familiar with although molluscum contagiosum virus which some of you may have heard of is also a member of this family and here's its DNA inside a little protein coat and having these special structures here so immediately a little bit more complex than one might think and then here's herpes viruses again the DNA's is centered inside this protein coat with an envelope surrounding it here's adenovirus the cause of sore throats and pneumonia looking very much like a lunar lander landing device with its fibers sticking out from the surface here's the polar viruses human papilloma virus which you've read in the paper recently there's a vaccine now against this virus this viruses that cause several strains of cancer carcinoma of the cervix very important now to have a vaccine against that Mars falls into this category and then lastly the parvo viruses the cause of slapped cheek syndrome some of you may have had children with this disease where the cheeks of the child now are quite red minor minor disease unless it occurs in a pregnant woman who can transmit the virus to the fetus but that's another story the the point I'm making is not so much that you remember which organisms these are but look at the size difference at least a tenfold difference in size and the complexity of the structures and a similar situation here on the RNA viruses the viruses of measles and mumps fall into this category this is rabies virus it looks very much like a bullet that's the way we describe it here's influenza virus poliovirus and human hepatitis A virus fall into this category the thought is again that these viruses vary significantly in their size and complexity they are not just simple little items much more complex than we normally think of here's influenza virus very important these days a to the threat of avian influenza so here it's RNA coiled up in the center surrounded by a lipid envelope and then these spikes protruding from the outside and that's the way the virus attaches to our human cells through these proteins that protrude from the surface so contrast I'm just going to give you two examples contrast that with the appearance of Ebola virus the cause of hemorrhagic fever in Africa of own lethality of perhaps 50% so completely different in its appearance long rods very long rods with the RNA you can almost see the central structure here and the RNA is coiled up inside in a helical fashion again just to give you a sense that viruses differ greatly in their size and and complexity okay with that introduction of the organisms that not only i but others in this lecture series will be speaking of let's move on to the infectious disease and begin discussion of that so why do people get infectious diseases so I structured it such that there are factors that are innate or belong to the organism and then factors that we have host factors that the whole the host factors will be discussed in the immunology lecture but I just want to touch on the fact that we have both innate defenses defenses that are present all the time that are active immediately and that are nonspecific in the sense that they don't care whether it's bacterial X or bacteria Y or virus X or fungus they recognize it as foreign and engulf it and kill it that those fall into the innate category and then the adaptive counters are antibodies and T cells which are exquisitely specific for that particular virus and even that particular strain of that virus remarkably powerful and highly specific so we basically have the front line here the initial defenses that say whoa wait a minute I recognize you as foreign and I'm going to engulf you and destroy you but if the dose of the organism the numbers of organisms or if the particular virulence and I'll define that on the next slide is is so great that it overwhelms our host defenses then as it says infectious diseases occur when organism factors overcome host defenses it's fairly obvious but I just thought I would say it for the sake of completeness so let's talk since the the focus of this talk is on the organism and let's talk just say a word about dose and virulence factors so the dose obviously is the number of organisms and again oh ISM at the higher the dose the greater the risk of disease not nothing much new there but what is new and interesting is virulence because this is now the ability to cause disease which varies greatly from organism to organism and within a particular organism and that's going to be a recurrent theme and I'll emphasize it as we go along and it depends on the or individual organisms ability to produce these various variants factors such as the capsule pili exotoxin endotoxin glycocalyx these are technical terms I'm going to define them show them to you I just want to introduce those terms now at the beginning and they are often encoded on plasmids which are extra chromosomal DNA and I'll again I'll emphasize I'll show you an example of that but the thought is that here's the DNA of the bacteria and that is sort of what it begins with or the starting point and then new pieces of DNA can enter the bacteria through a variety of mechanisms that I'll describe and endow that organism with new characteristics new traits new ability to cause disease new ability to make virulence factors that makes the organism a more aggressive pathogen more aggressive in its ability to cause human disease so this is one of the most remarkable images and all of biology and all of medicine so this is e coli and it was ruptured and the DNA was allowed to to be you exit the organism and then stained in a particular way in an electron micrograph so all of this is the DNA of the bacteria just think about the packaging aspect of all of this DNA has to get back it came from this organism something that's quite remarkable in and of itself but wait a minute what's that and what's that and what's that are these plasmids are these extra chromosomal pieces of DNA that can carry the genes that encode the virulence factors that I've just been talking about the answer is yes they not only encode the variance factors but also encode antibiotic resistance so when an organism becomes resistant to an antibiotic it is often because a new plasmid has entered the cytoplasm of this organism and now has encoded has now caused the production of various enzymes that provide antibiotic resistance so again here's the DNA of the organism but there are these additional pieces of DNA that this organism and this organism only has acquired okay say a few words now about how we think about infectious diseases because it tells us a little bit about how you as a person can think about it but also how physicians think about these diseases so the incubation period you've all aware of that term but let me define it and it has some implications so it's the period during which there are no symptoms but the organism is growing and often spreading systemically throughout your system into the bloodstream and into various internal organs and sometimes even to the skin so a lot now can happen but but there's but there's no symptom you don't even know that you are going to be sick now comes the prodromal period very important idea that there are nonspecific symptoms of fever loss of appetite just not feeling well muscle aching and we are often able to communicate the organism during this stage it's a very important concept in terms of public health but an ie people typically go to work at this time people interact with with friends and so on but yet are often communicable ie capable of spreading the organism at that stage okay then comes the stage of specific illness you get a rash weasels chicken pox we get a cough ramonja diarrhea from some viral or bacterial pathogen depending upon the organism we are also communicable at that stage and then we get better symptoms are resolving but now the antibody amount of antibody in our blood is increasing and this is often used as evidence that we have been infected with that organism a very important idea that during this time let's say we have a diagnostic dilemma we've tried this and we've tried this and tried to diagnose and we've taken blood and analyze this and allies that and we haven't made a diagnosis so what what not not common but but an important idea when when needed is that we take a blood sample now and then a week later we come back and take another blood sample and ask whether the amount of antibody against that specific organism has increased it is significant to a significant degree and if so you were in that patient was infected with that organism at this time so an important idea that during the convalescent period if there's a diagnostic problem then the rise in antibody titer can be very useful as far as diagnosis intuitively you would think of persons getting better they are less communicable that that is certainly true and then the period of recovery where health is restored but we should always remember that there are some diseases where a chronic carrier state occurs and you're all familiar with typhoid mary who have the salmonella typhi organism in her gall bladder and then excreting in the bile into the GI tract so that she was now spreading the organism to she was a cook okay and services preparing the food and an outbreak of typhoid occurred in people who ate at her restaurant people with hepatitis B and hepatitis C are often carriers of those particular viruses so we need to be aware that at least with some diseases people can be chronic carriers and think of the public health important public health implications of that and of knowing that and haven't been of educating the person about health habits and and hygiene washing hands not sharing toothbrushes and things of that sort okay does my patient have an infectious disease its address that question for a moment so fever well we all recognize that fever is an accompaniment of infection however let me quickly point out that of the big three diseases big three categories of diseases infection autoimmune diseases and malignancy and cancer the something like 80% of all diagnoses are fall into that category all three can manifest with fever so fever is a very nonspecific criteria well what about inflammation I'm going to say more about inflammation in a moment but you all know that something is red swollen tender warm well those of you know someone with rheumatoid arthritis and autoimmune disease know that the joints in fact are inflamed yet this is not an infectious disease this is an autoimmune disease so even inflammation is not by definition or very useful as far as making the diagnosis of an infectious disease is concerned well what about an exudate or a discharge an extra day let's say pus was and I'll have more to say about pus in a moment so we have an abscess was filled with this fluid or exudate is the technical term or sputum someone is coughing up sputum or your thrill discharge or a vaginal discharge now we're getting into something that is more specific for infections and then lastly an increase in the white blood cell count in the blood is often occurs in infection so putting all of these together if a person has this constellation of findings I think we can with some confidence using these criteria saying that we should investigate this problem from the point of view of an infectious disease okay let me move now to the address inflammation since we have infection inflammation and immunity let's talk a little bit about information for a moment so inflammation is the body's response to the presence of an infectious agent the affected area is red warm swollen and tender and the P show you rather dramatically this arm is inflamed even you would admit that it could make that diagnosis so this is streptococcal cellulitis step 2 cockle skin infection where the streptococcus organism has now gotten both into the skin and under the skin and has spread rapidly throughout the entire arm love you let me go back for a moment what inflammation is a two-edged sword on the face of it you would say no this is terrible this information is is causing pain obviously discomfort limitation and motion but it in fact has a beneficial component so well I don't understand how could that be in that the redness and swelling reflects increased blood flow bringing phagocytes and antibodies to the site of infection so in fact it is part of our host defenses to bring these beneficial components phagocytes antibodies there are other things complement variety of other components of the blood to the site of infection however it is detrimental in that the symptoms of some diseases result from the inflammatory response but the underlying idea is that this is a beneficial host response trying to combat the infection so how does this work so here on the left is the normal situation the non inflamed so this is that tissue here is the very small blood vessel carrying red or oxygenated blood and here are the capillaries and here this is the surrounding tissue and occasionally there's a lymphocyte or macrophage macrophage means big eater macro big phage to eat so it is a big eater in two senses one it can ingest and kill many different organisms and but it is also a large cell a macro cell larger may be half again as large as many of the normal white blood cells okay so this is the normal situation so now if should there be the presence of a bacteria or offending organism the area now becomes inflamed the arteriole is larger so more blood supply that's why it's red it more blood supply that's why it's warm more blood is coming to the area and more white blood cells now a migrate exit the blood vessels and now get into the tissue where putatively there would be a bacteria here or there and now this neutrophil and I'll describe just describe the various blood cells in a moment but this is our major in addition to the macrophages our other major phagocytic cell now he migrates and now engulfs and hopefully kills the offending organism here are the capillaries and in the process now of this inflammation fluid leaks out from the capillaries into the tissue spaces now causing the swelling so we have explained the redness increased blood supply the warmth increased blood supply and then the swelling from increased fluid migrating from the blood vessels out into the tissue spaces depicted here so again inflammation being now had looked upon as a beneficial act on the part of the host to defend against these organisms so here's a good example of inflammation it's now an abscess which is a walled off area of pus this is from a recent issue of the New England Journal but here's this man's inner arm it's obviously inflamed there's an abscess containing pus inside this lesion okay and so it better leaders show you another example now of pus I want to address what that word means so here's a brain cross-section of a brain containing an abscess actually see two of them and notice that the inner lining of this abscess is a walled off area of pus you can see the thick wall here and this greenish mucoid material which in fact is the pus so what is pus something might have asked yourself maybe not every day but occasionally it's a mixture of bacteria human cell debris and phagocytes so the phagocytes that are coming in there to engulf and kill the organism and the phagocytes almost always are neutrophils and again on I'll describe those in a moment but one question that I always get from the medical students is why is it this yellowish green color well turns out we know the neutrophils contain or possess an enzyme called myeloperoxidase that's involved in the killing of the ingested bacteria myeloperoxidase makes the hypochlorite known to you as bleach so bleach very important oxidizing agent killer of microorganisms so inside these neutrophils is an enzyme that making a tiny amount of hypochlorite a bleach that's actually going to kill the ingested microbe and well it turns out that Milo peroxidase when you purify that enzyme and have an intestine is green and that's the reason why pus has that yellowish green color I promised you a description of some of these blood cells very important in infection so here are the neutrophils this is the terms the names are based on the color of the granules in the cytoplasm of these cells because they and the color is determined by the contents the nature of the contents so here are our red blood cells and here is a neutrophil by the way these are the lobes of the nucleus the nucleus now instead of being one nucleus and now has segmented into these into three examples here sometimes it's force sometimes it's two but notice now that the granules by which these cells are defined are named have a kind of neutral beige kind of color when stained with this particular dye okay so these are our major phagocytic cells along with the macrophages but these are the neutrophils that are in highest concentration in our blood some 60% more than half of the cells in our blood are these neutrophils so very important phagocytic cells defending against infection should an organism get into the bloodstream it's there to engulf should an organism get into our skin the again the neutrophils migrate from the capillaries into the infected tissue but there are other cells that you may know and recognize or at least are aware of their effects so these are of cinah fills because the granules in the cytoplasm here's the nucleus but now here are all these granules are color eosinophils defend us against worm parasites but also are involved in allergies so for those of you who have hay fever or allergic conjunctivitis or allergies of once or another if we took a sample of let's say the nasal discharge from you with allergic rhinitis and looked and stained it we would see many eosinophils because those are the organisms that are in those are the cells that are involved in allergies as well as defending against word parasites and then lastly are the basophils again because now these granules in the cytoplasm are stained purple with this particular dye and these are now involved those of you who have had allergies skin allergies where let's say food allergies and you have a hive you have a swelling in your skin these are the cells that release the histamine that causes these the swelling in the skin okay so sufficient to at least have an awareness that there are these different cells that perform different functions the most important from our perspective from an infectious point of view are the neutrophils alright so we've discussed the various organisms that cause infectious diseases we've talked about inflammation a little bit about infectious diseases let's move on now to how bacteria look getting to the topic of how bacteria cause disease so here's a model of a bacterium let's start at 12 o'clock and go around so here's the chromosome lying free in the cytoplasm single strand of DNA single circular strand of DNA and now coming around here's the outer capsules one of our most important virulence factors it says okay I'm going to coat myself with this special covering such that when that neutrophil comes along it can't gain can't get purchase it can't really engulf sort of slides off we fight back with antibody against the capsule that now has a handle on it so to speak that allows now the phagocyte to engulf because we have receptors on the surface of the neutrophil for the antibody that now allows it to engulf that will be talked about in the immunity lecture but just want to introduce that idea suffice if a bacteria cannot predict a capsule it cannot cause disease it is engulfed and killed by our neutrophils very efficiently so that possessing a capsule so-called anti phagocytic factor the capsule extremely important as far as endowing and organisms the ability to cause disease next is the cell wall this is the site of action of some of our most important antibiotics such as penicillin x' and cephalosporins they prevent the organism from making the cell wall so I'll just mention that in passing it will be discussed I'm sure next week at the next lecture not so much not so important as a variance factor for us tonight but but extremely important in the life of the organism here's the cytoplasm where the proteins are made various other energy components are made moving around here's the plasmid I talked I emphasized the importance of that this extra chromosomal piece of DNA that encodes two very important proteins some of these various factors the exotoxins in particular that I'm going to talk about in a moment and also the enzymes that provide antibiotic resistance you've heard I'm sure whether that Staphylococcus or Neisseria going to really cause of gonorrhea that they are now resistant to penicillin why because they have acquired a plasmid that now encodes an enzyme that Cleaves the penicillin ring and destroys the antibiotic so plasmids again very important both from the point of view of encoding virulence factors as well as encoding antibiotic resistance Vaghela knots not so important it allows certain organisms to be motile but not important for our discussion conclusions typical also ribosomes these site of action of important antibiotics but not important for us at the moment but the pilis this hairlike projection that protrudes from the surface of the organism that allows the bacteria to attach to our cells extraordinarily important organisms that cannot make a pillows are not pathogenic we cannot cause human disease they must have the ability to attach to our you kozol cells to our surface cells in order to colonize and then to continue the infection so on these various structures in the bacteria the capsule and the pillows are extremely important variance factors I'll have more to say about several others in a moment ok so let me tell you a little bit about bacteria in in general they have three forms there are round spheres I'll show you electron micrograph images of these in a moment the round spheres are called I streptococci staphylococci gonna i noumic oxide these are all different bacteria that have these round spherical shape there are rods called bacillus for example the cause of anthrax is called bacillus anthracis a large rod then there are eco lies also a rod-shaped vector is something that you are familiar with and then there are the spiral-shaped organisms called the spirochetes Treponema pallidum the cause of syphilis it falls into this category so let's have a look at some of these so here's an electron micrograph of the streptococci so here's a round organism and remember they are in Chains I showed you those chains previously here's another round one round round round organism so the eye are the spherical organisms and we recognize these in the microscope very important fundamental property of these organisms these on the other hand are rod shaped by the way the weather that's round or elongated is determined by the nature of the cell wall by the particulars chemicals amino acids and the sugars that are in the cell wall determines the shape whether it's spherical whether its rod shaped so these are clearly different these long rods another long rod it's on so these are the Visayas happens to be Coley and then lastly here's the spirochete of Treponema pallidum looking very beautiful like a corkscrew very unusual shape the organism of Lyme disease you've heard of that Falls it looks like this falls into this category so these are the three fundamental shapes that we recognize let's talk now about how bacteria cause disease and getting into some of the specifics so the fundamentals of pathogenesis pathogenesis to produce disease the genesis of patho the genesis of disease so I like to think of it in terms of transmission our organisms transmitted to us and then what happens once they are transmitted they need to colonize they need to attach to our cells and grow there make more of themselves and then they invade tissue inducing either inflammation which by itself causes the symptoms of the disease or they produce toxins which again mediate the disease and I'll give you several examples of each of those talk a little bit about transmission so here in this image we have the infected host with these various organisms and we have human to human transmission and then sort of non-human so we are aware that contact personal contact with Earth's shaking hands sexual transmission or by sneezing coughing can transmit organisms to the new host there's also the form of indirect contact where food is now contaminated by the person who is infected and then transmitted and whether it's needles or variety of other inanimate objects all of these now can transmit there's also organisms whether they're insects whether they're infected animals very important source of infection for humans and whether it's household pets or domestic animals and forms of cattle and sheep or whether it's rodent in terms of rodent fleas and so on then now can transmit the organism the key is we must be aware how the organisms are transmitted to us in order to break the chain of transmission absolutely critical idea we would much prefer to prevent disease than to have someone get sick and then treat them with antibiotics or so one of the fundamental ideas in prevention aside from vaccines and sometimes prophylactic drugs is to recognize how the organism is transmitted and to interrupt that chain so we need to know where the organisms come from they're usually acquired from an outside source I noticed that usually because I'm going to contrast that a moment called the reservoir so the reservoir is out there whether it's animals or soil of water most often other humans and animals but also soil food water can be involved however infections are also caused by bacteria that live in or on our skin mouth and colon they are called the normal flora so right now on our skin in our mouth in our gastrointestinal tract that those of you are interested in topology recognize that the inside of the colon is actually outside of the body strictly speaking that the that the if you can think of it this way that the bacteria that are in the colon are really fine they're they're just having a grand time about replicating and feeding off the the goodies that we eat but they do not invade or they they are do not cause disease on they invade the body see what I mean let's say enter tissue inside leave their space in the lumen of the colon in the in the in the open area in the Bell and then enter into the that's when they cause disease okay so we all have these normal flora we all are on our skin we all have the same kinds of organism and in fact they actually do a good job in providing what's called colonization resistance that is to say the sites on my skin are occupied by these nice staphylococci the good form and it says to any pathogen that might come along sorry this site is occupied I'm binding to this receptor and there's no room for you okay unless I get a cut or something where the organisms now the bad Staphylococcus Africa's aureus now is implanted into my tissues sort of subverting the the nice colonization resistance that's provided on my skin or that's provided on my GI tract or my mouth and so on so these normal four actually have a reasonable function and we all again we all have them and so long as they remain in their appropriate place we're fine with that and they are fine with us okay so that's a little bit about transmission and this is sort of the classic picture is this going to transmit organisms uh so just imagine how many viruses can be on some of these little particles so we need to interrupt the chain of transmission right so this is what is recommended so just to make a dramatic statement in that regard um also interrupting the chain of transmission depicted here the crude death rate for infectious disease United States from 1900 to 1996 so here we are in 1908 hundred people died of infectious diseases private 1,000 population per year and notice that the remarkable decline has occurred over the next 40 years or so even before antibiotics were introduced here's penicillin in the mid 1940s so why does that happen we don't have antibiotics whose little smallpox vaccine was the only vaccine that was readily available because we interrupted the chain of transmission because we disinfected we chlorinated the water supply because we had proper sewage disposal because we refrigerated food because we pasteurized milk all of these precede these public health procedures interrupted the chain of transmission and provided remarkable decline in the incidence of infectious diseases in this country even before antibiotics and other medical measures came along okay so we've now let's say putative Lee transmitted the organism organism now needs to colonize and grow so the bacteria attached by the pill I I mentioned those before very important these protein fibers that stick out from the surface and I'll show you those in a moment two receptors on our human cells and we do that right now so here's the pneumococcus streptococcus pneumoniae d by far the most common cause of pneumonia overall cause of pneumonia but which is one of the leading causes of hospitalizations and death here's this pneumococcus and all of these fibers are the pillow that now provide the attachment of that organism to our respiratory tract cells and allow the organism to attach and then continue to grow so very important again if you don't make a pellet that organism any mutant that doesn't make a pilis is non-pathogenic cannot cause disease because it cannot attach then there's this technical term it's likely complicated but it's called a glycocalyx that helps adhere it to the surface this is sugar Cody and let me describe the importance of that let's say we have a damaged heart valve and we have some organisms from the mouth that now get into the bloodstream and have the potential for causing endocarditis a very important potentially lethal potentially fatal disease but picture now let's say we have a damaged a quartic heart valve and each time the heart contracts this volume of blood now swooshes past the the damaged part of the aortic valve so here's an organism from my mouth gotten in the bloodstream and now is trying to hang on for dear life with its pilis and shoosh you know all the the that organism is going to be dislodged from its site on the valve unless it plasters itself down so the organisms that cause endocarditis are the glycocalyx producing strains so in the same organism but unless it makes this particular what's called a slime layer the glycocalyx or or protective coating over itself it will not cause disease because it gets dislodged it might colonize momentarily but the force of the blood dislodges it so that's the most dramatic example of the importance of this virulence factor here's the capsule we talked about that before that protects the bacteria from phagocytosis organisms that do not produce a capsule cannot cause human disease and then lastly the bacteria form a biofilm that protects the bacteria from phagocytosis and antibiotics and it's a combination of these various factors let me just show you an image of that so this is now the inside of a catheter placed in a person intravenous catheter and some organisms from the person's skin in the process of passing the catheter through the skin some organisms have now gotten onto the into the tip of the needle and then into the catheter so here are some bacteria and here are some yeast much larger organisms but you see this sort of a matron a film of all this material now that has coated the inside of the catheter that is now when it gets a little bit larger and now we'll cover these organisms and limit the ability of antibiotics that are in the bloodstream now to get to these organisms and kill them also limit the ability of neutrophils to get to these organisms and kill them so the very very contemporary idea of the importance of this biofilm in the ability of bacteria to cause disease okay so now that we have inducing inflammation the presence of the organism detected by the macrophages the big eaters that I described that then secrete proteins technical terms cytokines that say okay neutrophils I'm in trouble here saito meaning cell and kind meaning to move I want you to move on neutrophils out in the bloodstream I want you to move to do the area here where I am trying to engulf these bacteria we need to we need some help in from the big eaters at the end and from the neutrophils so now ingest and kill the organisms and as a result inflammation by the processes that I just described earlier now occur and the resulting fluid and the exudate now we're getting to the detrimental aspect of inflammation where the fluid and the exude a decreases organ function and the most dramatic example of that is in pneumonia so let's say your important for the oxygen that I inhale to get into what's called the alveolus the little sac that's right next to the bloodstream and to exit the capillary and the oxygen has to diffuse across that membrane and get into the bloodstream to be picked up by the red blood cell and then taken to wherever it's going to be oxygen they're going to be used well if this inflammation now blocks the passage of the oxygen from the alveolus into the bloodstream that's not good right that's clearly not so you become short of breath that's one of the main symptoms of pneumonia and it's because the oxygen is not passing into the red into the capillaries into the bloodstream into the red blood cells properly now some so some bacteria cause disease solely by producing inflammation others produce disease by producing toxins and many of the toxins cause disease by themselves even in the absence of the bacteria themselves and I'm about to describe how that happens so we're now moving to the sort of last phase of the talking and that is to talk about how the toxins work so there are two basic forms the endotoxins and the exotoxins the endotoxins are Lippo polysaccharides so they have a lipid component and a sugar component saccharides as in Sakura sugar substitute but I want to press that point the Lippo polysaccharides in that are in the cell wall of the bacteria I'll say more about that and typically the bacteria need to be present in order for this to occur and so let's talk about the endotoxins for a few moments so Whipple polysaccharide abbreviated LPS and no toxin major cause of death people who enter the hospital with severe diseases bacteria in the bloodstream go to the emergency room go to the intensive care unit often die from endotoxin being shocked and so it's the main cause of septic shock fever shock meeting hypotension low blood pressure and this is a technical term and I'll describe it but it has a fairly long name disseminated intravascular coagulation abbreviated di C so what that means is that in the millions of capillaries disseminated through all over the body we're having clots form tiny little clots that are inside the blood vessels and so now think about if you had a clot there you can't have oxygen exchange capillary is blocked and so oxygen can't get into the liver or the kidney or the brain properly and so the function of that organ deteriorates and declines and that that's why people die is we can handle the fever we can support the blood pressure variety of methods but people died because from septic shock from this endotoxin mediation because of the clots that are in these tiny blood vessels that prevent the oxygen from getting into the tissue and now the organ and tissue can't function you can see how that could happen and I'll just mention that the lipid portion is actually the most toxic part let me just for a moment contrast what we call gram positive bacteria and gram-negative bacteria not getting into the specifics of why they are but simply to say that in the grand positive bacteria there is a very thick cell wall and no endotoxin no lipid Lippo polysaccharide outer portion whereas in the gram-negative bacteria they have a very thin cell wall and an outer membrane that is the place where the endotoxin is located so if physician nor you hear people talk about gram-negative shock or gram-negative septic shock or gram-negative rods sepsis or any of those terms they describe the idea that these bacteria now have on their outer surface this outer membrane that contains this Lippo polysaccharide that is the cause of the fever the hypotension the low blood pressure and the di see the disseminated intravascular coagulation that actually is the lethal event and then just to close this segment here's a blow-up of the gram-negative cell wall and here is the thin peptidoglycan the thin cell wall and now external to that is the interior the bacteria here is the lipid and the polysaccharide portion of the liberal polysaccharide the endotoxin that i just talked about and just to finish endotoxin experiences two main mechanisms it activates the macrophages to produce these various factors interleukin 1 tumor necrosis factor nitric oxide that mediate these the fever and the shock and also activates the coagulation cascade to cause the disseminated intravascular coagulation so this now is the constellation of symptoms that compose septic shock one of the leading causes of of death in hospitalized people just to emphasize the drama of this disseminated intravascular coagulation this patient had meningococcemia a gram-negative caucus with endotoxin and because of the these tiny little clots in the small vessels in the fingers also in the toes but not depicted the this person now has gangrene of the digits of the external portion which is this this person will lose the scan Grannis portion of these on these fingers and clearly this is inflamed it's on because of the tiny than millions of tiny clots in all of these small capillaries dramatic example of disseminated intravascular coagulation okay so that's endotoxin let's describe now some exotoxins these are secreted polypeptides so a completely different kind of molecule and they can cause the disease with the exotoxin alone you don't even need the bacteria you can purify the exotoxin leave the bacteria behind inject the exotoxin into automatic person but let's say an experimental animal and reproduce the disease so that's an interesting aspect these are now secreted by the organism I just want to give you a couple of examples of this and then a few last ideas and we'll close so here's botulinum toxin known to most of you as Botox this is what's being injected into your wrinkles okay so what's the remarkable mechanism of action by this organism called Clostridium botulinum so here we are at the neuromuscular Junction the place where the nerve ends and the muscle begins so i'm contracting my arm because i'm sending a signal down this nerve I'm now releasing the acetylcholine mediator it contained in these vesicles it goes across I'll describe that subscribe in more detail here now goes across the synapse technical terms with a space between the neuron and the muscle cell and now attaches to receptors and I can contract my muscle so here on the left hand side of this diagram is the normal way my muscle is contracting here's the acetic holding the red circles and contained in a vesicle it's just sitting there and along comes the impulse down the nerve cell and says okay release acetylcholine vesicle moves to the surface here comes the acetylcholine across the Sinan's synapse here's the acetylcholine receptor and now I can contract my arm botulinum toxin on the right hand side and here's the toxin binds to the surface of the neuron with this purple binding protein now is incorporated into the cell and the enzyme which is a protein something that cleaves proteins and is highly specific now Cleaves one of the releasing proteins names not important but it now is a protease that destroys this yellow protein and sorry the vesicle can no longer release the acetylcholine and I get paralysis I can't move my arm similarly that's what's happening in the injection in the wrinkles is that the muscle is relaxing and it smooths out the skin and that's the plastic surgery or plastic aspect of a cosmetic aspect of the use of Botox and that's the mechanism so again we are getting a what's called a flaccid paralysis and I want to contrast that with tetanus where you get what's called a paralysis spasm and why does that happen tetanus we're all familiar with tetanus we've all been immunized okay this organism now also produces an EXO toxin and but now we're in the spinal cord and we're blocking the mediators that different from acetic choline that now activate the inhibitory neurons I don't want to get into too much detail but let me let me just describe this so my ability to contract my arm is dependent not only on the ability to contract but also for the ex these muscles to relax otherwise I couldn't do that right so something is telling these triceps and these other muscles relaxed at the appropriate moment well it's these inhibitory neurons that say okay triceps relaxed because I want to contract my bicep tetanus the the tetanus exotoxin prevents this inhibitory neuron from acting so the if the inhibitory neurons are inhibited only the excitatory neurons are active and I get tetanus or technique or paralysis illustrated by this unfortunate gentleman whose powerful extensor muscles of the back so he's touching the ground by his heels and the tip of his head from the extrude ordinary power of these two paralysis of these powerful extensor muscles in the back okay so now here's a question for you I pose this to the medical students this is a cartoon from The New Yorker honey let's lay off the Botox for a while shall we very cute but wrong right because she should be flaccid not uh-huh so now you can um next time you see New Yorker cartoon you can think about that's okay we're one more toxin and then a few last thoughts and then we'll close so here's cholera very important disease kills millions of people worldwide extraordinary diarrhea watery die prolonged watery diarrhea and also exotoxin mediated so here's the binding proteins and would keep our eye on the red one that's the actual active portion of the toxin and we are in the lumen of the intestine in the open portion here's the Yoko's the rest of the body inside binds to the surface of the intestinal tract and in injects or introduces the active molecule which now stimulates adenyl cyclase enzyme very important enzyme to make more adenylate cyclase for those who you know about this you know it's an important signal transducing agent important mediator that that transmits the signal from the surface of the cell into the interior often to the nucleus to tell it to do something very extraordinarily important but in this case this adenyl cyclase enzyme is turned on irreversibly it just keeps making cyclic AMP E and there is a protein in in the intestinal cell which under normal circumstances is very valuable it has allows water to leave and a certain amount of sodium a certain amount of potassium in order to keep the balance of the cells and the balance between the ion content and the blood and the ion content in the stool it's functioning normally but when it gets turned on irreversibly in large amounts tremendous amounts of water and sodium and potassium so on leave the tissue and end up in the lumen of the bowel and now manifest says watery diarrhea persistent day after day of a basically watery diarrhea that that dehydrates and kills the person all mediated by this exotoxin that activates the adenyl cyclase in just a word about cholera so here are three views this child obviously is comatose eyes are sunken requiring intravenous fluids in order to activate here's a child who better off able still able to drink able to hydrate and no doubt will recover and then lastly this is called a cholera cut such that the person with cholera is too weak to get up and literally go to the bathroom so there's an opening here and the diarrheal stool persistent effluent just passes into buckets essentially underneath the cot until the person either gets better or doesn't get better so that's a sort of a unique aspect of cholera okay we are coming to the close I want to just end with two important ideas and that is I mean I mentioned that exotoxins now can are made by the organism and can be liberated by the organism and now get into cells and do whatever damage they have to do the botulinum toxin or the tetanus toxin or the cholera toxin so that's depicted here on the left hand side where here's the organism and here's the toxin and now it binds the receptor gets into the cell releases the enzyme that's going to attack the target and do whatever damage if it's going to do however some organisms clever Devils that they are have said well this is inefficient to just release it out I'm going to bind to the surface of the human cell and inject my toxin directly much more efficient much more effective so this is the technical terms pathogen dependent entry not not not important for us except to know that these organisms that have acquired this particular injection mechanism are much more virulent than the ones that do not mind you but having said that botulinum toxin is the most toxic substance known to humankind molecule for molecule because it's an enzyme and can keep on working it is the most toxic substance known as I say okay lastly okay I want to describe how a an organism that is present in our intestinal tract as Eureka coli that you all know can live perfectly normally in in a mutually compatible relationship with us and yet be an extraordinarily important cause of disease you say well how can that be what's depicted here it's because something I've touched on before it has acquired new genes Lou DNA so watch what happens here's this ordinary E coli but now if here's a plasmid talked about that before and now if this organism has acquired this plasmid that makes this particular protein it now can cause dysentery dysentery is bloody diarrhea often fatal or cancer we can be fatal alternatively it can acquire a different set of genes and now cause watery diarrhea you all know about Teresa when we go to certain countries and we get this watery diarrhea syndrome from eating something that's this ecoli that is making this different toxin that now is causing watery diarrhea also familiar with hemolytic uremic syndrome on the spinach and I have my little article here that I want to close with that several people died from this ecoli on the spinach why did they die because they were infected with this particular organism that has now acquired yet different DNA that has now allowed it to encode completely different virulence factors and now is the cause of hemolytic uremic syndrome severe bloody diarrhea and and again potentially lethal here's another example ecoli is the most common cause of urinary tract infections particularly in women but in in men as well why because it has acquired yet different in proteins that now on the surface allow it to attach specifically to the urinary tract of Atheneum of the bladder so no wonder it causes urinary tract infections it can anything attached to our bladders wall that's why it has this particular strain has the ability to cause urinary tract infections and then lastly it is the second leading cause of neonatal meningitis meningitis in newborns why because yet a different piece of DNA has now been acquired either as a plasmid or from a bacteriophage and back to a virus that infects bacteria that now has a capsule a specific capsule that endowed with the ability to cause neonatal meningitis so they bought the the thought that I'd like to convey is that organisms differ markedly in their in their virulence and their ability to cause disease and so now so then when you read in the chronicle a dangerous eco live bug blossomed this year you'll say oh I remember that fellow talk told us why this happens it happens because that particular eco lie now has acquired some new genes by one of these mechanisms either the plasma or phage of some of these others not and and it endowed it with new capabilities to cause this particular severe disease so in closing where have we been we've described the various types of microbes worms protozoa fungi and bacteria and viruses we discussed the concepts of infectious diseases incubation period prodromal period we described the beneficial and detrimental effects of inflammation and what causes these effects we discuss bacterial structure and its relationship to pathogenesis we discuss the fundamentals of pathogenesis to transmission the colonization of Asian toxin production discuss the structure and action of endotoxin to varied we describe the various actions of several important exotoxins and lastly we discussed how one bacteria can cause many different diseases depending upon the nature of its virulence factors and so thank you very much for you so we hear microorganisms evolving along with us and what about other organisms what about other prime primate subtend and they're the one example that's coming to mind is human immunodeficiency virus in US and simian immunodeficiency virus in monkeys for example so independent evolution here is this virus that's killing the helper T cells in these monkeys and chimpanzees and other primates and at least to all intents and purposes until the call of 1981 when it first really became obvious to us but but there's ant antibody evidence that we were infected perhaps a decade earlier than that that that this organism was causing disease in monkeys and then somehow changed in its nature or in its ability to infect us by mutation of that particular monkey immunodeficiency or simian immunodeficiency virus and now allowed it to infect us so yes there are there are many bacteria and viruses out the field of veterinary infectious diseases enormous okay but yet we continually interact whether it's avian influenza virus whether it's a Lyme disease whether it's those of you remember the hantavirus outbreak in the Four Corners area of the Southwest where the rodent population because of the El Nino had more rainfall more piñon nuts more rodents more rodents in contact with people living there more rodent droppings that contain the virus more rodent droppings that dried out and were inhaled by the unfortunate people who got hantavirus pneumonia and in 24 hours they were dead oh and similarly the berkeley graduate student who went up to the sea Arras to investigate some ecological project and rented a cabin and swept out the cabin swept out the rodent feces and 48 hours later was dead so it's sort of dramatic stories but it illustrates that that there are animal pathogens continually circulating there and and we then now sort of intrude or infected by either because of dose the large amount that was swept up or because the organism has changed just enough like the avian influenza var has changed just enough in its surface proteins to allow it now to infect ourselves what are the origins of viruses you know this is yes so often I have pondered did viruses come first or do have to have cells in order to have viruses because viruses can only replicate in cells the answer is I don't know intuitively you would think that you would have to have a cell and then somehow this piece of noop this nucleic acid acquired the ability to replicate independently almost maybe plasmid alike and then don't know but but acquired the ability to coat itself with certain proteins and then somehow acquire the ability to leave that cell and infect another cell it's a lot of hand waving going on here but it's hard to imagine a virus as we know it existing prior to sell something that may have been self-replicating nucleic acids but but if a virus can only replicate within a cell seems to me it would have to have started as some ab marren form of nucleic acid from from a prior previously existing something
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Medical_Lectures
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Introducing_MRI_The_Basics_1_of_56.txt
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most of the people here are Radiologists Radiologists and training whatever you consider yourselves uh but there are some people here who are using Mr and they're not using it really for clinical work at all and um or in clinical research perhaps and the the first thing that I always like to uh start out is just to make sure that we're all on the same page about a few issues um and there's really you know there's really a lot of similarity between the the goals and purposes to which we're going to try and harness and utilize this this technology regardless of of what we're doing here so um you know the Radiology residents who are the the bulk of the people here are really I would imagine are mostly interested in you know real clinical diagnostic applications of Mr which is good I mean that's that's one of the wonderful things about MRI but we have people here that are also interested in really addressing some pretty detailed research questions now if we think about what it is that we're trying to do regardless of what the question at hand is so if we think about a clinical question and if we have for example and this is just a schematic right so if this is a I'm already impressed see it was worth leaving bin Wanger Auditorium okay so so this circle I'm drawing here is just a schematic of let's say the abdomen uh another thing I should say is so I'm a neuro radiologist I'm really interested in the brain other parts of the body other than the extent to which I guess they support my own existence they don't interest me that much but I try to be you know as even-handed and equal opportunity as possible I'll give you examples from Imaging in other parts of the body uh but you know ultimately the question is the same regardless of the body part we're talking about but so for example if if this is a an image of the abdomen and let's say we have an organ in here the liver and in this person there is a tumor right in the liver so a clinical question that we might want to ask is can we detect the presence of this tumor it might be a primary tumor or a metastatic tumor whatever it happens to be so can we make an image of this using MRI and detect the presence of this tumor so I just want to think of a and this may seem very basic but I think it's important just to be clear about what our questions are when you look at an MR image and you want to detect the presence of that lesion you're essentially asking the question right can our image and these images are basically arrays of pixels right they're digital images just like the kind of images you get out of your digital camera so it's an array of pixels and our question is can we detect something about this part of the liver where the tumor is that is different from the part of the tumor where there from the part of the liver where there is no tumor okay so that really requires us to be able to do two different things first of all let's say that the tumor is occupying this location in our Digital Image so if the signal right that we detect from the location where the tumor is right let's say we'll give it a number let's say it has a measurement of 10 units of something if that's the same as what we detect in the areas surrounding that tumor it's going to be invisible right so even though it's there if our image can't tell us that there's something different about the signal or what we measure at that location in the image we're not going to be able to see it so the first thing that we're going to need to to be able to do to detect the presence of this lesion is be able to detect a difference in that signal intensity now even if we have an image that can clearly show us that there is let's say a 10-fold difference in the signal between what it is we're trying to detect and the normal surrounding tissue what if we have a different scenario where let's say we have something very small that might only occupy let's say part of the volume of that pixel in our image so when we look at these images just like if you take your digital picture on a on a PC and you blow it up really big right so the pixel is what the pixel is and we only get a single measurement of this signal from that location in the image and what happens in this location is that our signal is going to be some average of all of these things together so if we have something very small right the mean signal intensity might be close enough to the normal surrounding tissue that we'll be unable to detect it and this right all the Radiologists know is something that we call a partial volume effect or partial volume artifact so there are two things that we're going to always be trying to do once we get to the point of talking about Imaging one is to generate a sufficient contrast resolution in the image meaning a big enough difference in the signal between the thing we're trying to detect and everything else and the other is that we're going to always be trying to get an image that has a sufficient right spatial resolution in other words that it has a there's a there is a fine enough grain to that image that we can actually detect these tiny small areas of pathology now both of these things are going to be true if we're trying to make it sort of a gross anatomic assessment like this or for example if someone is interested in doing some right right functional Imaging study in the brain where they're trying to detect let's say the part of Vortex that is functioning when you're tapping your fingers together and some other Associated area it's all really the same thing we're trying to generate an image that has sufficient contrast resolution and sufficient spatial resolution and what we're going to talk about this week is really different ways that we can vary the nature of the image in terms of its contrast in terms of its spatial resolution to do this job and the thing about Mr that makes it very different from other modalities like CT and like ultrasound is that we have a tremendous array of tools or parameters that we can adjust to change this nature of the image right for example uh for those of you that do other types of Imaging when you think about CT you probably never hear about adjusting the spatial resolution of your CT images right there are things you can do to change the field of view but it's not a parameter that we tend to think about uh it's it's relatively fixed from image to image similarly when you think about a CT image of a patient let's say of the liver there really isn't a whole lot that you do to change the intrinsic contrast of that image you can inject a contrast agent but the image is what it is we can display it differently but in Mr we can actually change the nature of the image so that putting a patient in the scanner taking them out 20 minutes later we can have several different images that have dramatically different types of contrast okay so that's part one
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Medical_Lectures
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The_Diabetic_Foot_Exam.txt
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[Music] today I'll be giving a demonstration of the diabetic foot exam the term diabetic foot refers to a constellation of physical findings and medical complications in the foot arising as a consequence of impaired sensation due to diabetic neuropathy impaired blood supply due to concurrent peripheral vascular disease impaired wound healing and secondary infections due to the relative imos supression of diabetes and last impaired mobility in the chronically ill diabetic which prevents him or her from routinely inspecting and caring for his or own own feet the diabetic foot is one of the most common causes of Hospital admissions for diabetics and the leading cause of amputations in the United states predictors of future need for amputation include prior amputation foot ulceration impaired arterial supply high hemoglobin A1c physical deformity and neuropathy there are three major components of the diabetic foot exam they are inspection assessment of arterial Supply and assessment of neurologic function hello Miss Park my name is Eric strong I'm going to be performing the diabetic foot exam on you today as a demonstration for some of our trainees do you have any questions as with most examinations you should begin the diabetic foot exam with inspection with both legs exposed to above the knees look for prior amputations and assess for General foot hygiene look for dryness and cracking of the feet superficial fungal infections and good Nail Care in adequate hygiene is a risk factor for developing infections even mild skin cracking or a superficial fungal infection could serve as a potential portal of entry for more serious bacterial infections assess the hair pattern an asymmetric absence of hair or abrupt change in hair pattern as one moves distantly down a leg suggests arterial insufficiency look for calluses which may be a sign of improperly fitting Footwear and may precede ulcer development ulcerations may be due not just to be repeated trauma but also arterial insufficiency due to concurrent peripheral vascular disease or from concurrent Venus stasis and the presence of an ulcer does not necessarily indicate infection signs of infection include surrounding aemma tenderness and perent drainage osteomyelitis is likely if either bone is visible within the ulceration or if the bone can be reached by examining the ulcer with the sterile probe also inspect for joint deformities these include metatarsal subluxation which is usually seen with the first MTP joint the first MTP joint can be vertically subluxed or more commonly laterally subluxed which creates a pressure point on the joint's medial surface this is what is known as a bunion and is formally referred to as Hu abducto valgus deformity bunions are not caused by diabetes per se but can complicate the treatment of the diabetic foot another pair of related deformities are the claw toe and hammer toe which are due to combinations of vertical subluxations of the MTP pip and dip joints which often happens to toes 2 through five simultaneously the more dramatic deformity seen in diabetes is called neuropathic arthropathy or more commonly the sharod foot which is a pattern of destructive changes due to repetitive trauma to insensate joints early features include edema warmth and athema surrounding affected joints later features include overt joint deformity joint dislocations pathologic fractures and overlying skin ulcerations neuropathic arthropathy is not specific for diabetes as it's also seen in other ideologies of peripheral neuropathy including alcoholism HIV and syphilis the most extreme form of shako foot is the so-called rocker bottom foot deformity in which the normal arch of the foot becomes everted leading to a dangerous pressure point in the midfoot before leaving inspection be sure to examine the patient's foot wear abnormal patterns of wear may suggest improperly fitting shoes and or an abnormal gate which can be an early sign of neuropathic arthropathy the next part of the exam is an assessment of the vascular Supply this begins with checking the temperature in the legs ensuring that they are equally warm at this point the formal diabetic foot exam would also assess capillary refill in which the nail of the great toe has compressed and the time to reuse the nail bed is measured however this exam finding has been demonstrated in the literature to be of little value in assessing for peripheral vascular disease which is consistent with my own personal experience and thus I omit it then we check the per contr pulses always start assessment of pulses as dist as possible and only bother with more proximal pulses if the distal ones are abnormal the most distal pulses in the leg are first the dorsalis pedus pulse which is found just lateral to the extensor tendon of the great toe and second the posterior tibial pulse which is found posterior and inferior to the medial malis in the event that either of these is abnormal you should follow it with a assessment of the poil pulse this is found in the midline within the poil FAA behind the knee and is best felt with a hand wrapped around the knee from the front and pressure applied while the knee is relaxed in moderate flexion when it comes time to describe the strength of a patient's pulse despite occasional insistence to the contrary there is no specific standardized numerical scale to use therefore pulses are best described qualitatively as absent present but diminished normal or bounding any abnormalities that suggest the presence of peripheral vascular disease should be further investigated with measurement of the ankle brachial index the last of the three main sections of the diabetic foot exam is assessment of neurologic function beginning with sensation this park I'm next going to evaluate vibration sense in your feet with this tuning fork can you hold your arm out and can you feel that yes and do you feel it stopped yes okay I'm going to check the same thing in your feet now with your eyes closed okay all right so let me know when you can feel the vibration and then let me know when you can no longer feel it okay yes it's gone on the other side yes it's gone m park next with your eyes closed I'm going to lift your great toe uh either up or down and I want you to tell me which direction you think it's being moved okay up down down up the other side up up down up great the sensory exam typically ends with a monofilament test in the conventional monofilament test the patient lies supine with eyes closed while the examiner touches various points on both feet with the monofilaments using enough pressure to cause it to buckle the patient indicates when he or she feels the monofilament touching them despite some claims to the contrary as with the pulse scale there is no one standard as to the specific spots on the foot where the patient should be touched however a reasonable list of locations would include the plantar surfaces of the first and fifth toe and of the first and fifth MTP joints Plus plus or minus the center of the heel the test is generally considered to be abnormal if the patient is unable to feel the sensation at any one site the conventional monofilament used in this test is called the 5.07 monofilament which defines the amount of force the monofilament imparts to the skin before it buckles Miss park with your eyes closed I'd like you to tell me when you feel this monofilament touching the bottom of your foot yes yes yes yes yes yes yes yes yes yes right great thank you some clinicians also include assessment of Pain by randomly touching different parts of the foot with either the sharp or dull side of a broken cotton tipped applicator or tongue depressor if the prior sensory modalities are normal and the patient has no sensory complaints I personally emit the section as well the next component of the foot f focused neuro exam is a check of the ankle reflex this is most commonly done with a patient sitting upright but anyone who is actually attempted to elicit this reflex in this position can appreciate it is awkward and difficult to perform therefore I advise checking the ankle reflex with the patient supine with a hip externally rotated knee flexed and ankle in mild dorsy flexion in order to stretch the Achilles tendon the final component of the exam is a assessment of gate it should be assessed for smoothness and symmetry gate abnormalities may be indicative of neuropathy Andor early neuropathic arthropathy well miss park that's the end of the exam and I just want to let you know everything was healthy and normal thank you that concludes this video on the diabetic foot exam I hope you found it interesting and useful the
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Medical_Lectures
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8_Women_and_Heart_Disease_Mini_Med_School.txt
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[Music] Stanford University uh also very high percents just to get a sense of what is the population looking like this is all us data but these are all us data by the way uh and at this point we see that men are uh exceeding women slightly when we get into these a higher groups but hopefully you've been also paying attention to your age group as you go along here just to get a sense of how you feel was kind of I'm over on this side so I'm looking at the 80 plus but if you're going over here you can see uh how this is increasing very very rapidly as we go through these 20year increments so when we talk about cardiovas disease it is the leading cause of death among women in the United States and actually it's the leading cause of death in most developed countries uh one in two deaths so half of all women will die from cardiov vaster disease using those broad definitions kills more than 500,000 women each year kills six times the number of women that breast cancer kills and almost twice as many women as all forms of cancers put together um and one in Four Women actually have coronary heart disease uh some one in four deaths in women are from cor uh coronary artery disease so just to get a sense it's not just for older it's not just an issue of older women it's the second leading cause of death for women 45 to 64 and it's the third cause of death for younger women 25 to 4 before so it's a huge issue and a lot of women don't know it we still have this problem that people still call it a man's disease and I do a huge study of men on osteoporosis where everyone calls that women's disease one out of every three hip fractures is a man so we have to get rid of this gendered bias about these diseases they can B basically get men and women and we need to be becoming educated um now when we talk about this there are clearly differences by different ethnic groups uh in this particular slide we can see the black men in uh blue and the white in Red so you can see there's minor differences there much larger differences when we look at white women in yellow and black women in the uh light green uh and this is now the incidence of a heart attack by age race and sex and to just look at the first heart attack you see the same thing that white women are clearly the lowest um as we age it's increasing uh black women are next and and then we have the two male categories um now some of the issues about heart disease we're going to talk about some areas that really are just now coming to light uh one of the big problems is under detection so there's a lot more heart disease out there than we realize in women and one of the problems is symptoms so women have kind of different symptoms than men they're more subtle versions of male symptoms uh they also report other things abdominal pain back pain pain nausea shortness of breath and fatigue and other than shortness of breath I think most of us experience many of those things often so how do you distinguish that from heart disease and that's a real challenge now I put quotes around typical because that's a real male perspective this is the typical male symptoms uh and it's just a little hard to figure out what is typical but if we look at younger women and one of the things I'm going to be pointing out to you is that there's a really big difference between heart disease and younger women and older women uh heart disease and older women is very much like heart disease and older men but when we're talking about younger women we have some really different issues that have gone undetected until the last five or so years um chest pain is most common but it's a little atypical it's a chest heaviness a pressure tightness or squeezing but 58% of women have pain in the jaw or shoulder 38% of women sweat um 29% experience nausea 29% of shortness of breath 21% have indigestion or heartburn and they often are sent away with Pepto-Bismol or something like that when they're having a heart attack uh and then weakness and fatigue is also 8% um now an interesting point when we talk about gendered medicine is the fact that there are delays for women every step of the way if a woman starts to have this experience of a heart attack she may call 911 and it turns out that the ambulance takes longer to get to the woman than to to the man they basically are not aware that women could be having something that needs urgent the same urgency so 52% more likely that they're going to have a delayed uh ambulance coming to them uh once they get to the ER the door to balloon time which is from the moment you arrive until they actually get a a a balloon into your artery is much longer in women so we have all of these problems that basically we need to fix because we need get everybody to recognize that this is something that could be happening uh another quick uh set and then we're going to get into a little more meet is that it turns out we have lots of different reactions to medicines and they have different preventive benefits so for instance Asprin which we recommend to men at a low dose uh does reduce uh coronary heart disease in men but it turns out from the data we have so far and lots of people debate this um it's not true in women uh but yet we do recommend that you take a low dose aspirin from age 65 on on the other hand aspirin has been shown to reduce stroke in women but not in men so these are some really interesting issues um also we know that some of the drugs increase arhythmia in women much more than they do in men and there's actually a particular kind of of uh arhythmia so quinidine actually causes it's an anti-air rhythmic that causes this really slow Q wave so this is called Tad de and and uh another one to just quickly mention is atrial fibrillation it's much more common in men but if women get it they're much higher risk of getting a stroke and having some bad outcomes and these are the kinds of things we're looking for now there have been huge campaigns I think if you've been through San Jose airport in the last 10 years you might have seen some of the red dress campaigns because they actually had this set and there's some interesting political battles between the national heart lung blood institute's red dress campaign and the American Heart association's red dress campaign but they all seem to recognize that a red dress is a great way to make the point that we need to bring heighten awareness to uh women's cardiovascular disease and basically when we look at now this is deaths in thousands so this is not a percent or a uh rate it's actual numbers uh what we've been aware of is that these deaths in thousands are have been dropping in men over time and for quite a while they weren't dropping in women and that was really seen as being an alarming issue the good news is that they are now dropping now to some extent this relates a little bit to the interaction of smoking and heart disease because it turns out that men were smoking at much higher rates and then they've been slowly dropping and in World War II women started smoking and so they just started to get the the uh consequence of that the heart disease and that was really kind of hovering here and now they're dropping because their smoking is dropping so that's a little confusing how these data look but this is actually a really interesting slide that's looking at uh 52 countries that go from a really low rate of cardiovascular disease so the far end of this Japan is the third one in so Japan has a really low rate uh and we get up to the higher ends where we have Northern Ireland and Scotland and the US is in here somewhere somewhere close by and what you see is when we go from a low rate to a high rate nonetheless the difference between men and women it turns out to come out to about 2.24 whether you're talking low rates or high rates so it's just sily steadily climbing where what we see is the male rate relative to the female rate for these diseases so it's a really robust it's not just one sociocultural experience it really does seem to have some probably biology basis um now that said um I'm not going to become a man and and the men in the room probably aren't going to become women although we can do that um nonetheless the point I want to make is that within ourselves within each group we have the percent of total deaths due to different diseases and so the first thing I want to point out is that for both men and women the number one cause of death is coronary heart disease and they it exceeds all cancers put together and then you can kind of March your way down so stroke is a has a higher uh um totality it has a higher rate of death for women than it does for men that's at least a part because men are having deaths from accidents at higher rates and and lung cancer at higher rates so all of these death things take into account competition you can only die from one thing and so which one is going to get you but it seems that heart disease is getting uh the vast majority of or a much higher percent of both men and women now that's the overall picture so that's from birth until the end uh I do want to point out that this is true in every single race ethnic group in the US so heart disease is a leading cause of death in whites blacks and Hispanics now it's not true in Asian but it actually is the case that they have much lower rates if I was not looking at percent within the population if I was just looking at the rate and they actually have lower cancer rates than the other groups but the total effect of cancer that said there is no individual cancer that comes anywhere close to coronary heart disease so this is pretty uh Universal also true very true of men across the board so this is something that all race ethnic groups can recognize that we need to be paying attention to heart disease now to just home in on the fact that most people most women the majority of women are dying at age 80 up uh here's where the big toll of heart disease and the big numbers of deaths is occurring uh we don't have as many men dying by now because they already are gone and we'll get to them but you still see this very high rate uh and it certainly exceeds all cancers in these older groups I will point out that Alzheimer's disease appears in the 80 plus group for both men and women and stroke does uh and one thing I'll quickly say is I'm not going to spend a lot of time talking about Alzheimer's but when you actually start to look at what do people believe for preventing Alzheimer's it looks exactly like preventing cardiovascular disease it's all the same stuff so people are basically homing in on the one thing we can do is try and keep our blood vessels healthy so that that we have our best chance up there um when we talk about uh the younger groups so the middle-aged uh and slightly older men uh we actually do have all cancers exceeding uh heart disease in those groups but then again no individual cancer so lung cancer uh doesn't come anywhere near heart disease accidents actually exceeds lung cancer by the way so that's a big part of our prevention message is trying to prevent accidents uh and then really since the focus is women I wanted to really highlight the fact that um I'm not giving you enough time to study all of these I think they go online so you'll be able to look at them but to make the point that uh many more uh Men by the time we hit 48% of men had already died only 4% of all the women female deaths have happened and this is also true when we get into the next age group this would be 18% for men on the last slide and 11% and it's not really until we get into the 60 to 70 that we're kind of equaling out on the percent of deaths but that's really just um all math but what I wanted to point out here is that a lot of people have HED in on this uh when you kind of curtail the data so that you have a point and you go to 40 to 40 and it's a category and you put one point there and then you go to the next point you get this really steep uh increase and many people have tried to associate this with menopause and in fact when I was going through a lot of my uh uh graduate or postto work um every article would start out women are protected against heart disease until they go through menopause and then all protection is lost and we just saw that over and over again and that's a really interesting thing if you're interested in women's health and so um I started to try and find out well is that true is it really that menopause is a risk factor and there's there are several analyses now but there was one done actually uh back in 19 98 and the British population that tried to examine this and here you see men starting off with their heart attack heart disease this is actually death so coronary heart disease death they're having their deaths about 10 years ahead of women there's menopause there's really no evidence when we put this on log scales that menopause is making any kind of real impact and if I showed you breast cancer you would see a very clear impact basically the breast cancer rates are very steep until menopause and then when you don't have your estrogen on board they start to they're still going up so age is still higher is a big risk factor but they really you can really see the inflection from menopause but you do not see that with heart disease so that seems to be a myth that we're probably not going to get out of the literature for a long time and an interesting point actually to go back here you can actually see that the 15 to 19 group there's already more heart attack more deaths from heart disease very but more in boys than there are in girls so already at at puberty and an interesting phenomenon at puberty is that at puberty the HDL in males drops and that's the reason that we have this 10 milligram per deciliter difference between men and women it's not because of estrogen even though estrogen can raise your HDL the sex difference that we see in this particular lipoprotein is really happening in males going through through puberty the LDL Situation's a little more complicated and I don't have time to go into that but because of this myth a lot of attention was taken to menopausal hormones and the idea that the way to prevent heart disease in women was to give them menopausal estrogen and there were lots of observational studies U of various quality that were uh weighing in on this now the most interesting thing is that right when I got interested in this question it was 1985 the New England Journal of Medicine published side by side and I'm sorry these aren't lined up quite right but the famous Framingham study which I'm sure many of you have heard of and the famous nurse's health study and they had absolutely opposite results so the Framingham study said that women who were taking menopausal hormones had twice as much heart disease and the and the nurse's health study said they had half as much heart disease and so it came out as a controversy which is really exciting so Sophie knows that's the kind of stuff we love to go for like who's right um and so any rate the bottom line is the the bulk of evidence and because of St studies like this we now have to register trials because we've learned since that there were a lot of trials that never made it into publication we have publication bias any of you who know Johnny anidas who's now the director of the our prevention center can tell you we have so many problems with what makes it into the literature and what doesn't but at any rate the the literature was pretty heavily uh presenting the idea that estrogen was protective and there was almost no data on estrogen and progestin which just so you know if you have a uterus you take both you take estrogen and progestin to protect the uterus against cancer of the uterus and if you have a hysterectomy you take just estrogen so we actually had a big question I'm going to give you a little bit of information about that and then we'll get back to Classic risk factors uh but in that I did spend a lot of time working on this question one of the issues with observational studies is they're biased because of your preferences the the women who will take hormones and keep on them feel better on them and lots of women start them and get off them and so what you have right off the bat is people who are reacting to them differently and we know that back in that old literature the majority of women who were taking hormones most of them stopped two to three years so the average the the nor the majority of women did not continue taking them whereas the women who continue taking them was this healthy user group as a group they were less obese they were more physically active they went to see their Physicians to get their cholesterol checked they also were getting mamogram so we were detecting a lot more breast cancer so that was just kind of a fluke of the testing situation uh but at any rate we were kind of left with is it really protective or is this just a coincidence from the data and I can tell you that we I spent a lot of Time marching my way through these trials the first one we did was called the Pepe study and this study was complicated it had five treatment arms we actually designed it with eight to begin with because we wanted to study the patch as well but for reasons I won't go into we didn't we weren't able to do that but it was by today's standards very small was the biggest study ever done of this question at the time so this was published in 1995 and the bottom line is that things that we were calling traditional risk factors seem to have some favorable effects HDL cholesterol the good one went up uh fibrinogen uh didn't go up LDL went down the bad the bad cholesterol um we had some neutral no effect on blood pressure insulin but we also had unfavorable results that people didn't want to talk about and in fact triglycerides was one that because it hadn't been shown in Framingham or the lipid research Clinic which was one of the original uh the two trials that had men and women it didn't show up to be a risk factor in men it was just discarded as not important uh and what I'd like to show you is that in fact the LR r c was able to demonstrate so this is an old uh paper so it's an a an old uh graph but basically what the lrc was able to show is that in women who had a combination of high triglycerides and low HDL they had really high risk of coronary heart disease and that wasn't seen in men so we never even got triglycerides built into our modeling uh I will also point out that HDL was really important in both the Framingham and the lrc for women uh to the point where it didn't matter if you had a low or high LDL what really mattered was whether you had a low HDL low HDL gave you high risk for heart disease as a woman and uh High HDL gave you low risk and it's also true of men we just don't have good drugs for it so you hear only about LDL and statins and things like that now one of the questions that people asked in this hormone Arena was about women who have heart disease should we put them on estrogen to prevent another event because they're at the highest risk and so the first big trial we did with real outcomes heart heart outcomes was the hers trial uh and this hers trial was only for women with heart disease uh it was a four-year study they were randomly assigned to a combination of estrogen and progestin in the form of conjugated ecoin estrogen and mroy progesterone acetate which is Prem probe which was the most commonly prescribed estrogen progestin combination at the time and still is uh or Placebo and um as we had seen in peepe we got these what we would call good outcomes for HDL increasing the good one decreasing the bad one again this increase of triglyceride which we do not see as a favorable event but in fact after four years there was really no difference in the overall heart disas disas the cardiovascular events however in the first year there were 52% more heart attacks in the women who went on these hormones and I should just let you know that the year before we published this the American College of cardiologists put in their recommendations that if you have a woman who's had a heart attack you should put her on estrogen or estrogen and progestin to prevent a second event so that was a recommendation for cardiologists and it turned out that in that first year more women on those hormones would have had a heart attack so we really were going the wrong direction on preventing heart disease in women with this motto now lots of people criticize that and I'll just tell you that within the next year and a half all kinds of people came out ready to publish the fact that yes well we studied the same drug and we saw no benefit by angiogram we studied a different estrogen 1 beta estral for stroke and we actually saw early Strokes no benefit uh other group did a patch looking at this at an estrogen with a different kind of progestin also seeing possible early harm no benefit uh the wave study was using angiograms they also had vitamins possible harm so over and over and over within a year and a half boom boom boom was the idea that no if you have cardiovascular disease you don't start estrogen but what about um women who don't so I can tell you at this time lots of people people were very upset they were really holding on to the idea that hormones are pre prevent are protective and gave us all kinds of scientific reasons why this should work and so the question is okay that's great that that's a good reason but does it and so we had a puzzle to solve and we basically I I put in quotes HRT we used to call it replacement hormone replacement therapy as if all of women needed to have it I don't do that anymore we just call it hormone therapy now but at any rate we knew that it was great for taking care of menopausal symptoms but we didn't really know does it is it a good thing for preventing diseases of aging and that really was what was happening to women so Women's Health was very much about get them on estrogen because you want to prevent heart disease and strokes and um there was concern about blood clots because estrogen has been shown to increase blood clots but certainly for bones it was seen as being beneficial there was concern about breast cancer many people believed it was good for memory and Alzheimer's disease and we didn't know about these other things and so the Women's Health Initiative happened and this was a huge effort it was actually a woman cardiologist who was the director of the uh NIH at the time and she went to Congress and made an appeal that we would get the funding to really look at women and heart disease which hadn't been done so this was a huge study I don't have time to take you through all of it um but bottom line is if if a woman had had a hysterctomy she would basically be put into the estrogen only trial she doesn't need progestin she doesn't need to protect a uterus but she would be assigned to uh the estrogen uh in the form of Premarin or Placebo and if she still had her uterus she'd get the combination of estrogen and progestin or the placebo so this was hers but on healthy women and then estrogen only and the hypothesis was based on all the observational literature we had at the time and to just kind of do it in a schematic way uh basically we had stopping rules if in fact the belief that was on pretty much everywhere that we could prevent heart disease and we could actually reduce heart disease by 30% then the idea would be that we would stop this if that benefit occurred because that would not be fair to give to the placebo women on the other hand we also were worried about breast cancer and if in fact um if was harmful then we would want to stop for harm but we actually built into this the whole woman perspective we weren't going to stop just for benefit to heart we weren't going to stop just for harm to breast you had to have an overall picture that it was either good or bad so it was really unique and how it was done and this is why it got so much attention I think but we didn't know what was going on with stroke the observational literature was all over we didn't know is this good or bad what do we do about that uh and I will just quickly tell you that this was meant to be a study of older women it was not about menopause we already knew estrogen's great for taking care of hot fleshes we wanted to know is it good for Women's Health and so we had the goals for women in their 60s to uh 79 which was going to be the majority and we had cut we had limits about how many younger women 50 to 59 we would bring into the trial and this is true for both trials so one is the estrogen and progestin trial and the other is the estrogen only trial trial and the other thing is that we were very determined to have minority women in the study and um an interesting phenomenon is it turns out that Minority women get hysterectomies at younger ages in the United States than do white women by the time all is said and done I think it's pretty equal but because of that we had a much higher percent of minority women in the estrogen only trial and I'm only saying that because there's a few caveats I'm going to hit you with in a moment but the first thing is these were two different trials and the estrogen and progestin trial so the trial where if you remember we didn't have a lot of observational data on this combination we just had the data on estrogen only but we were giving every woman with a uterus this combination and so the first thing that happened was the preset cut stopping point for breast cancer was hit much earlier than we expected with a combination that would not have stopped the trial because by that time we actually had fewer hip fractures and we had uh a surprising decrease in coloral cancer and those would have balanced each other out if in fact our cardiovascular hypotheses were not completely wrong so the reason why this study if you remembered it's already been 10 almost 10 years now um got a huge press because instead of preventing heart disease we had 29% more heart attacks and we had 41% more strokes and we had had the blood clots that we weren't surprised about but the overall picture was clearly one of harm and we were doing this as our primary prevention Focus for women so that was a big mistake um I will also quickly tell you that we had a study within the study of the older women so women 65 and up uh looking at probable dementia and we actually had one within each of the trials and to everybody's surprise instead of having less dementia we had actually had twofold increase this is the active group relative to the placebo the stepping by the way is because we were only doing the cognitive tests once a year so you didn't find out until you did the test that whoa these women were twofold higher more likely to get dementia on this hormon Dementia or Alzheimer's um well you know it's probable dementia and it includes Alzheimer's about a third of it was Alzheimer full-blown Alzheimer's and about a fourth of it was vascular dementia so it's a combination now this got a lot of press uh and so uh not only did it get a lot of press about we shouldn't be giving this to older women but lots of attention finally to menopause which we won't go into right now but that was before we even had the estrogen only trial finished the estrogen only trial did not see an effect on heart disease and did not see an effect on breast cancer or colorectal cancer but what happened was we end ended up having again a huge increase in Strokes uh and some blood clots uh and that even though the benefit to hip balanced it out uh this was funded by the national heart long Blood Institute they weren't ready to continue a study on stroke so the bottom line is we learned also with dementia that this was a much a higher wasn't quite significant dementia and something called mild cognitive impairment so the bottom line was this was not the right thing to be doing to prevent heart disease and just quickly say a few things if we look at where did the two Trials come out the same they showed no benefit for heart disease and then for the combination it was early harm for stroke and blood clots both studies showed harm both studies showed benefit to Bones which continues to be a challenge for us and both studies showed uh harmful har harmful events for dementia what was different was breast cancer was actually increased in one in the combination estrogen and progestin but not in the estrogen only and then colorectal cancer was different and the global index which was this balance of all the information together was one of harm for combination therapy and neutral for estrogen only now basically we solved the puzzle I'll tell you we also saw that gallbladder disease was increased on the on the hormones and incontinence was increased but um a lot of attention went to subgroup analyses for heart disease and and to just kind of walk you through this a little bit there was no difference between the estrogen only group and the placebo group in terms of the total events but when he broke it down by age group a lot of people focused on these younger women having fewer heart attacks relative to Placebo and even though this wasn't significant a lot of attention went there now I can tell you the pharmaceutical company played a role in that um and nobody paid too much attention to the fact that what was also driving that interaction was much higher rates or at least higher rates in the older women on the active hormone on the active estrogen relative thebo um but lots of a lot of attention went into that and we came up with there was this timing hypothesis that came along and I only bring it up to let you know that when you really look at what the timing hypothesis says it doesn't say that younger women benefit it says that compared to older women they're better off that older women are clearly harmed by going on this and that's really what the message should be but that isn't quite what we hear and I can tell you that why did the two trials differ there's lots of reasons it's not just because one had estrogen and progestin and the other had estrogen we had much more obesity in the estrogen only group it turns out that in the population women who have had a hysterectomy are more obese part of the reason they're getting hysterectomies is because of uncontrolled bleeding which happens with obesity so there's other reason reasons that these two trials differ uh including the hormone uh history but I don't have time to go into that uh all I want to say is that uh when we actually put the two trials together we saw some other things that I think are important for women to know and that is that uh women who are starting the hormones close to menopause seem to be not so there's really not the same danger clearly as there is starting at distant from menopause and so we feel that yes we can give some reassurance we wouldn't say stay on it the rest of your life but what was really interesting is that it turned out that older women who had hot flushes and about 10 9 to 10% of women continue having vasomotor symptoms on into their 70s that group of women were the most harmed by going on the estrogen and progestin so it's not something that older women should start if they're having those particular problems and stroke was increased regardless of age um now I will tell you that lots of women came on their hormones at that time we also made our women come off and so we were able to see what happens when you come off and just to summarize a long story and make it short the cardiovascular risk disappeared within two and a half years so we no longer had increased strokes and blood clots uh well there's a little question about stroke but there was no more problem with blood clots and I won't take you through all the rest of that uh similarly with estrogen only now with estrogen only we do still have this hint of some benefit for the younger group Group which is where the literature was so I'm going to tell you right now there's something different about older women and younger women and you're going to hear a lot more about it now um I will tell you also that the American Heart Association viewed all of the data and basically said in their most recent uh guidelines that look at Women's issues specifically that interventions that should not be promoted for prevention uh include menopausal hormone therapy we should not be prescribing this to prevent heart disease and any age group uh there are some other ones I'll just quickly mention antioxidant supplements vitamin E C and beta carotene we have no evidence that they're beneficial to women to prevent heart disease and in fact there's some suggestion of harm in certain subgroups F folic acid with or without B6 or B12 supplementation and then I've already mentioned aspirin if you're younger than 65 so these are some of the kind of current guidelines now to uh make a little segue here uh when we start talking about causes of death from a prevention perspective you really want to know what's the behavioral cause what can I do to prevent these diseases so this is basically looking at lifestyle factors and I'm going to talk a little bit about them uh tobacco is the number one bad thing for men and women combined poor diet physical inactivity next and then we have alcohol consumption microbial agents toxic agents Motor Vehicles those accidents and Fire firearms for the suicides which is a really high cause and there's a really interesting recent analysis this pulls all the men and women together I'm going to show you the sex differences in a moment but trying to look at what is attributable uh what are the um deaths attributed attributed to uh when we look at cardiovascular which is in Black cancer which is the stripe bars diabetes and yellow with dots uh respiratory which is blue uh other um n uh chronic diseases uh and um non-c communitive chronic diseases and injury and so what you see is smoking is leading as the cause of death when we pull all the chronic diseases together then comes high blood pressure and that's all about cardiovascular so I already showed you how high high blood pressure is especially as we age and when we break this into men and women uh I'm going to focus on women and those of you who want to look up here can look up here but you'll see that the number one cause for women is high blood pressure this is a really important thing for us to get under control uh the next one is smoking which is basically impacting cardiovascular cancer uh diabetes and respiratory uh same with men but it's a much higher problem because of higher smoking in men then comes high blood pressure in men so it's just flipping the two uh then the next one one for both groups is well okay the next one is for men is overweight obesity for women it's physical inactivity and so I'm going to talk a little bit about that today and then comes obesity and then high blood glucose I'm not going to walk you through all the well I guess I will high sodium LDL cholesterol uh dietary Omega fatty acids um High dietary fatty acids sorry yeah fatty acids and then comes low vegetables and fruits uh and so all of those are black pretty black for cardiovascular so what's your risk this is essentially the kind of question that you would ask and we actually have special algorithms now to calculate your risk uh the Framingham 10year risk score is one of them but uh I'm going to kind of walk you through some of the issues for smoking blood pressure cholesterol uh I put in blue the ones that are sexually different so for both men and women has your father or brother had a heart attack before the of 55 that's premature that's basically suggesting some genetic uh impact uh has your mother a sister had one before 65 that's premature so the 10year Gap and then the other one that's in blue is are you 45 if you're a man um for men that's a risk and for women are you 55 so we have this 10year difference that we kind of build into our risk algorithms and here you see this so age is clearly the the number one risk factor and there's nothing we can do about it hopefully hopefully we'll all get to live as old as we can be but this is actually dying here so don't you're not on this chart if you're succeeding on the on that scale but what you see is men are taking off as I've shown you before before women are uh in terms of the impact of age um probably you've heard about cholesterol in one of the other lectures I'll just kind of remind you that when we talk about total cholesterol it really does doesn't mean very much because there's a good component the HDL cholesterol essentially is removing cholesterol from the arteries it takes it back to the liver and you basically repackage it in a different in a healthier way uh and we do talk about uh having low levels uh for I'm sorry HDL so there's an optimal of 60 or more um we talk about less than 40 for men but I really think that less than 50 is the less than 50 is what we should be talking about for women I think there's a sex difference there that we should be building into our algorithms um LDL cholesterol is the one that's actually depositing in your arteries and so this is where a lot of the management risk management goes we do have really good drugs for treating that we don't have such good drugs for the HDL and then triglycerides is this interesting thing that it does matter for women but it's not given very much credit because it doesn't seem to be as important for men um when we talk about about blood pressure I'm just going to point out that this slide to orient you is here's white men white women black men black women Mexican-American men Mexican-American women and that um African-Americans have much higher blood pressure than do the other two groups and in fact when we talk about hypertension African-American ancestry is a risk factor by itself uh obesity uh overweight for all of us sedentary lifestyle alcohol um sodium in take we really want to get that down and insufficient potassium so these are some of the major risk factors for this huge problem now it turns out in women um the Women's Health Initiative also had a huge observational study so we have lots of other data besides the hormone trial but there was they defined prehypertension and that was actually shown to be independently Associated so prehypertension is essentially you're not yet hypertensive oh I don't have the yeah here we go so prehypertensive instead of being above it being up to 139 over one over 140 over 90 where 120 to 139 over 80 to 89 it turns out that that's also giving you increased risk uh certainly not as bad as being full-blown hypertensive so this is your risk of getting a coronary heart disease over the course of the followup study and you can see that prehypertension was about 66% more uh like to be getting a heart attack or one of these events so it could be an MI it could be a stroke it could be heart failure it could be cardiovascular death uh whereas hypertension was 2.89 and we don't have data quite as good as that for men now because we actually have this big study of women U one of the points I want to make with this slide is that uh for both men and women and you can see here that this is giving you a a much higher scale for men than for women so the blood pressure issues actually is quite important for men as well uh this is giving you the kind of now I guess what we'd be calling prehypertensive borderline and then high this was old data when we had even higher rates that we tolerated now we we pretty much go start with the green but you can see that it's a very dose response situation for both men and women and it's true for coronary heart disease stroke is increased with blood pressure peripheral vascular disease is increased with blood pressure and and congestive heart failure in both women and men um now in terms of the smoking rates uh women do smoke less than men uh not quite as uh good for whites uh white women I uh have higher rates uh than the other groups uh and also blacks are not that far below but the good news is Asians are really low for women uh and also Hispanics and so this is just the smoking rate uh that we know is very much related to heart disease now an interesting issue for women is if women have diabetes their risk for heart disease is much higher than if a man has diabetes so this is a big sex difference that has been showing up over and over this is actually pre-diabetic so this is actually just impaired fasting glucose so even within that whether you use old definitions or young new definitions what you see is that uh for women and for men your risk for coronary heart disease is much higher if you're diabetic uh or if you're fat impaired fasting glucose and for women cardiovascular disease as a big group uh so this is also now seen as an independent risk factor to have these high blood sugars uh and also if we talk about uh risk factors for prevention it's really hard to do everything at once and so what we kind of recommend to people is to realize that as you add the risk factor so you go from pry no risk to total cholesterol being elevated that increased the risk for both men and women uh cholesterol and blood pressure increases it even further add smoking you increase it even further add the bad uh glucose but bad blood sugar you increase it even further and if you actually have left ventricular hypertrophy you're really in in a really high place men and women look quite the same in terms of adding these up and what I was going to say is if you can't do it all at once anyone one of them will reduce you and move you back down focus on where you can to just reduce your risk and that's true for men and for women uh when we talk about diabetes it's kind of interesting because it depends on what race ethnic group you're in in terms of whether men have more or women have more the case of whites uh diabetes is higher in men than it is in women in the case of uh blacks and Mexican Americans it's higher in women than it is in men uh and these uh these women also have higher obesity as a group and so that's part of it now as I mentioned if a woman has diabetes she has much higher risk for cardiovascular events with one exception and that's stroke so what you're seeing here is women the if a woman is diabetic compared to a man their risk going above this line of one and my pointer is kind of dying out here this line of one you see that women have higher risk for all cardiovascular disease also for coronary heart disease for heart failure for intermittent claudication and it's really only stroke that is higher for a male diabetic than a female diabetic so diabetes is a really serious issue for women uh this is kind of saying the same another version of that we've done a much better job of bringing diabetes down in men so men with diabetes actually we've been able to control their risk for heart disease so if you can see uh basically uh in 1971 81 uh they were much much higher than they were by the time we're in the 1988 2000 whereas women nothing changed and this ratio of not diabetic to diabetic uh really it's slower in women than it is in men but it's not improving whereas men are so this is a a serious issue for us so in summary those risk factors age cholesterol blood pressure smoking diabetes and having had a heart attack before is a huge risk factor and then the other said is premature is family history which we've mentioned already and I want to say a little bit now about lifestyle factors and something called metabolic syndrome metabolic syndrome is having any three of the following five so you see what they are but I can show you here um this is a classic case of uh what would be at high risk for metabolic syndrome so abdominal obesity for men it's a waste of of 40 in for a woman of 35 in if you're metric you can read that um triglycerides of 150 or higher LDL cholesterol below 40 for men below 50 for women so the metabolic syndrome does build the 50 in instead of the 40 blood pressure not hypertensive yet but close 130 over 85 fasting glucose not diabetic but high and if you have any three of those five then you have metabolic syndrome and I will point out that when we talk talk about women with metabolic syndrome they're much more likely to look like this than like that and what we know is that women who have the Android obesity pattern the male pattern for uh obesity if they were matched for obesity level so they had the exact same body mass index this woman would be much more likely to have a heart attack to have coronary heart disease to have high blood pressure diabetes low HDL high triglycerides gallbladder disease specific cancers uh then the woman with a gynoid pattern and we've got lots of really interesting data now that essentially the fat that this woman is putting on is primarily subcutaneous L much of it is in her thighs and in her hips and that turns out to be a pretty safe place to put extra fat um on the other hand this woman and that man have most of their fat intraabdominally in a fat Depot that will release directly into the portal circulation which goes right to your liver and causes all kinds of metabolic problems and so there's a different physiology depending on where you put your fat and this is one reason why men have higher risk because they tend to put their fat in the male pattern but if a woman is in this particular kind of pattern she's at much higher risk and this is generally the woman who's showing up with these younger heart attacks and other kinds of problems and by the way you can't change that where you put your fat that's a genetic uh predisposition and here you're seeing that if we talk about someone with metabolic syndrome uh the the likelihood that a woman's going to have cardiov vaster disease is substantially higher than for a man but for both men and women this is significant and I know I'm showing you a lot of kind of data that you probably don't completely follow but uh this confidence interval means that these are both significant and the men the women are slightly higher um I think you all know that we have an obesity EP emic in the US uh and that we know that it is associated with some of these risk factors that I've already talked about so we are very clear that uh obesity increases our risk for hypertension three-fold relative to normal normotensive it increases our risk for the low HDL and the high triglycerides and it's three-fold higher risk of diabetes if we are obese relative to uh normal weight uh and when we talk about the Rel reltionship of obesity to physical activity which I spent at least 10 years trying to sort through and finally just gave up and said you know what they're they're just so highly related um they're not they're also related to all kinds of obesity comorbidities so physical activity and obesity um have some similar physiological reciprocal physiological activities but if we look at women when we first of all more adult women in the US have obesity than men the good news is they're putting it more likely they're putting it in the safer way but the bad news is it's still having an impact so that if we were to look at a woman whose normal weight and physically active and we set her as our reference group if that woman the normal weight woman is less active she increases her risk of uh this is a multivariate risk for coronary heart disease U 32% and if she's really sedentary that goes up to almost 50% but if she's obese and she's active um she still is at higher risk 2.5 but if she's obese and sedentary she's up to 3.44 so these interact and again if you can't lose weight exercise because you get some benefit from that and weight loss is a lot harder than going out and doing the exercise um uh in the Women's Health Initiative we also were able to show by quintiles just walking that the women who so we set here the women who were at the lowest walking level as our relative group and that what you're seeing in this is the decrease Risk by becoming more and more physically active so the women who were in the highest quintile had more than 50% reduction in their likelihood of getting coronary heart disease over the course of this uh followup period and we actually had enough power uh statistical power in the Whi to say that this was true of blacks as well as whites uh it was in the same direction for the other ethnic groups but we didn't have quite as many and also it was true across the body mass index groups so that was also true within obese women who were Physically Active had half the rate of coronary heart disease as the obese women who were sedentary so again Being Fit even if you're fat is a much better thing to do than just having both of those things out of whack um the data on physical activity continues to ACR uh we have lots of good data to say that we can it lowers your risk of early death of coronary heart disease stroke blood pressure lipids diabetes uh cancers preventing weight weight gain and helping you with weight loss and depression uh and also cognitive function uh and so I just want to kind of be sure you know what the guidelines are I think most of you know that we talk a lot about getting um the the aerobic activity moderate level aerobic activity but there's new literature coming out that's fascinating on inactivity the fact that when you sit without standing your risk of coronary heart disease goes up uh except that uh there are four additional pieces if you can't do the 150 minutes a week that we recommend for physical for modern intensity then do what you can um balancing exercises to prevent falling or built into older adults uh relative intensity basically means means that what might be a a slow walk for me might be feels like a jog to you and that's all that matters that basically you go with your own relative experience and check your own heart rate and eventually as you train you'll be able to go a little faster but not to real not to think that a three m per hour walk is what I have to do that might be the max of your particular um Fitness level until you get more fit uh and the only other thing I'm going to say about the new guidelines is that uh when they talk about uh weight control uh in all the guidelines they kind of say more is better so we don't talk about 50 150 minutes a week is good that's the minimum and more is better and more is also better for weight so that's all I'm going to say about that now as I mentioned we've got this obesity epidemic so we've got uh lots of states that used to have relatively well that was pretty high obesity tend 14% but now it's you know outrageous where you know some states are actually this is even old now we have some states that are up to 30% of their population is obese not just overweight obese but what's going right along with it is the DI the diabetes Trend so we're seeing almost within one year of the states converting to having a higher percent of their population being obese having a higher percent of their population having diabetes and so what I wanted to show you is that we have some evidence that lifestyle actually helps men and women uh there's actually several studies but the best one that we hear a lot about is the diabetes prevention trial the dppt uh which had 68% women uh 45% of the whole study were minorities uh they were a mean of 51 years of age and these did not yet have diabetes but they had this high blood sugar that we I showed you a slide that that's associated with increased risk for coronary heart disease disas and they had a goal of losing 7% of their body weight um and doing this with a diet and exercise and the diet was a hard healthy diet we change what that means all the time so that's why I'm just going to not talk about that at the moment but uh vegetables is a great thing and low fat low saturated fat I don't think anyone questions that uh and so uh basically this was a almost threeyear average followup and they were looking to see how many of these people actually converted or developed uh diabetes me melodus and the interesting thing was there was a placebo group a group that was taking metformin which is the leading drug for treating um the blood sugar or this lifestyle intervention and the bottom line was lifestyle one heads over Tails over the leading medication and that was true in men and women so what you see here is the placebo and red this is everybody so those were the data I just showed showed you um the bottom line was the physical activity was the 100 uh minutes a week that we recommend modern intensity or vigorous you can mix them up too you can do 75 minutes of vigorous or you can mix them up U they lost 12 pounds on average this was the metformin dose and what you see is in both men and women that metformin was successful but not nearly as much as diet and exercise so this is kind of a whole new phase where we now have good randomized clinical trial data that diet and exercise lifestyle will reduce our risk for this case diabetes now there's a huge study underway called look ahead which has taken diabetic men and women and they're coming near the end they're trying to prove that by having diabetics do the diet and exercise and weight loss we can prevent heart disease so that study will be coming out in the next couple years so look for that um so the bottom line is when we talk about uh prevention uh how do we lower CHD risk uh we don't want to change our we want to keep aging um at least I do um and but we do want when we talk about LDL cholesterol this is where diet plays its biggest role we want to have low saturated fat High vegetables fruits whole grains I did stick on here meds because it's pretty hard to not get out of the doctor's office without that prescription U but you could try and talk them down a little bit and say could I try PR diet and exercise for 6 months and see how it goes um and that's what I would encourage you to do uh for HDL we talk about weight loss and exercise we don't have a great diet to take care of it for blood glucose we talk about weight loss and exercise and reducing simple carbohydrates for high blood pressure we talk about weight loss and exercise for some reduce salt and reduce alcohol and smoking stop or cut down so weight loss and exercise is kind of the the big story and the most amazing thing is oh I was sitting thinking like Oh I thought that I had uh come to the end but I didn't okay good so symptoms we're going back now to women and heading into the issue of uh women report greater symptoms they often uh they they have a greater number of less common symptoms so it's not the same thing if you have a heart attack it's not the same thing that most men will talk about this elephant standing on my chest some women will have that especially older women but there's really a lot of other things so they'll talk about shortness of breath nausea vomiting transient nonspecific chest discomfort arm and shoulder it's usually in the left but sometimes it's in the right abdominal pain and digestion back pain neck pain jaw pain headache fatigue dizziness all these things and women could report many of those and none of them may give the tail none of them may be the give the giveaway now I can tell you that men do report more chest pain and belching and hiccups so uh I don't know if that's because we're trying to Snuff it or what but at any rate uh there you go that's one of the sex differences um but also both of them are pretty likely to have exertional symptoms so if you start to exercise um but in fact women will have more pain when they are at rest and during sleep and with mental stress I'll also note that for menopausal women or premenopausal women they may have worse symptoms in their menstrual periods um now an interesting issue is for women who are having these events these uh symptoms 95% of women report these before they've actually been diagnosed before they have the heart attack and I can tell you one of the things that we're going to move into now is the fact that many women will go and they'll get a full work up and they'll be told there's nothing wrong with you go home and then they have a heart attack because there was something wrong with them but we didn't understand it um most women will have an average of five symptoms the most common is fatigue sleep disturbance shortness of breath indigestion anxiety well we have those things all the time so it's really hard to distinguish that from a heart attack uh only 30% will report chest discomfort um and this will go on for a month before they actually have their heart attack so this is a challenge and this is something actually Stanford is really aware of uh Jennifer treml if any of of you have heard that name has a wonderful women's heart health clinic uh so she really caters to women's cardiovascular issues and I'm going to show you some of the things that she does now some of the things that are a challenge is all of our tests are male driven so the test that we have to distinguish and different and and determine heart disease the the exercise treadmill turns out to not be very good for figuring out that women have a heart attack it's much lower specific for women compared to men really high F false positive rate um and so often a woman will get a special kind of exercise test but if she just gets a regular exercise test you're going to miss it um stress echocardiog cardiography and nuclear profusion scan the sensitivity is similar but it's not great uh and so on and so forth uh that when we talk about some of these other issues we just don't have good tests now actually Jennifer has some good test so I'll mention a few of them but I want you to know this term acute coronary syndrome it seems to be what most cardiologists are calling um these uh Mi so when we talk about stemi we're talking about uh my cardian farction so this is either St elevation or non-st elevation and we go through a series of asymptomatic to then stable angena and going talk about angena in a moment uh to then unstable angena and then we move into the acute coronary syndromes of the non stemi and stemi and the reason why I want to show you that is that when we talk about women compared to men um as a group they come in they're older so we already talked about the 10year age difference um they have more comorbidities they've got a lot more hypertension they have higher cholesterol they have diabetes they basically fall into a group where you have to manage lots of other things things besides just their heart disease they're much more likely to be depressed um before and after their diagnosis they're more likely to have heart failure than a man they're more likely to have a history of chest pain angena uh and more severe angena they're less likely to present with this um the St elevation Mi and they're more likely to have no obstructive disease and that's what we're going to get to in just a moment I'm still going to go through a little bit more list here um they present later so that's kind of the older thing they're slower to receive treatment that's not their fault that's the medical system doesn't realize they need to be treating these women um they're less likely to get the guideline based medical therapy um now a woman who's got heart disease aspirin is fine we're not talking about primary prevention we're talking about managing a disease that's there and also they're less likely to get statins um they're less likely to have an angiogram and to get revascularization but there's some good reasons for that I'm going to show you in a moment um they have much higher rates of severe bleeding especially with some of the procedures when you get an angiogram they actually thread a catheter they usually thread it up through your femoral artery uh it actually turns out I'll show you some data that going through the radial artery is a much better thing for a woman and actually better for a man and Jennifer treml has taught all of the uh cardiologists at Stanford to do it that way um they're less likely to be referred for cardiac re rehab and they really need to go to cardiac rehab now a lot of women don't find it a very good environment because they're kind of you know one or two women in the group uh and so this is a little bit of a challenge but they really should be referred to cardiac rehab and uh I'll show you that they have more death so this is this radial versus femoral access issue and basically um there have been some studies now to show that you have this major bleeding that occurs with the radial and it's uh guess I don't have the scale here but this is men and this is women so for women it's much much higher bleeding but also when you go through the the rist to get this catheter threaded you see that men also benefit tremendously as well so this is a case where in trying to solve the problem for women I think that we've really done men a great service and I think that that's what we're going to keep having as we do sex difference research uh this is kind of showing the same basic thing that basically bleeding complications are much higher in women than they are in men um and uh actually these are two different kinds of things so this is looking at uh the angioplasty so you can see this difference so again women are much higher than men but also the radial versus femoral also vascular complications are much higher in women so these are big issues and these are issues that I think Stanford's really taking on in a serious way now there's an interesting Paradox and I alluded to it earlier um and this Paradox relates to the fact that although I've already shown you that when we talk about uh having coronary events women versus men that men are uh clearly having them at earlier ages than women but there's another side to this story and that is if a woman's had a heart attack and is put in the hospital the younger she is the more likely she's going to die than if she's an older woman and this relates to the fact that we're just blind to the fact that young women could possibly be having heart disease we just don't get it whereas older women we kind of ready for the fact that hey this could be a heart attack uh and this is actually showing the odds ratio for death during hospitalization for a myocard infarction uh in women versus men and what you see is these younger women have two-fold higher rates and then as they age they get closer and closer to men so this is a big issue that we need to start to understand how to identify these heart these problems in young women and this is kind of one of the original studies to look at this Vino uh where you see in the young women so this is women in red and men in the blue that this is the death rate this is uh myio cardium infarction uh this is actually looking at uh the hospital mortality so women who died compared to men and you see that young women are much more like to die than young men but as we get older then it kind of evens out and so this is an interesting issue where we've got now this hormone story that's a little bit different for young women versus old women and then we have the entire medical management problem which isn't just medicine so it's not just gendered medicine I'll show you some data to show you that we've got some different biology but uh this is looking at that sex ratio this is kind of the same basic story uh looking at the hospital deaths and that this is actually your odds ratio so if you're younger than 50 uh and you're a woman you're two two more than two fold more likely to die because of your myocardium farction after you're in the hospital than a man of your age uh and you're you're better off if you're 80 plus which again was when most of them are happening but still uh and this is kind of the same basic thing just kind of giving you another picture of that uh when you even when you adjust just for all kinds of things uh so this is one place where a lot of attention is going now uh and this is true when we talk about coronary injury also bleeding younger women just have a much worse outcome than do older women uh in our hospital setting in our medical system and this is kind of more of the same now looking at uh major in hospital complications uh and so what you see is younger women in black they Fair the worst younger compared to younger men uh older women compared to older men um it's not so bad for these complications up here um now this I don't expect you to really see what's going on here so I'm here's the story but bottom line is uh 74 population samples were built into this huge uh database this is the line of identity so anything that falls on that means that it's the same um it's kind of equal and basically had 13331 anen cases uh in um a huge sample of women and uh a huge sample of men and the bottom line was chest pain is much more prevalent in women than it is in men uh and this is something that uh now you see it's kind of hard to actually pull that out of those data but the ratio is about 1.4 uh among Americans uh non-whites are higher than uh whites uh but the this chest pain thing is a really interesting phenomenon many women will go in with this chest pain and what we do with chest pain the first thing that usually happens is you get an angiogram and the angiogram turns out to very often not reveal any coronary disease in women whereas most of the time it does in men so 87% of the time that a man has a chest pain and he goes in and he gets an angiogram they actually find an uded artery and it's about 50% % for women so they get sent home and this is actually something that generated a huge study that women who come in with these symptoms of angen are less likely to have obstructive coronary artery disease and so they're sent home with a clean bill of health but in fact we a study was done called the Y study so women's es schic stroke evaluation syndrome evaluation trying to understand what is this chest pain why do women have it what is it and it basically required getting brand new technology to try and understand what is the real pathology going on here uh it's not benign they often end up with a heart attack and it's a serious event now this is just a schematic um I'm sure you've seen better from previous lectures but to kind of make the point that as we get older we start out with normal vessels we start to get fatty streaks in the lining of the vessels um these fatty streaks by the way they see them now in two-year-old so it's um I don't know if it's because of our bad diet or what a fibrous plaque starts to develop uh then you get an occlusive plaque where if you go to put an angiogram in there you actually see a 60% stenosis and that will be then your definition of a plaque you won't have a heart attack with that but if it ruptures then you end up with blood clotting and you have your heart attack or you have your unstable angena your coronary death your stroke your critical leg es schema at this level you're already getting your effort angin claudication and the reason why I'm showing this is that if you went in with a an angiogram into this vessel um here this yellow is the plaque and you actually can see that it's blocked and you would see a block there and the cardiologist would put a stent in and you've probably seen that now in one of the other lectures uh and if you actually were looking on the other end of The Vessel and you watching the pressure you you basically would see that there's this huge pressure drop right where you've got the blockage it turns out that these young women uh and these women with angena where they go in and they don't see this uded artery um they go in there and they just say oh women have thin smaller arteries when in fact they've got this whole case this whole sheath of plaque narrowing the whole thing but there's no individual blockage and that's a real hard thing to stent now I can tell you Jennifer treml is really good at I'm sure she's trained many other people but what you can see from the opposite what you see from this is that this is a slow steady drop but the bottom line is you're going to have a heart attack here just like you're going to have one there you don't have enough blood and there's now kind of a new um movement ahead to redefine this we call this generally coronary artery disease there's a coronary vessel blocked and where uh a lot of folks are going now with this one is they want to call it esic heart disease because you still end up with no oxygen at the other end es schia but you don't really have this kind of blockage that we are used to seeing and so the angiogram is not going to pick this up very well uh what they actually have now is several other methods um before I say anything let me just say that when we talk about this we don't with the case of the classic ruptured plaque where you end up with your blood clotting and then all the whole uh arteries uded this is very characteristic of older women and men um you have a thin fibrous cap that basically has a huge necrotic core infiltrating with foamy macroasia so these are those things that are chewing up uh the area whereas the younger women have what we call erosion instead of explosion they call it erosion where you have a thrombus over a base of Rich smooth muscle and the necrotic lipid core is often absent question are the pla esic artery disease still you know I don't think so he asked is the plaque that's in the um the the more female type that we're the young female type that we're calling the esic artery is that as unstable I don't think we have enough data to really talk about the stability factor of this this is kind of new stuff and the whole idea of eskee es schic heart diseas is really the last two years people have been saying we should rename this um now some of the other causes of it it's not just that problem what they've also found now with these women with this type of esmia is that they have endothelial dysfunction actually both men and women have that uh and actually women have a better endothelial function than men uh generally speaking it declines with age but these young women who have this particular problem they often have tests they can do to see that that can be part of their chest pain and end uh there are special tests that you can do to te to find that um I guess I didn't I took my ivis picture out but I I'll go back for a second and just say that what uh what they're doing now with these is there's something that's called intra intravascular ultrasound where you actually can put a a ultrasound you can put a a guide wire down through the artery and you can see the the uh Lumen you can see the the the walls of the vessels and you can really see that there's this plaque all the way along and so uh there's some wonderful pictures that you can see but you can pick this up very nicely it's an expensive test but um it's a good way to find out that you've got that problem and then the other one I'll just quickly mention is vasomotion that there's a test you can do where you put acetal choline in to see if in fact the artery will spasm as I mentioned that 160 well this one is um a study of 163 women that had suspected es schia 75% of them had no uded artery so they did the angiogram they would have been sent home with a clean bill of health and then it turns out that they show that they have this particular vascular reactivity and that 50% of them also had endthe dysfunction so we're finding out that the pathology can be detected we just need different tools and the angiogram is not really good enough to do the job so this is where we're now talking about esemicolonr they have more preserved systolic function but they have more symptoms lower function and more adverse outcomes after they've had the Mi and the cardiac procedure if they're younger probably because our medical system just doesn't know how to deal with them and that's where a lot of our focus in the cardiovascular institute's interest is so rounding uh coming coming to a close here uh is es schia based on microvascular disease so in addition there's one more piece to the story and that is the small vessels so in the case of men we focus on the big vessels the big coronary arteries and what we're seeing with a lot of these women is that they've got disease out in the small vessels I took that picture out I should have kept it in um but basically uh it's much more common in women and some of the things that you might know about Rod's disease have you ever heard of that Soo disease is something that's much more characteristic of women it's a disease in which you actually have this vascular sensitivity that if you put your hand in the refrigerator of a super at a supermarket you can actually get frostbite because it just completely shuts down and so it's very common in women compared to men um also migraine is much more common in women than men there's a real hormonal uh relationship with migraine actually it depends on whether it's uh with aura or without Aura I'm not going to take you through all that a coronary spasm is more common so the bottom line is we have different physiology different pathophysiology uh particularly when we're talking about younger women um retinal microvascular abnormalities predict cardiovasular disease in women but not in men so this is a place where we're getting more and more information that I think will be able to do a better job for women and with that it looks like I could have kept you know another 20 slides but I didn't so um uh basically I'm going to end with uh the bottom message that applies to men as well as women when we talk about prevention for coronary heart disease and when we talk about when we talk about prevention by the way I'll just quickly say something um in cancer we talk about primary prevention trying to prevent ever getting it and we say the same thing for coronary artery or for heart disease and then in cancer we talk about secondary prevention which is the screening and detecting and diagnosing um whereas in heart disease we call secondary prevention you've got the heart disease you're trying to PR another one uh and then tertiary prevention we talk about in cancer where you've got the cancer it's detected and now we're trying to control it and manage it and keep you out of a crisis and that's kind of Where The Chronic Health the chronic disease model is going we're talking more and more about tertiary prevention where if you've got the disease we're trying to keep you out of Crisis and so um when we talk about this it doesn't matter what level you're at whether we're talking primary secondary or tertiary we we really do have lots of evidence that whatever you can do physically you should that moving is is is good and we move very little now everything is buttons I mean you all remember when we used to wind up our car windows Nobody Does that anymore I mean that was one little movement that we had and there's just like one after the next when you start to think about how you just don't do anything anymore you just push buttons everywhere so be physically active U maintain a healthy weight vegetables I think is more important than fruits and whole grains but those are good too restricting saturated fat limiting your salt we have data on this for women we have data on this for men um as I said I I uh took out a lot of slides and that's okay because I think you probably had enough adequate calcium and vitamin D vitamin D is a big story now we don't really have enough data on how that relates to cardiovascular disease and men or women uh limiting alcohol intake minimizing stress um didn't bother me that we moved the rooms that was fine um get enough sleep that's one that I don't do but hopefully you do I can tell you the Stanford students don't get that one uh and uh know your risk factors so just out of curiosity how many of you know your HDL cholesterol level a few of you don't how many of you know your LDL cholesterol good how many of you know your blood pressure how many of you know your uh blood sugar okay how many of you know your BMI your body mass index good okay you're clearly an enlightened group here uh and that's great and then the the important thing is how many of you are getting enough exercise how many of you are are watching your weight okay so the bottom line is you all know what you need to do and what I'd like to recommend is that you go home and do it for women I think what we realize is that we just need to keep learning more about us so that we can be sure that they learn what they need to do to take better care of men so they can take care better care of us too thank you so actually I remember the reason I took them out is that I was told that there'd be lots of questions so here's the time yes it is so actually we're doing this amazing study right now men um oh she she said that she asked me about sleep apnea so it turns out that sleep disordered breathing and sleep apnea C are certain things more important for Stroke versus coronary heart disease and other things so sleep sleep disorders are highly related to coronary disease you know I'm not sure we really have a good cause and effect relationship there there's an association but I'm not sure which way it goes whether it's the heart that's causing the Sleep problem or the sleep that's causing the heart problem and in some case the sleep apne is highly related to obesity that's probably where you saw it uh and in that case the Obesity Factor probably is causing the sleep apnea because the weight on the chest and when you go to sleep at night you actually have like this whole phase of of muscle relaxation you just you're kind of paralyzed actually and so um it's just harder to lift that weight off of the chest so that's at least one of the explanations for that issue other questions you make a comment on the relationship between coronary um so vascular dementia is basically expressing you know when I had that one slide where you could see the blood vessel and then I'm showing all the different things your blood vessels can have this plaque in them whether it's in the brain whether it's the cerebral vessels whether it's in your leg and it's the peripheral vessels or the coronary vessels and so when we talk about vascular dementia essentially you're just not getting enough oxygen to the brain um that's the main thing which is really different from Alzheimer's disease where there's a really different kind of pathology there are these Tangles and things um the answer is we don't know I mean right now we're just discovering that it's true oh the question was why is it that women have this sheath of plaque and men have this block plaque now the first thing I want to just remind you is that older women tend to have the same kind of disease as old older men so it's not just men versus women it's the younger women who are coming in with this chest pain and these women who have angena so the answer is we don't really know uh and we also know that it may be that and it may be the microvascular disease so the small blood vessels uh and it may be the endothelial dysfunction so there's at least three different possible pathologies and I don't think we have very good data yet on how often those happen I can tell you I've mentioned Jennifer treml a few times she's actually doing this great study the Y study only had women and then the data came out and everybody said oh women are so different from men but it's like well how do you know you didn't have any men in the study and so she's actually doing a study where she's looking at men and women about 40% of um the men that she's seeing will have this cor about 40% of the people who have this coronary this nonobstructive coronary disease are men uh that's said almost all the time when men have chest pain they have this blockage but at any rate um she's found it really hard to find enough men in the study to even look at that so we don't have enough data to say why are women different from men because we don't even know yet that they are once you're in that particular category of disease that probably wasn't a very good answer but um the ANS is I don't know but we we're going to find out I mean hopefully we'll find out you talked about the young women going into the hospital and uh having heart attacks basically is that because they were undiagnosed or no they were admitted for a heart attack so these women are admitted for an mi into the hospital but then what I was saying is that women who are admitted into the hospital with an MI um young women die at much higher rates than young men who are admitted with an MI well probably for the reasons I've been saying so one thing is they're misdiagnosed uh well in this case they they're having their heart attack but they're not treated right they're not given the same quickness um the door to balloon time that I mentioned is a really important factor and so you've got to be acting quickly and they don't act quickly enough so by the time they're in there they've just you know too much time has gone by they're not they're not treated as aggressively when they have the procedures they have more bleed leing they just have more complications so it's um multiple reasons so this sounds like an area where Physicians need to be trained and educated better yes whereas many these other things people yeah so he's saying Physicians need to be trained we all need to be trained because the biggest Pro a big part of the problem is even before the the ambulance doesn't come for for you the woman doesn't know she's having a heart attack so she doesn't call the ambulance and so we've been trying they used to say that oh the first she's getting dinner ready for the family and doing this and doing that so we used to talk about that and you no one really had the data to prove that so we stopped saying that that was true but it might be true but at any rate um women don't recognize that they're having the symptoms then when they call and they have these diffuse symptoms it doesn't sound like an elephant standing on the chest I'm not going to rush over there for you then I get to the the uh emergency room and you have the same problem so it's delay delay delay and this is a big focus and I can tell you when you say education that's really the heart association's red dress campaign has been trying and actually we started this in 1989 in 1989 the AHA had a big campaign called uh heart disease is an equal opportunity killer trying to let people know that it could get women too and that didn't work so the red dress C CA campaign came later so it's a a long hard process to educate the public that women are at risk for heart disease and then the Physicians Reas the hospital and have a surgical intervention because the blood vessels in women are smaller and the whole surgical uh apparatus is not to the fact that the blood vessel smaller we already heard that for the Pediatrics is it true for women well I think that that actually I can tell you since I've been doing this for so long that was where a lot of the focus was in the beginning they were all talking about the smaller vessels and they were using pediatric catheters uh for women so they were you know recognizing the need for the Pediatric catheters and they were using those but I think that we've now moved to the point where that could be part of it for some women but it's not the whole story and then we have other issues like when we start talking about transplanting Hearts you know there's a lot of really interesting stuff that Stanford has done showing that when you transplant a woman's heart into a man and a man into a woman it's just not as good as when you're transplanting them in the same sex because you have to change the blood vessels because that heart's not ready for that blood vessel so there is a lot of size to take into account but I actually think the pathophysiology is going to take a lot lot more time and attention for us to resolve as well in the back yeah you know as I mentioned I didn't go into the diet part there's like another whole prevention talk that I was saying to Sophie we really should have a prevention talk um I I try to put as much of that into that um you know it it would be better to get Christopher Gardner to tell you about all the gumes and all the different kinds of foods because there's a lot of debate and challenge what I will tell you is the direction that the American Heart Association and the nhlbi are clearly going is not focusing on individual Foods as much as diet patterns and so that you're really talking about how your legumes and tofu and those other things would fit into the overall scheme of what you're eating uh and actually um one of the slides that I showed you where they were taking apart the different blood pressure and to and tobacco and those different pieces the same uh the same authors uh maaf faren has just done a really interesting paper in the New England Journal of Medicine on diet and the different parts of diet that really make a difference and I'll just tell you the thing that was really interesting was yogurt turned out to be like like a really one of the strongest good things to eat which none of us are quite clear why uh that's true but it's an interesting U mafarin is his name so that would be a place for someone who interested in diet but diet is another whole talk anything else okay well thank you very much for more please visit us at stanford.edu
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Medical_Lectures
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The_Necessity_of_the_Immune_System.txt
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[Music] Stanford University good evening everybody wonderful to see you again let me just get a quick show of hands how many of you are new for the first time for this quarter all right well welcome my name is Phil Pisa I'm the Dean of the School of Medicine and this is our third quarter so for those of you who are just joining you've missed 20 incredible presentations but don't worry we have 10 more to go and they're all going to be wonderful as well I should say in passing that as you if you are coming for the first time you're going to be leaping into new fields that are really quite exciting and interesting and each one of them in a sense is sort of self-contained but for those of you who have had an opportunity to be with us either from the beginning or from the winter quarter I think as we get a little bit further you're going to begin to see some connections which is often what begins to happen in medicine I often tell our first-year students when they're beginning that it's like parachuting right in the middle of a knowledge field and you have to kind of look forward and backward and around before you begin to get some of the intersections of the information that really brings what we deal with alive and well so a couple of practical things first we will be meeting here each Tuesday for the next ten weeks we'll start pretty promptly at 6:30 we'll end pretty promptly at 8:20 and respect your time they'll always be an opportunity David for any of you who have questions that aren't answered during the presentation to come forward at the end and have an opportunity for some additional dialogue we want to foster obviously your knowledge and hope that you will be great ambassadors this has turned out to be we're told the most popular continuing education course in Stanford's history at least measured at least measured by the number of you now the way you rate it at the end of the day can have a different nuance on that but we're pleased to be part of this the reasons that we're doing this began because we were last year celebrating our fiftieth anniversary from the time that the medical school moved here from San Francisco and it was a seminal time for us but I must tell you that this course also began thanks to a Stanford undergrad who came to my office one day Eric lifer and said he had heard about mini med schools and why weren't we doing one true undergrad stimulus and I reckoned and told him that I'd actually participated in one of these programs decades ago and was well aware of how significant important they could be and we decided to go from what is often the kind of minimal response that many medical schools do where they'll have a handful of lectures and call it a mini medical school to a three quarter event so I think we've covered the waterscape on this one I'll also tell you that if you're interested in going backwards and looking at what we've done the first quarter is all online and it's on YouTube and I think an iTunes and really quite doable I've looked at some of them so you can enjoy those and last quarters will be up relatively soon we purposefully delay a bit just to keep the action live before you in addition Cathy Gilliam who I'd like to have stand up and acknowledge who has been literally my partner in crime she is the senior adviser to the Dean but we work very collaboratively and when this idea arose I asked her to work with me on it and she's done a splendid job in helping to keep everything together and moving forward and she tells me that the syllabus for the rest of the this quarter will be online tomorrow and as you likely know we'll be covering a variety of different selected topics this quarter we're going to start out with immunology and I'll come back to that in in a moment and then we're going to move in a couple of directions actually move is the correct term because we're going to have a session on movement and then we're going to talk about what happens if you don't move enough which is obesity a real problem in this country in fact it's been forecast that the two things that could change longevity in the United States from the current projections are one [Music] unreconciled here this evening and the second is obesity so these are real issues and we'll want to talk about them and then we'll go to something that I'm particularly concerned about which is aging and all of the science that relates to it and for those of you who are here last quarter you'll remember the very interesting presentation that we had about memory that relates also to to that process and then we'll kind of end with a trilogy of topics that relate to cancer cancer biology and some new innovations and we'll finalize the the quarter with the fusion product that has been quite characteristic at Stanford which is the connection between stem cells and cancer so you'll have a pot pourri of presentations again each contained within themselves but you'll begin to see the intersections between them now I have to tell you with regard to our presentation tonight that when I was in medical school decades ago immunology was a to lecture course there wasn't very much known and the reason it was to lectures is one was on b-cells and the other was on so-called t-cells and they actually had names associated with them but this has been an area that has evolved enormous ly and is perhaps one of the most complicated areas in all of medicine because the immune system may rival the nervous system for its intrinsic complexity but as you'll hear tonight it really it really stands at the center piece of what protects us what defines us it's what gives us our own personal identity protects us from all the organisms around us you'll remember from the first quarter that while you may be comprised of around 10 to the what 13th or 12th cells you've got at least a log more bacteria and germs living inside and outside you and it's the immune system that creates the balance of power within them if the immune system gets out of whack it leads to autoimmune disease if the immune system is altered it doesn't only increase the risk for infection but in my world it also increases the risk for cancer so these connections are important and you'll see how much the immune system discussions are going to play into our subsequent presentations next week on autoimmune disease and the one that follows on vaccines nomen vaccines has probably been one of the things that has so transformed human history in the last last many decades and I can't think of a better person to lead off this presentation than my friend and colleague David Lewis and I say that because he's a pediatrician and you know I come from that ilk and so that's always a good sign at least he'll be nice I don't know how smart he'll be we'll see how that goes but nice-nice is the usual term that goes with Pediatrics David grew up in California he told me to say he grew up in a log cabin but I don't really believe that but he did migrate east as many have as I've done these introductions there was a kind of phenotype that he merges he was at Yale for his undergraduate work and then moved back here to do his MD degree at UCSF and then moved a bit north to Seattle to do his training in pediatrics and immunology one of the great fonts of pediatric immunology is at Seattle Children's Hospital and I think David was well mentored there he now directs the immunology program in Pediatrics both at Stanford and Lucile Packard Children's Hospital and is really an expert on congenital immunodeficiencies meaning when you're born with a risk for not having a completely formed immune system and there are many consequences that unfold from that but I also have the pleasure of interacting and working with David on the infectious disease service so he kind of crosses that line between the immune system and infectious diseases and I think that's going to be where he starts his saga I'm with you tonight so let me introduce David Lewis well thanks for the kind introduction I think I'd like to start with a joke about how a lot of non immunologist medical people view immunologist so this guy is robbing a bank he sees these two really well-dressed gentlemen he says I'm taking you hostage and they go into this getaway car and it's up this very steep hill with a very sheer escarpment on one side and it's an old car and it's kind of underpowered and they can hear the sirens getting louder and louder he says I hate to tell you guys this I'm going to have to throw one of you out of the car and I don't know how to decide how to do it maybe I'll base it on what you do for society so he points the gun at the first guy and he says what do you do with this is I'm a cardiologist he says what's a cardiologist he says well like if your baby is born with a hole in his heart I make sure that the surgeons patch him up or if your grandmother has a heart attack I make sure she leaves the the ICU he says that sounds pretty important he says what do you do he points the gun at the next guy who guy says I'm an immunologist he says what's an immunologist well and then the cardiologist says just shoot me I don't want to hear any more of us so so you are now for warm if anyone wants to leave now it's fine okay so what I'd like to do is is really provide an overview first why do we need an immune system then talk a little bit about innate immunity and some historical observations that then led to the thought over a thousand years ago that there was a special form of immunity that we call adaptive immunity and then how innate an adaptive immunity worked together to protect us and and really properly integrate the immune response and then a little bit about how things can go wrong and how we might fix it so kind of a tall order but we're going to try and do that so immunity really is a biologic term that describes a state of having defenses against infection so the focus really of why the immune system came into being is to protect us from some bad pathogens that are out there and we really live in a microbial world we may not believe it as much now because of modern society but our immune system has had to deal with all kinds of potential pathogens and I'm going to briefly because I think it's so important to understand the immune system I'm going to actually introduce the bad actors if you will first and though I'm a peace-loving person and I've been accused of being a war monger if you really look at the best metaphor for the for the immune system it is a defensive strategy and a lot of military strategy that is independently evolved in society once civilization reached the point of being relatively complicated and you had a city-state that you needed to defend against these Marauders very simple what we've really done in humanity is come up with in parallel the same strategies that the immune system has been using for hundreds of millions of the big decisions have to be if there is a microbe out there is it a friend or a foe or something that you can ignore if it's a foe how do you best mobilize defenses to contain it and the immune system makes a lot of distinctions about whether the foe is inside cells of the body or in the outside in the humors you know the circulating fluids of the body different strategies are needed for these two locations and then finally once the foe is eliminated you have to just like with the military somehow get them to demobilize so you're not using all your resources inappropriately and becoming just a giant lymph node at the end of the day so bacteria are some of the most important pathogens so I'm showing you here these round Staphylococcus aureus this is a form of this is called mr si that's that's highly resistant to a commonly used antibiotic this bacteria forms these pus containing infections frequently a less frequent infection but one that we've unfortunately had some encounters with in in this millennium is the anthrax bacillus which is this sort of long as they're called Boxcar like bacteria they can form these ulcers where there's a lot of dead cells that are forming this crater as they're being eliminated some some bacteria have really had a disproportionate influence on the evolution of the immune system meaning that they're so important that the immune system has really had to tie up a lot of its major defense mechanisms just to keep them at bay one example is Mycobacterium tuberculosis which is the cause of tuberculosis and you can see the bacteria they stain what a lot of the infectious disease people call little red snappers they're inside cells these are bacteria they can cause these cavitary lung disease particularly at the tops apices of the lung the bacteria are very smart like to hide out inside cells of the immune system they particularly like cells that are called mononuclear phagocytes and they actually live very happily in these cells and these cells live for a fairly long time so this is actually a way of evading the immune system so you need to get other components of the immune system to wake this cell up to eliminate this pathogen and even today with all of our therapies almost 2 million people are dying every year from tuberculosis and 30% of the entire world is infected another example of a of a disease that's not gone that that at least historically even in the United States and and more developed countries used to be a really big problem and it still strikes terror I think into the hearts of many of us as clinicians is something called Neisseria meningitidis or the also called the meningococcal so this is also around sort of bacteria in parts of Africa it still is causing probably one out of a hundred infections in in the population in certain countries and the mortality even the United States with all of our technology can still be as high as 10% but as but as dr. piso mentioned we are also awash in bacteria that are in our guts you heard that there's at least tenfold more bacteria than there ourselves in the human body and some of these are actually quite helpful they're helping us digest foods or helping produce vitamins and we call these these bacteria commensals in that the relationship is that we get something out of these bacteria being in our body they don't get too much other than it's sort of a free ride but commensal comes from the sharing of food but the important thing is that you don't want the immune system to inappropriately attack these these bacteria and that's actually one of the things that I think we really have a long way to go in in fully understanding I'll give you a little bit of Vince that's recently come from a few studies as to how the immune system tries to distinguish between these commensals that are harmless and true pathogens in the gut okay so we've dealt with bacteria but there are other kinds of organisms there are fungi which are more closely related to us than bacteria in that they have a nucleus in their cells that makes them a little bit harder to treat because the kinds of antimicrobial therapy tends to overlap somewhat in side effects on ourselves with with those that inhibit them this is an example of one that we see mainly in in patients who have a problem with their immune system particularly with t-cells not working which we'll talk about called Pneumocystis this is what was one of the major causes of death from HIV infection and still is unfortunately in many parts of the world where there's limited access to therapy to prevent this infection so this is a very important infection in terms of human mortality even now then we can move up a little bit in evolution to parasites that are a little more complicated than the fungi there are some that are single cells in this case malaria you can see that malaria is still endemic in much of the tropics and amazingly enough there are still about almost half a billion cases per year in the in the world and it kills between 1 to 3 million people a year unfortunately most of these people are children that live in sub-saharan Africa that die from falciparum malaria so this is one of the reasons that there's really a push by the Bill and Melinda Gates Foundation to try and get a a malaria vaccine if you look at all the things that we could do to improve humanity's lot this would probably be at the very top in terms of just the impact on the on the total number of people now some parasites as you know I guess this is a nice Pleasant after-dinner conversation is that is that they can be actually quite large and become these worms that like to inhabit the gut and sometimes a respiratory tract I'm using for an example a worm that's called schistosomiasis again this is particularly prevalent in Africa some parts of Asia as well this shows you a little larvae of schistosomiasis and what I want you to notice are all these little white blood cells that are attempting to eliminate this larvae and you can see how large these worms are relative to white cells so the immune system for some pathogens that are so large has to come up with a completely separate strategy to get them out of the body if possible which we'll talk about finally viruses I'm sure Harry Greenberg when he talks about vaccines because he and I are both very interested in viral next things are going to talk in more detail but viruses are kind of like software viruses and the in fact that they're just an incredible diversity they can be small they can be large but unfortunately just like computer viruses are almost never beneficial they replicate inside cells you can have a acute viral infections such as influenza where in relatively short order the body is able to rid itself completely of the virus until the next time that you get a brand new infection say for example a year later these are obviously very important for example influenza probably kills about up to five hundred thousand people a year in the world in a typical epidemic but when we have a pandemic that it affects a larger number of people in the population it can be substantially higher thankfully the current pandemic that we're finally seeing light at the end of the tunnel in most parts of the world is probably not going to be that bad compared to some of the others that for example cost three million deaths or the 1918 pandemic that caused its estimate between 30 to 50 million deaths but unfortunately these pandemics caused a disproportionate number of deaths in otherwise healthy people particularly children so it's always a concern that we don't have better ways of controlling these pathogens there are many viruses that have a very different kind of history once you're infected with them you never get rid of them so the herpes viruses are kind of the classic example of this and I guess there's the joke about what's the difference between true love and herpes viruses and the answer is herpes viruses are forever so cytomegalovirus for example affects probably if you look at the world population overall in this country it's not quite so high but probably on average about 85 to 90% of humans by the time they die will be infected with this virus and it can cause serious problems in pregnant women by causing the fetus to be born with a number of congenital anomalies particularly deafness and possibly can can cause a mental retardation so what's going on here in terms of what led to the development the need for the development of an immune response well probably when organisms were only living for a brief period of time it wasn't a big deal but once you have a multicellular organism that needs to be around for a while to reproduce it needs the protection from invasion with these microbes because from the point of view of the microbe a big multicellular organism is like a bunch of goodies that they can just grow on and and themselves reproduce and in the case of mammals or homeo therms the thermostat is on and that makes it even more attractive for them to grow that much faster so the innate immune system developed first and it was poised to act immediately in the event that an intruder came into this multicellular organism and innate immunity goes back to the very beginning of multicellular organisms you can see it in sponges sea urchins insects as well as ourselves the characteristic thing about the innate immune system it's ready to go all the time but it's a stereotyped response so the response it gives the first time it sees a particular pattern will be the response it gives the second third or fourth time it doesn't really learn or adapt its immune response to to the to the organism and it uses clues that that there is a pathogen around by having receptors that are able to detect the difference between pathogen derived products and things that are normally produced by the own host cells what's interesting is that in many cases trauma or damage just by itself will it will activate these innate immune mechanisms and the way I look at this teleologically is that in the past during evolution most of most of the time during human development when trauma almost always occurred in a non sterile setting you might get bitten on the arm by the saber-toothed tiger you might have been wounded in battle with a spear or whatever but there's going to be dirt associated with it it's only very recently that you might actually have trauma that might actually be a sterile thing where it's not introducing microbes so I think a lot of the parts of the body that deal with trauma for example clotting to protect you there's been a linkage to at the same time turn on the innate immune system because you're really most likely are going to need it so innate immunity includes just the barriers of the body for example the skin is a barrier for entry of micros for example and many of these what we call epithelial linings of these of these tissues that have to be crossed for the organism to get into the deeper tissues actually produce natural antimicrobial peptides these are small chains of amino acids that are very potent at actually inhibiting both bacteria and fungi and this is really a field of immunology that still I think somewhat in its infancy and could be much more exploited for therapeutic purposes one of the oldest cells again goes back you can see these in sponges and other multicellular fairly primitive organism is the phagocyte and what the phagocyte does is it takes up the microbe puts it into a compartment that's shown here as the phagosome and then has ways of killing i in a neat and clean way inside the cell so it doesn't cause a lot of disruption of normal bodily processes one of the things that the phagocyte also does is it puts out little proteins that we call cytokines that are sort of an alarm to bring in other cells that are even better at killing than it might be the so-called professional killers and to cause what we call inflammation so inflammation is basically just an accumulation of white blood cells at a site of infection or some kind of damage now the key discovery I think for innate immunity was really made not until 1997 by Charlie Janeway jr. and and meds etaf when they discovered that there are these special receptors that innate immune cells have called toll-like receptors toll has nothing to do with like a toll road toll means cool or far out in German and it's just the name that these receptors have this interesting effect in fruit flies so a guy who's a postdoc said well I'll call them Toru and so now these receptors are similar to the toll receptors of Drosophila but that we call them toll-like but they're very important because they recognize specific products that are unique to two pathogens but not to the body's own cells so for example there are these things there's something called LPS that's made by a kind of bacteria that's called a gram-negative bacteria and this tells the body that there's been an infection with this particular bacteria triggering it and triggers the cells to release some of the same meteors that I talked about with mononuclear phagocytes or phagocytes that is helps set the alarm off to bring in pathogens to I'm sorry to bring in white cells to control pathogens so there are a number of these toll-like receptors some are specialized for recognizing the nucleic acid of viruses many are on the cell surface are specialized for recognizing bacterial and fungal derived products but once they set off the alarm with these inflammatory cytokines they bring the white blood cells that are normally just circulating around in the bloodstream and bring them across into the tissue to form collections of white blood cells that you may have experienced if you've had an infection as pus where you have this collection of intense collection of white blood cells these white blood cells called neutrophils are particularly professional killers for eliminating bacteria and fungi compared to just a generic phagocyte they are really about a hundred to a thousand fold more effective at killing now there's another system of the innate immunity that's that's very old and important and just to show you the diversity of mechanisms so this one involves tagging the pathogen for destruction by the phagocyte so the way this works is that there are proteins in your blood called complement proteins so this is a we all have substantial levels of these proteins all of time and they're constantly slowly being cleaved into smaller proteins that tend to stick to the surfaces of cells it can either be the pathogen if it happens to be around or your own cells if it happens to be on a pathogen there will be this amplification reaction to make more and more of these cleaves products and this also helps attract phagocytes to the site of where this is being generated and at the same time you also get the ability of the phagocyte to take up the pathogen much more effectively so this is called opsonization and the way I like to remember it is opposite this is quoting directly from George Bernard Shaw the doctors dilemma obstinate is what you butter the disease germs with to make your white blood corpuscles eat them and that in fact is what compliment does this compliment protein when it's tagged on the pathogen there are receptors for it that make it much more efficiently taken up by this phagocyte leading to the end of the pathogen and again another piece of the complement proteins that are cleaved helped attract more of these profession or killers to the site of the infection so you get both of these effects working together to help control infection there's a also a nifty thing that the complement system can do it can actually make holes in bacteria that where if the reaction is allowed to go to completion and these holes will actually cause the bacteria to swell up and burst now why doesn't this happen to your own cells the reason is that your all of your cells they do get tagged by the same complement proteins but you have multiple proteins on the cell surface that actually are there for one purpose and one purpose only which is to inactivate the complement system most pathogens haven't figured out a way of doing that so this is one of the most primitive ways of distinguishing between yourself and something that's not yourself that might be a pathogen there's also recently discovered something called an inflammasome so there are certain pathogen derived products that may get inside cells and this is another alarm system similar to the taluk receptors that will produce inflammatory cytokines that will help bring in more white blood cells in this case these also are very potent at inducing fever which is characteristic as you know of infection okay so that's all I'm going to say today about innate immunity though I think when you really look at clinical medicine probably our patients suffer infectious consequences from problems with it with an eighth immunity that we unfortunately in this country have the luxury of inducing with things like chemotherapy but adaptive immunity of course is really captured immunologists attention for a long time we've known since 430 BC that there was a funny phenomenon that sometimes occurred with very serious inflect infections so in the plague of Athens people were dying right and left but it was noticed at that time that if someone was lucky enough to recover from the plague they could then go into the population of people who were already you know were afflicted and not worry about getting the infection again so there was something about there was some process that was allowing there to be long term protection if you survived the initial infection and then of course more recently was discovered that you could use vaccination where you vaccinated with cow pox a relatively mild disease found on cow udders and you could then protect people from smallpox and again the the protection seemed to be for a best one could tell close to life long so one of the key things about adaptive immunity is that there's a learning process but once the lesson is learned there's durability of the immunity and durability of the protection and we now know that there really are striking differences between adaptive immunity and innate immunity so first of all innate immunity as I told you is ready to go right away it's on all the time unfortunately for adaptive immunity it takes probably about five days for you to develop the kinds of responses that are characteristic of adaptive immunity and unfortunately certain infections may win the upper hand you may die of Ebola virus infection or of meningococcemia before your adaptive immune system gets a chance to protect you the adaptive immune system unlike the innate immune system where I told you that it's going to be the same the first second third enth time it actually changes with repeated exposure it gets better and better if you get repeated exposure and the innate immune response is pretty much limited to the number of receptors you have of these total number toll-like receptors flam is ohm receptors and it's still a finite number it might be a hundred different tricks total in the body in terms of dealing with infection yeah is it is it constant throughout our life or does it sounds good so the question is innate immunity constant throughout your life are there differences with age there are some subtle differences with age but they're they're important ones so in newborns it seems to be somewhat slow to get started and then you kind of go through this honeymoon period where probably from childhood well into being elderly it's pretty constant at the extremes of age you may actually see again some decline in in innate immune function but again it's relatively subtle yeah the people who survive during the plague that would describe condition for what would be featuring about them to survive were diagnosed well it would be great to know but I think part of the problem with the plague of Athens is we don't even know what the actual microbe was that caused it so we really would only be speculating so that the question was what allowed some people during the plague of Athens to survive and others not but even if even in in current if you will modern plagues like Ebola virus outbreaks in Africa we often don't understand why there's such variable courses there are some immunologists that that like to believe everything can be explained by genetics perhaps but I think we still have to you know prove that I think the thing that's amazing yeah go ahead what about the so called immune enhancing things like vitamin C and echinacea first of all you know they work and secondly what do they do okay so the question is whether there are do these so-called immune enhancing agents like vitamin C echinacea work well all I can tell you is that most of them have very subtle effects in if you if you look at the evidence kind of based medicine and do immune ology I think they have a fairly modest impact on innate immunity they may have some moderate benefit but probably not in the in the bigger scheme of things anything like for example the impact for example of whether you're on an immunosuppressive drug or nice nothing like that that level of impact is true David that control trials of vitamin C and akinesia have showed no benefit yeah control trials so dr. B's was asking what's seen in control trials I have not seen any you know evidence that that there there's a significant effect statistically so so the important thing though about a DAP of immunity is it's highly tailored to the specific pathogen so if you get encounter if you're one of those unlucky people you travel to Africa you get Ebola virus it will come up with an Ebola virus specific immune response if you never encounter a bolivars it will come up with something else depending on what you encounter and one of the great mysteries for a long time was how is this specificity achieved so I'm going to try and I get to that and I think again one of the reasons that you really needed the development of adaptive immunity as multicellular organisms live longer and longer there was a greater chance that you're going to have repeated infectious threats that it would be an advantage to respond to and then in the event that you got second bout of this infection you could deal with it much more effectively so particularly with vertebrates we're now lifespans were now approaching you know years to decades you could see that there would be a clear advantage so the adaptive immune system involves the generation of many more receptors that are specific for what we call antigen that is that these are things that turn on the immune system that are derived from the pathogens literally millions of receptors to deal with any possibility of any kind of infection that you're going to encounter the adaptive immune system doesn't know tomorrow what that infection is going to be so it's approach has been generate receptors for everything receptors so receptor is something that can bind on the surface of a of a adaptive immune cell there's two kinds called t-cells and b-cells and I'm going to talk about both of them this can actually bind the antigen which is again some kind of product from the pathogen and turn that cell on it delivers a signal to the inside of the cell telling it it's time to get activated it's time to divide and it's time to carry out an immune response so to generate these receptors there's a unique pathway that's called vdj recombination the names kind of a lousy name but we're stuck with it it stands for variable diversity in joining but those are just the name of the segments that are used to make up these receptors and this is unique this system only works in two kinds of cells T cells and B cells it has no other role in the body other than to generate the adaptive immune system and what happens and you're going to learn more about stem cells I guess at the end of this quarter they can give rise to these early T cells or these early B cells the t-cells developed are called t-cells because they develop in an organ called the thymus which is in the neck and they undergo this this process of vdj recombination to generate literally millions of different receptors each cell has a different receptor the b-cells do the same thing but they're called B cells because they differentiate into the mature B cells in the bone marrow so B for bone marrow but the same process is used to create literally millions of different receptors that can see anything that the immune system is going to be thrown at it how does this happen so what these receptors do is you can think of them as sort of a lock and key thus getting a fills question in that a receptor will happen to have the right shape if you will to bind something from a pathogen to trigger that path to trigger that cell to divide and to become active in the immune system so because this is one kind of cell that has a unique receptor we call it a clone so so all descendants of this clone have the identical receptor okay and this is actually very important part of how the adaptive immune system works you start with a very rare cell that has a particular kind of receptor and you select that one to become much more prevalent to deal with the infection in response to it being able to recognize the infection so the thymus a lot of us are very interested in the thymus because it turns out that of the T cells and the B cells the T cells are kind of the master regulators of the whole show they kind of boss the B cells around so cells from your bone marrow enter into this gland called the thymus where they where they become mature T cells and then they leave the thymus and move out to the rest of the body where they can do surveillance for infection so we call these when they're outside of the thymus peripheral t-cells because they're now in the periphery of the body not in the in the thymus t-cells are very important for sensing infection inside cells B cells mainly function against pathogens that are outside cells and we'll see how that how that works in a second T cells are odd though they don't what they actually detect is they don't go inside cells to recognize pathogens what has to happen is that the infected cell or a cell that's taken up the pathogen derived material in the case of T cells they really only see one thing they see proteins and actually just fragments of proteins from pathogens that are taken up by these cells or that infect these cells these are ground up into little hotdogs like small pieces that we call peptides that are placed on these particular molecules whose only function is to present these things to T cells so T cells are only able to see little fragments of protein from pathogens in the context if you want to use that term together with these these MHC molecules I don't have enough time to tell you why they're called MHC molecules but if you want to come afterwards I'll explain it but we're stuck with the nomenclature so this is really very different that as we'll see of what B cell see but this is what T cells see and what they do is they will sort of do surveillance of all of your cells and if they come across a peptide with an MHC molecule that their particular receptor called the T cell receptor that's made by this VD j recombination if it can bind this with a high affinity triggers the T cell to become activated and a trigger adaptive immunity the last thing I'll do with Delmon cloture then hopefully we'll be done for the night with the with the bad stuff more or less is that there are two other molecules on the surface of the t cell that divide them into two kinds two major flavors we'll have to talk a little bit about some other flavors of cd4 T cell but for the purposes of tonight I'm only going to talk about one flavor of something called the cd8 T cell so you'll hear that a lot that their cd4 and cd8 T cells these are the two major subsets of T cells and I'll talk a little bit in a second about what they do but the cd4 T cells are really the master regulators of of the immune system they carry out a whole variety and orchestrate a whole variety of immune responses and they also boss around the B cells as well to some extent the cd8 T cells the only thing you really need to remember about the cd8 T cells is that they're licensed to kill they are professional killers and they kill one thing well which is viruses they are able to do that quite well cd4 T cells have a lot of other functions that they have to do so again it's this T cell receptor that both the cd4 T cells and the cd8 T cells have that recognize these little peptides sort of like hot dogs in a bun on the MHC molecule that's what is determining whether the T cell is going to get turned on or not so you can think of it as sort of a lock and key mechanism it's that shape formed by that particular peptide on that particular it makes a molecule that determines whether or not the T cell it's turned on so here's the problem if you've got say on the order of 10 million different kinds of T cell receptors so each each T cell is going to have or B cell for is going to have one unique receptor and there's 10 million of them and then like probably each one of those there might be 10 copies so 10 cells with that particular t-cell receptors circulating in the body how do you generate all of that when there's we know that there's only 20,000 genes in the entire human genome if we use the genome to encode each one of those receptors we'd run out of the genome pretty quickly so how does that happen and that is the is the mystery of vdj recombination that's now been solved and what's done is that you take little fragments of the of a gene that could become a t-cell receptor in this case and you select randomly from among all these different little segments in a particular in an individual t-cell it's sort of like a the menu at a restaurant you're going to select one of these segments one of these yellow segments one of these blue segments and in that cell you're only going to put together those three to form the final t-cell receptor so it's like a total deck of cards where you're selecting randomly and this is a totally random process now that's going to get you some a number of different kinds of t-cell receptors if you look at probability theory but it's not going to be enough to give you ten million so how does that happen I'll get to that in a second so what actually has to happen in order for you to piece together these genes is you actually have to make cuts in the DNA of the gene these little segments and stitch them together so there are special proteins called recombination activating gene or rag proteins that do this now this is a really radical solution to this problem because you never want to mess with your your genome unless you absolutely have to so there has to be a pretty darn good reason that this is evolved and the adaptive immune system apparently is worth it to actually mess around with your genome by doing these doubles stranded cuts and as part of the joining process more randomness it's it's almost like a random number generator you just put in random you have four nucleotides that you can piece together add on to these these cuts and this actually allows you to generate more than ten million potentially up to ten to the eighteenth possible different kinds of t-cell receptors or V cell receptors so it's this randomness that's inherently built into the system that's allowed to generate what we call this highly diverse t-cell receptor repertoire where you have all of these different cells expressing this this incredible array of receptors and I'll briefly mention that these cell receptors do the same thing so I'm not going to spend a lot of time same kind of randomness they just happen to do it for slightly different genes but it's exactly the same process now this this pattern of random recombination from little fragments must be important because it's actually we now know as of two years ago it's happened at least twice in the evolution of vertebrates people like us with backbones other animals with backbones so if you happen to be a jawless vertebrates like a hagfish you've done it with a completely different set of segments but nevertheless it's independently evolved at least twice same thing you've stitched together this highly diverse set of receptors we use the vdj recombination system from everything from sharks on up and all all all jawed vertebrates use the vdj resist system to make both t-cells and b-cells so one of the downsides that I have to mention in passing is that when you do these double-stranded breaks of DNA there's always the risk that there might happen to be another double-stranded break of DNA somewhere else on another chromosome and you can actually get a what we call a translocation where you have an inappropriate joining between one chromosome and another and this is actually an important cause of certain kinds of tumors but again this is relatively rare so for the species overall the advantages of having an adaptive immune system outweigh the risk of cancer the other problem though is that remember that the T cells don't know really they don't intrinsically know what they're saying in terms of they can't tell whether that little peptide in a hotdog bun it could be from anything it could be a protein from your own body it could be a protein from a pathogen and remember they're randomly generated so there's no control over what their particular binding affinity is going to be for so you could for example just generate by chance a t-cell receptor that's going to bind insulin from the pancreas and cause diabetes if that T cell gets activated and starts attacking your tissue so there has to be a way of purging the body of T cells that have this reactivity with your own tissues in other words with peptides that are derived from your own proteins like insulin and other proteins and how is that done so this is really I think one of the most remarkable findings in the last ten years of immunology it turns out that in the thymus gland that that a part of the thymus actually expresses all kinds of proteins that are found in the rest of the body and there's a particular time and place where the T cells are put through whether they're reactive with your own proteins so in other words you can find evidence of insulin protein being made in the thymus it's not that insulin a role for the thymus it's just to get rid of t-cells that can react with that particular protein and you literally express probably thousands of genes that you ordinarily wouldn't Express in the thymus except for one reason which is to purge the t-cells before they leave the thymus and start acting in the rest of the body so that they don't cause autoimmune disease and I'll show you an example what can go wrong when that doesn't happen and just in passing if you look at what the thymus really looks like where we talk about thymic epithelial cells these are cells that form this kind of meshwork that the developing t-cells have to kind of they have to go through like a sim so they have to interact with them and this is how they actually get interrogated for whether they have a self reactive t-cell for a particular peptide and they get rid of them we think about 25% of the t-cells and otherwise would be released from the thymus just get eliminated at the very end and we and we use the term tolerance because this is allows the the t-cells to be in tolerance with the your own tissues of the body so it's called a tolerance mechanism and this just shows you a little schematic of what actually happens the developing t-cell if it happens to have a very strong reaction with a peptide on a thymic epithelial cell it's triggered to die and it never leaves the thymus if it doesn't have that reactivity it's allowed to live and it can go out and be ready to deal with a pathogen so if all of this happens so there's a lot of quality control in making T cells most T cells that develop in the thymus very few of them actually ever leave because of all these things that have to be just so but if they do get out they can now form what we call the Nighy antigenically naive to our it's a fancy word but what it basically means is these cells are now sitting outside in the periphery waiting they've never actually encountered a foreign antigen but they're now waiting and to encounter they're ready to go if you happen to get that Ebola virus infection tomorrow now there's another problem you've got say maybe 10 T cells that can recognize that Ebola virus peptide and you don't know let's say that the Ebola virus enters into your body in your great toe but most of your t-cells happen to be at the top of your head at that time how are they ever going to find each other it really is a true needle in a haystack so there is an amazing system that has evolved to allow the T cell and the the antigenic peptide the thing that will turn it on to find each other you have specialized cells called dendritic cells because they have these things that look like dendrites but they're really a special kind of white cell and these are really the sentinels of the immune system they sit in all the tissues quietly waiting for danger or infection to turn them on if they get that danger or infection signal some of the same pathways we talked about with the innate immune system like the toll-like receptors turn them on they will be very effective at taking up that the these foreign proteins grinding them up into peptides and putting them on the surface of the cell and at the same time they go to specialized train stations if you will called lymph nodes where they now have a better chance of meeting t-cells because these lymph nodes are places that t-cells like to hang out so a Tisa will enter into a particular length node and and interrogate all these dendritic cells see if it's antigen happens to be around if the antigen is there the t-cell recognizes it it stays put it starts dividing it becomes what we call an effector cell well it carries out an immune response and now you have adaptive immunity but if within 24 hours or so it hasn't encountered it it actually moves it'll just randomly leave that lymph node go into the blood go to another lymph node so it's this elaborate shell game that allows your whole repertoire to it's constantly undergoing surveillance of your body for foreign proteins this is part of the reason that there may be a lag with the adaptive immune response as opposed to the innate immune response because they still have to find each other and it can take some days sometimes for that to happen so the immunologist view of cardiology getting back to my joke is that really the heart is the pump for lymphocytes to kind of move them around from one place to another so it is location location location just like real-estate again if there's not the encounter the diesel says I'm out of here and it goes to another site and we call this lymphocyte recirculation so this obviously is a very important part of the immune system but it's amazing when you think about how much energy is spent just being poised for the t-cell to find the antigen so again it must have an enormous adaptive advantage to evolve this way it may seem incredibly wasteful but such as the the nature of some of our most intricate defenses as we know in military terms so this is what we would call a pretty high-end expensive defense mechanism so the other thing that is characteristic of adaptive immunity as opposed to eight immunity is the memory recall the plague of Athens so once you've gotten over it you're able to have an enhanced immune response that can that can maybe save your life or give you a milder infection how does that work well as part of the expansion of these cells so so when a t-cell gets activated by its by its particular antigen that it sees let's say the Ebola virus peptide the numbers of that cell expand enormous Li and then after the pathogen is dealt with there will be a contraction back down to lower numbers but it will never be as low as when you start it and it's that persistent higher frequency of cells to it to a pathogen that allows you to have a memory response so you can actually use this to determine what people have actually been infected with if you can show that there's more cells that react with a particular pathogen than is characteristic of what we call a naive T cell repertoire you know that they've already been exposed to that pathogen and we use this all the time and immunology to characterize it this is shows you in diagrammatic terms you get the initial big expansion of the number of T cells or the number of B cells and then when with the resolution of the infection hopefully you get a very slow decline but it can take years or decades in some cases memory really is that durable a five year old that has for example smallpox at 80 years of age will still have detectable T cell immunity and B cell immunity to smallpox so it's quite remarkable yeah could you use this for prime identification so you just take a person's blood find everything uses - and say that's the person just as well as the fingerprint or a DNA you'd have to know what they actually have been exposed to but I suppose you could you could determine where they might have grown up or some of their infectious history right is there enough unique that you can say that this person when you go I see yes so I think the question is more of a sort of forensic thing can you use is it is each person's if I may rephrase it for you if each person's sort of adaptive immune repertoire unique so that you can say this this drop of blood came from that that individual it theoretically you could do it it's just that it's so much easier just to use straight DNA but you could theoretically work yes active visit the immune system defined with the agent of saliva it does yet so the question is does the effectiveness of the immune system decline with age absolutely does one of the problems is that the thymus eventually becomes lazy fat filled and probably what happened is that from an evolutionary point of view we used to not live so long so there probably wasn't any need to have an immune system after age 50 because nobody lived to 50s so there actually is a problem now with what we call immuno senescence so for example after age 50 you make very few new T cells by the thymus and that probably is an important factor for example of why the elderly don't respond well to new infections that come on the scene like pandemic influenza diseases developed within this immune system so the question is how to autoimmune diseases valve in the immune system I could always just say stay tuned next week I will talk a little bit about it one of the problems is that I'll talk a little bit about defects in this negative selection process causing autoimmune disease but I think the a fair answer is that we really don't understand the pathogenesis of most autoimmune diseases in the kind of detail I think you're probably after one thing that's true is that you can have inappropriate triggers by infections what we call molecular mimicry where the T cells will be triggered by a pathogen say for example a streptococcal infection and it will trigger the immune system to attack related antigens that are found on the body like in the heart so that would be the cause of rheumatic heart disease but to be honest for most Ottoman diseases we don't even have that level of understanding but I'll be happy to tell you some theories if you want to afterwards they're fairly complicated yes they really what they see what they so in a person receives a bone marrow transplant they receive stem cells what we call metabolic stem cells from the donor and those have to give rise to an entirely new immune system so they give rise to new T cells and B cells and that's one of the reasons that as your thymus gets older it's difficult to do bone-marrow transplants and get good t-cell recall t-cell reconstitution if you're over 50 because you still are depending not just on that bone marrow but on the bone marrow going to the thymus to make those new T cells and that's one of the limitations of bone marrow transplant yes are you going to get into the effective stress on the immune system and its ability am I going to get into the the impact of stress on the immune system probably not tonight but if that was covered last quarter god help us that's great because that's a whole there's a lot of interesting things I'm not going to get into tonight one is stress the other as phil was mentioning that there's some very interesting impacts of obesity on being pro-inflammatory it actually turns out that obesity is interpreted in a way by the innate immune system as sort of a danger as something that's gone amiss so it's actually a very inflammatory and that's one of the reasons that obesity is associated with things like heart disease and atherosclerosis because those are those are linked but I unfortunately could cover everything so but I do want to give you some idea about the importance of what the immune system can do against pathogens so let me just give you some examples so your cd4 T cells I think are the most remarkable example of what of how the immune system has evolved to deal with all these nasty critters out there so it turns out that you have these funny named cells at the bottom here T helper one cells or th 1 cells T helper 2 cells th17 cells what are these things each one of these are a type of cd4 T cell that has evolved really to deal with different types of pathogens and each one involves a process where it secretes these proteins called cytokines that orchestrate other non T cells other kinds of cells in the immune system to carry out the immune response and this is really I think one of the more important achievements of understanding of the immune system that wasn't probably known when Phil was in medical school or even even there wasn't any inkling that any of this stuff is going on so there's unique cytokine profiles as shown here and what's also interesting is that these cells sort of have this these marching orders to become you know to carry out all this stuff where maybe one or two proteins in the cell that we call master transcription factors once they get activated everything is stereotyped the cell knows what to do it's quite remarkable one protein regulating fifty to a hundred protein so that it's done just the right way and all this is now known so let's start with T helper one or th one effector cells these really have one important purpose they deal with organisms like tuberculosis that love to hide out inside these these what we call mono nuclear phagocytes so these cells are great places to hide out because they're very wimpy at actually killing things that that get inside these little compartments called the phagosome they actually need signals from the T helper one cell to get revved up to kill tuberculosis and in the rare patients that have defects in just these two helper one cells this is the infection we see over and over again mycobacteria infection they also help orchestrate kind of a holding action to keep these bacteria from running amok so they may not be able to completely eliminate tuberculosis as many of you know once you get exposed to tuberculosis and get what we call latent infection where it may not cause disease it's really hard to be sure that you ever get rid of the bacteria but you can do things that the body can do things to contain this infection sure the bacteria try to get the upper hand and they form these sort of little forts where they have all these cells around the site of infection called granulomas and these granulomas are characteristic if you see this when you take a piece of tissue look under the microscope you know that there's been a t-helper one response and there's likely to be an organism like tuberculosis t-helper two cells even though they're just one number different completely different strategy they're really here for one reason to deal with those nasty big worms I talked about so the nasty big worms they're so big you can't really put them inside a phagocyte they're dwarf they dwarf the phagocyte they can't be taken up you have to use mechanisms to get rid of them that are quite different so what these cells do is there's their cd4 T cells but they make other cytokines that have a very different program they bring they make your gut contract so that it helps expel the worm or cough it up in the case of if it's in your lung they put a lot of mucus into your gut so that if that worm is trying to hold on to the to the gut wall it's harder for it to do it and they bring some special cells called eosinophils and these are Naevia Sinha fills because he goes was the goddess of the dawn and the histologies thought gee this kind of looks like this nice pink color you see when the Sun is coming ups we're going to call them eosinophils and that's the dye that stains the ESN dye that stains these cells they may look kind of pretty but they're actually the suicide bombers of the immune system what they do is they fling themselves on parasites and even though they're small they gather in large numbers in it and they are chock-full of very nasty toxic things that will tend to make that parasite unhappy and either try and leave the body or if they're lucky they may kill it so all these things are coordinated by a single type of cd4 T cell called the T helper - cell then there's another one called th17 cells this one's kind of a funny name it's it's mainly named because it makes us cytokine called interleukin 17 or il 17 and this is an important cytokine for bringing neutrophils remember neutrophils are really good killers of things like bacteria and this is an example of where a t-cell is helping orchestrate a response against bacteria that unlike TB these bacteria are found outside the cell most bacteria like to live outside cells and these these cells these th17 cells help orchestrate the the production of more neutrophils by the bone marrow bring them into the site of infection so that they they can help eliminate the pathogen now the you have to have appropriate control of the immune system each one of these mechanisms is kind of like a bomb going off in the body potentially if it's allowed to run amok and in particular in your guts as you've heard already you have literally millions and billions of bacteria probably all of us have about 50 species of bacteria in enormous numbers in our guts that are actually commensals so they're helping us they're making vitamins they're helping us digest and yet each one of these potentially encodes all of these proteins that are not our own proteins that have never been expressed in the thymus so you have not purged your body of potentially of T cells that would react with with the proteins from these bacteria so one of the I think outstanding issues with the immune system is why is the immune system so good at ignoring all this noise when it could be going after all these bacteria inappropriately and causing things like inflammatory bowel disease so we don't know the answer to it except that it seems that in the gut the default path wave for the cd4 T cells is if they counter a uninflated state where these these peptides are floating around taken up by these dendritic cells they actually are converted to what we call a regulatory cell that suppresses turns off the immune system so this is the normal pathway it's only in the exception of where that dendritic cell gets a danger signal that there's actually a serious pathogen around say that the pathogen is associated with tissue destruction that the cd4 T cell gets a different signal it becomes for example a T helper 17 cell but I think this is going to be a very fruitful area of study and it's still to me one of the most amazing things about the immune system of how good it is it's starting out friend from foe yes yeah so in so the so the question is what's the involved in the immunology of pregnancy so there probably are multiple mechanisms by which pregnancy is maintained because the the father is encoding the father's chromosomes but the baby inherits are going to potentially encode things that could lead to the rejection of the baby turns out that these regulatory cells are induced during pregnancy it turns out that one of the most interesting things I think in the last two years that's come out is that mother's cells almost routinely enter into the fetus at these these are the blood cells and they induce what we call regulatory t-cells of the baby as a fetus to turn off response to the mother cells and what's interesting is that persists for at least 18 years after you're born so it's a quite remarkable system and this was work that was performed by Joseph McCune at UCSF so so yes some of the same Meccan are involved in in keeping the mother and baby from rejecting each other okay so again some of the key signals for which of these cd4 T cells you become if you're at if you're what we call a naive cd4 T cell you're an undifferentiated t cell you don't know what you're need to do how does the immune system make that decision for you what it uses again are these proteins called cytokines there are many different types of cytokines but these particular cytokines come from cells like dendritic cells and other cells of the innate immune system and for example in a parasitic infection there will be particular cytokines that are induced by having that parasite around that will direct the cd4 T cell to become a T helper to cell there will be particular cytokines that mononuclear phagocytes when they get parasitized by TB they will make cytokines that help direct the cd4 T cell to become the T helper one so this is an example of where you have the linkage between the innate immune system and the appropriate outcome in terms of the adaptive immune system but sometimes there's a mismatch and this can be real trouble between the signals that are made and what actually is needed to deal with the infection and the outstanding example is in a disease that's related to Bourke ulos is called leprosy it's caused by the same gener a bacterium Mycobacterium so in leprosy you'll have examples of where though the best thing that you could have to control this infection just like with TB would be a T helper one response for some reason and we still don't know that exactly why in some individuals instead the cd4 T cells are guided to become T helper to cells which is if you've heard helps you deal with large parasitic worms but don't help you deal with these these bacteria that are inside your phagocytes and when you get this mismatch it's a disaster these patients have huge numbers of bacteria running amok and again it's another mystery where despite many years of study we still don't quite understand why in particular individuals there's the the crossing of signals that are inappropriate but probably a very important thing to figure out okay a little bit about the killer the killer T cells cd8 T cells are truly double-oh-seven they're licensed to kill so what they do is they have special granule contents so these are little proteins inside their cell that they release when they get triggered when they detect a virally infected cell so they detect it just the same way that cd4 t-cells do they only see in this case viral from viruses little peptides on hot dog buns if they happen to get the signal they will release these proteins that poke holes in in that what we call the target cell that they're trying to eliminate and then they bring in other proteins that trigger that cell to undergo something called programmed cell death or apoptosis so the cell suddenly shuts down and dies but it's done in a very clean manner the virus isn't allowed to escape it gets destroyed at the same time the nice thing about apoptosis is it doesn't cause a lot of collateral damage you don't get a lot of inflammation at the site of where it's happening it's like a surgical strike and you need in a typical viral infection as all of us that have had influenza are where you feel like your whole body is coming to an end you've got all this virus and this huge numbers you may need a lot of these foot soldiers to actually go out and check out all of your tissues to get rid of virus and for example in mononucleosis which is due to a virus called epstein-barr virus half of the cd8 T cells in your blood may actually just be seeing one kind of protein from that virus so you may need enormous numbers of these cells to control the the viral infection but again once the virus is under control you're going to demobilize these cd8 t-cells and you can see that you need more foot soldiers you need more killers to go out in the field and you need the generals the cd4 T cells so you don't have as many of these getting expanded you have an enormous number of the cd8 t-cells typically to deal with a viral infection and this is just a cartoon showing the kind of surgical strike I'm talking about the cd8 t-cell will march along let's say this is a group of cells in a thing called an epithelium and it says AHA t-cell receptor gets recognizes that there's a little foreign viral peptide on this MHC molecule gets the appropriate trigger kills the cell then it goes to the next cell sees if it has the same infection and so on so these cd8 t-cells can actually what we call recycle where they can Rev themselves up to kill again and again and again in a relatively short period of time and they're very efficient now the problem with viruses is they can infect virtually any tissue you don't know what virus is going to come along ten years from now we hope not but there might be a virus that has a completely new kind of tissue that it's attacking that we don't even know about so the immune system has got that covered as well because the particular kind of MHC molecules that display these viral peptides these are found on all the tissues of the body so the viruses can infect whatever they want but the cd8 T cells will eventually be able to find them and get rid of them and that's what one of the interesting things about these MHC molecules they really are very versatile and are there to be able to present these peptides from viruses in they may occur do you have a question it is a virus that's correct is the question was epstein-barr virus or her preserves it is or preserves and like all herpes viruses it is forever once you get it okay the other thing that cd8 t-cells illustrate is that I that part of memory is the fact that you after you've had an infection you always have for the rest of your life more of the cells that have that specificity for that particular infection ready to go the other thing about memory is that once a cd8 t-cell has you've generated response of cd8 t-cells you get these cells that are what we call memory cells that when they if they let's say that you happen to get influenza one year expand your influenza specific cd8 t-cells the next year you get another case of influenza and it's very similar you will have the cd8 t-cells that that come from the memory cells the ones that become licensed to kill again they're better they have better function each cell has the ability to do more things than the first time the cd8 T cell gets its license to kill so there's there's a difference in the quality of the of the immune response from what we call memory cells as opposed to from naive cells and that accounts for also why you're better protected the second or third time around now we think that cd8 T cells mainly are there as antiviral mechanisms but they do have the ability to kill tumors if a tumor in particular comes up with something that is not found for example in the thymus so that the T cell wasn't purged of that particular you know that particular T so wasn't purchase of reaction with that something new it will be interpreted by the T cell as as something foreign like a virus and they can potentially kill tumors and this is something that had that is being attempted to be exploited for for therapy and I actually think that there's very promising approaches by which to really unleash cd8 T cells in cancer I think there was a lot of skepticism but I think that there's growing evidence that even naturally with many tumors you can find seagate t-cells that are going to be too more reactive they often are suppressed by the tumor but if you can figure out how to bypass that they can actually be potentially potent weapons against the tumor I'll probably leave that for the lectures on on tumors so of course I have to talk about b-cells even though some may mean I'll just think the B stands for boring I actually think they're kind of interesting so B cells are also mediators of adaptive immunity but they have a very different role so remember T cells have this very unusual process where they recognize peptides bound to MHC molecules so very very strange and sort of counterintuitive B cells are very intuitive you can grasp right away how they work they secrete these proteins they have from the cell and they float around in all the fluids of the body and like the complement system they can tag the pathogen for destruction so it's much much simpler but again you have tens of millions of different kinds of potential antibodies are also called immunoglobulins it means the same thing and so you have to go through the same process of making these by this process of vdj recombination but the process is the same other than the fact that the genes are slightly different they are still put together exactly the same way where you kind of cut select from column a column B column C stitch them together with the same enzymes that are used for the teas cells you have the same double-stranded breaks by the rag proteins you have the same random nucleotide generator at the ends to make them even more diverse it's just that that these these genes encode immunoglobulin and not t-cell receptors so these have the ability unlike the t-cell receptor which is always stuck on the surface of the t-cell it's never secreted these can be both in a form in which there on the surface of the B cell and then they can also become secreted into the fluids of the body the other big difference between antibody and t-cell receptors antibody does not see peptide MHC complexes it actually sees 3-dimensional shapes also more intuitive so kind of like a lock-and-key system as shown here so it has this nice triangular shape that mat that complements the antigen then it's going to bind but it's not going to bind to this this shape over here because it's not a good match with its complementary shape the other thing about antibodies they're they're not limited to seeing proteins they can see any molecule any three-dimensional molecule shape so they're much more versatile in that sense of it's a sugar molecule on the surface of the pathogen they can bind it most of the antibody is actually produced these cells get activated in the same way that t-cells do where first the antigen binds to the surface triggers the B cell to get activated to divide and to start carrying out its immune function but but the immune function of the B so really is to just make lots of antibody and put it out in the fluids and it does it in the form of a special cell called the plasma cell shown here so b-cell immunity is also called humoral immunity because in Latin the humors are the body fluids and and that's that's app so you'll often hear that term humoral immunity but all it really means is that it's B cells and plasma cells that are secreting at bodies into the fluids so just like the complement system antibody can help tag bacteria and fungi for destruction so you have special receptors on on the professional phagocytes like the neutrophils that are the good killers that help optimize using the George Bernard Shaw terminology bring in the pathogen into the inside of the cell where you can now kill it and it turns out that antibody binding to these pathogens can also activate the complement system so you get a double whammy you get both antibody and complement both tagging the pathogen to help bring it in for destruction make it even more efficient as shown here so where do you really need the system to deal with what kind of pathogens are really the bad actors where B cell immunity is essential probably the single species of bacteria where you really need it the most is something called streptococcus pneumoniae also called the pneumococcus you may hear it as pneumococcal pneumonia so these bacteria this is the part of the bacteria that actually has all the met metabolism going on so these bacteria can live this way but in in the actual body when they invade as a pathogen they have this enormous goo you can see how much larger they are this is all an outer capsule that surrounds the bacteria proper and its main purpose is to help the bacteria evade the immune system this stuff is extremely slippery these capsules and complement really cannot tag it effectively the only way to effectively tag these so these bacteria for Destruction is to have an antibody that can bind the capsule then the neutrophils can take these things up and kill them so what we see is that in people that have problems with their bee cells they're going to have recurrent problems in dealing with these bacteria they're going to have pneumonia and sinusitis and other kinds of infections and this bacteria is going to count for 90 percent of the problems that they have so this also illustrates I like in this slide just looking at the specificity of antibody cell receptors they're exquisitely specific so if you for example get infected with measles virus you will make antibodies that are able to bind to the surface of that virus and prevent it from infecting a cell but this is not going to have any help if you get infected with an unrelated virus so the specificity is a great thing but unfortunately what it means is that you have to either get vaccinated or actually get the infection to have that great specificity and those of us that are parents know full well what that really means getting infected over and over again with the latest cold virus is part of the job description now the exquisite specificity though can be exploited you can make in the laboratory specific antibodies to almost anything if you want to and in large quantity and these can be used for therapies these are called monoclonal antibodies because they come from one clone so it's one type of antibiotic 'el molecularly and i think an outstanding example is using these for example to treat lymphoma b-cell lymphomas you can make an antibody against a protein that's on most b-cells and use this to eliminate the b-cells with the same mechanisms that would normally be involved with eliminating pathogens so this is a very promising way of for example cancer therapy and there are now literally dozens of monacle antibodies for various kinds of tumors but of course we'd like to be able to boost the immune response and this is what Harry greenberg is going to talk about I've been interested in using a particular technology to fool the immune system if you well into thinking that it's actually been invaded by a serious pathogen it makes sense to me anyway that that if the immune system interprets the vaccine is actually a real threat to it the same thing is like a natural infection it's going to do everything it can to put up a good fight and so we've come up with tricks ways of mimicking infections to fool the immune system into putting out putting out its best defense so for example you can make things that look like viruses to the immune system in that they rapidly enter into cells such as dendritic cells and they activate the toll-like receptors and the other receptors and turn on the immune system and to get these these walloping immune responses and I think we're just really in the infancy of developing these approaches to to enhance the immune response so when we talk about adjuvant these are really additions to vaccines to make them work even better comes from the Latin same Latin root for rejuvenation or juvenile you want that youthful vigor of the immune response in dealing with pathogens so health vs. disease so I often I obviously can't go into all the things that can go wrong with the immune system but just like Tolstoy said all happy families resemble one another but each unhappy family is unhappy in its own way the same thing with immunodeficiency each one is a special story we know of a hundred and forty genetic defects and each one is a story but they do illustrate the importance of the immune system and normal a function so for example you can have a problem where the only problem is that there's one protein missing that helps the neutrophil get across from the bloodstream into the tissue to form pus and to help kill bacteria and fungi and this protein is also when it's missing as part of your normal health of your mouth your gums are always a potentially going to be invaded by bacteria and the only thing that keeps them healthy is that the neutrophils have the ability to also go into the tissue and help control them so these patients lose their teeth prematurely this is a patient with this at 16 years of age and they have all kinds of recurrent infections but they never like pus you can have a complete lack of the adaptive immune system from a single gene defect remember that in order to make the t-cell receptors and the B cell receptors immunoglobulin molecules you needed to do these double stranded DNA cuts by these rag proteins there are rare individuals where there's a problem with the rag proteins they have a mutation in the genes encoding them they completely lack a thymus that's functional this is a normal thing as showing all these nice developing t-cells and this is all the the few t-cells that can be made in someone who has the deficiency of these rag proteins you can have much more selective problems so it turns out that there are certain genes whose sole function is to help the thymus express all of these proteins of the body as part of the purging process of what we call autoreactive t-cells so for example helping the thymic epithelial cells Express the insulin protein so that those can be used to purge the the developing T cells of the T cells that react with insulin so in fact what happens is if you lack this one gene that process is impaired the patients tend to get diabetes because the t-cells get out the thymus they find the insulin peptide in the in the islets of langerhans and they and they attack the whole tissue and they cause diabetes so quite remarkable single gene can do that so I want to end with this because i Richard vitamin though he's a physicist I think I like this using this quote with medical students because I like to teach immunology not just for its pragmatic purpose but I what he says here is what people are poets who can speak of Jupiter as if he were a man but if he is an immense spinning sphere of methane and ammonia must be silent so I have really tried to provide you with as accurate as I can the real immune system in with some scientific rigor because I think that really reveals its elegance more than anthropomorphizing about it so I hope you've found it interesting in that way and I'd like to oh yeah go ahead I'm Astrid the question is do these thymus transplants there is one center that does thymus transplants it's still in the early stages they've had some reasonable results it's for a particular disease called complete to gorge syndrome but it's not it's not something that's been reduced to clinical practice all over the world Gavin I see the blood transfusion you're deso deso mmunity if so the question is if you get a blood transfusion for me do you get potentially my t-cells as part of it if you get a whole blood transfusion the answer is yes and in individuals that don't have a normal immune system what's typically done is that that is irradiated with radiation to paralyze and kill the t-cells so that they can't potentially try and set up shop inside your body there is a phenomenon where that can happen called graph versus host disease where the t-cells will actually sit in your body and will start rejecting every tissue they encounter but if you're a healthy individual what will happen is your t-cells your t-cells will be able to get rid of the t-cells that are attempting to enter into the body because they will recognize them as foreign I didn't really have time to get into transplantation immunology that's a whole probably at least worth an hour our immune or not sensitive to their mother's immune but I understood that that babies who are breastfed and when they're first born and have their mothers immunities services well what what so the question is what kind of immunity do babies get from their mother so I didn't have time to get into it but during the last trimester of pregnancy there is a transfer from the mother of antibody across the placenta so that the baby is actually born with as high a level of antibody to anything the mothers had in her life and it's a wonderful protective mechanism it's one of the reasons that you may have noticed many of us may have noticed that the first six months are kind of like the holiday from infections that then suddenly start appearing with the babies and it's because these antibodies last for about that long the mother does not transfer T cell immunity however so it's pretty much just antibody the tolerance mechanism is probably very durable where the mothers some of her blood cells do get into the fetus and do induce this tolerance mechanism but it doesn't again doesn't have anything helpful in terms of T cell immunity it's really just to prevent there from being a rejection of the mother by the fetus now what's interesting is that if you get a transplant from the mother to the baby compared to the father in general the mother is better tolerated for that reason because of that that and again this wasn't even known until two years ago the mechanism so it's quite more yeah theory that our homes in a way are too clean and that children [Music] see y know what the mechanism with so that so the question is there's this hypothesis it's usually called the hygiene hypothesis and it's used to account the question is why is allergic diseases such as asthma eczema increasing in in developed countries first thing you have to know is that that I don't think we really know the answer and I think that that whether hygiene itself is really the cause is still controversial I've been accused of being kind of a naysayer of that hypothesis but I can talk to you about why I've been accused of that just publish what I see but but I think what's interesting is that allergy can really be seen as a mistaken q so what the immune system is doing is it's turning on the t helper to response all the mechanisms that I mentioned that help you deal with worms are being targeted against allergens dust mite cockroach allergen ragweed whatever all these things are triggering inappropriately these are otherwise completely innocuous compounds but they are triggering the same response as you get with a worm we now think we understand some of the things that they do they trick a cell that helps direct the T helper to response called the basal fill it misinterprets the cue but it leads to the to the production of allergy it doesn't answer your question but I think now that we understand really what fundamentally triggers allergy we may have a chance to see why having a lack of dirt and certain kinds of infections around might be leading to increased allergy I think until that observation last year we really weren't any position to put it all together one more yes fire so the question is what what is why is aids so causing such profound immunodeficiency what happens in AIDS is that the the virus very early on starts destroying cd4 T cells all kinds not just T helper one T helper to all of them and as well as ones that are freshly produced from the thymus and it's particularly good at doing this in the gut and even though your numbers in the blood may not look that low initially the total body number of cd4 T cells starts declining almost within two weeks of infection and it just goes downhill from there the virus also in addition to reducing the numbers has some other immunosuppressive effects on cells such as dendritic cells but essentially you're taking away the master regulators of the immune system all of these mechanisms that cd4 t-cells can do they're all taken away at the same time this would be two weeks of actual replication of the virus in detectable amounts so people now feel that the gut is probably the major site within about two weeks of it entering into the body you start to see the virus probably disseminates from the initial site of entry and then in the gut you have the smoldering infection again the blood really doesn't give you a real window on what's going on in the gut when people have actually been able to look more directly a lot of the immune destruction is happening in that tissue so and as you will remember for those who are here for the talk on the GI tract you'll remember those peyer's patches and the fact is the GI tract is actually on a surface area level one of the largest parts of the immune system in its own right so it's amazing to think about this so I think while you've been sitting here your own immune systems have been producing interesting things some cells responding others dying some to your neighbor some to yourself but it's been an active event I really wanted David to do what he did this evening this is not easy stuff I began by saying that when I was a medical student immunology was almost one or two lecture course and you had a compression of this the differences the level of sophistication that has taken place and continues to take place about the intricacies that this system is just unbelievable and by thinking about now some of the issues and lessons that you've had tonight I think it will make more relevant the discussion that you'll hear next week on autoimmunity or the one that follows on how vaccines work it will relate to aging in some ways and you heard some forecasts of that and then when we get to cancer it'll be quite clear how it relates to that both in terms of impact as well as in terms of therapy so I want to thank dr. Lewis for are getting us off to a great start making wonderful lecture and if you have immune fortitude and abilities you can come forward just don't get too close to him okay tonight see you next week for more please visit us at stanford.edu
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Medical_Lectures
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14_Biochemistry_Enzyme_Regulation_I_Lecture_for_Kevin_Aherns_BB_450550.txt
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Captioning provided by Disability Access Services at Oregon State University. Kevin Ahern: Okay, folks, let's get started! As you can see on the screen, I do not have grades posted. I will have an announcement about the exam at the end of the lecture today, however. So, that'll keep you tantalized. I spent some time last time going through most of the mechanisms. I finished talking a little bit about restriction endonucleases and restriction enzymes, which we also call them. I want to just say a few words about those and then we're going to move on to catalytic, regulation and control mechanisms. So, as I indicated last time, restriction enzymes are similar in mechanism to what we saw with proteases. We remember, of course, that restriction endonucleases are cutting DNA, not protein, and DNA, of course, has nucleotide building blocks, not amino acid building blocks. And further, we remember that the bonds between nucleotides are phosphodiester bonds, not peptide bonds. So there are some different terms, there are some different things there, but the similarities are that we're using an activated water molecule as a nucleophile to break phosphodiester bonds. That's a very important consideration. Students frequently find interesting the sort of story of restriction enzymes, so I'll spend a couple of minutes talking about that. Restriction enzymes are, as I said, enzymes that specifically cut DNA at specific places, and you might wonder why such enzymes exist, because we think of the genome as being something that we want to protect, we don't want to damage it, et cetera, et cetera. It turns out that restriction enzymes are produced in bacteria. They're not produced in human beings. Bacteria use them as a sort of primitive immune system. It's a very simple, primitive immune system. In the immune system, we have antibodies that attack things that are recognized as foreign. In bacteria, bacteria get infected by viruses just as we get infected by viruses. The viruses that infect bacteria are known as bacteriophages. And these bacteriophages typically have a genome of DNA very much like other viruses have genomes of DNA. They can have genomes of RNA, but most of them have DNA. And part of the infectious cycle of any virus is that the virus must attach to a cell and inject its nucleic acid into the cell. So in the case of a bacterium that gets infected by a bacteriophage, the bacteriophage has grabbed hold of the cell and it will inject its DNA, if it's a DNA virus, into the bacterial cell. The bacteria make restriction endonucleases as a protection. So they recognize and cut specific sequences, and if that invading virus has those sequences, which it typically will, because the recognition sequences are relatively short, typically four to six nucleotides in length, then the invader will basically get chopped to bits. And one of the questions that arises then is, "Well, why doesn't the bacterial DNA "get chopped to bits as well?" And that's the last part of the story I need to tell you about restriction enzymes. Restriction enzymes are part of a system that we refer to as "restriction/modification." The restriction part is the enzyme that does the cutting, as I described to you. The modification system I haven't told you about. The modification system is comprised of another enzyme that recognizes exactly the same sequence that the restriction enzyme recognizes. That is, if I have something that's an EcoRV, then that EcoRV restriction/modification system will have a second enzyme that recognizes GATATC, just like the restriction enzyme that does the cutting does that. Well, what does the second enzyme do? The second enzyme doesn't cut that sequence. Instead, it puts a methyl group in the middle of that sequence, a methyl group. So the modification enzyme does that. It's called a "methylase" and when we examine that, we see in the case of this particular sequence, there's the unmodified enzyme sequence there. There is the modified sequence, and we see that a methyl group has been put onto the A of the GATATC. That means that one will go on the bottom A, obviously, as well. And the significance of that is that single methyl group prevents water from binding in the right place to make the cleavage. So even though the enzyme can recognize that sequence, water can't get in there and do the attack and cause the bond to be broken. So when a DNA has been modified by the methylase, it will no longer be cut by that same restriction enzyme that recognizes that sequence. So, for example, this guy has been modified. This could be in the cellular genome, and it's protected from cleavage by the restriction enzyme. The question then arises, "Well, can the methylase get to the invader phage "before the restriction enzyme does?" and the answer is, yes, it can. And if that happens, then the phage will survive the being cut and go on and infect the cell. You say, "Well, why does that happen?" Well, it happens because there's no perfect system. Your immune system's not perfect. The bacterial system's not perfect. But, suffice it to say, this protection system that bacteria have is pretty darn good. If you've ever worked in a laboratory and you've tried to put plasma DNA sequences into a bacterium that has a restriction/modification system, you'll find it's very inefficient to do. Most of the things you try to put in get chopped up and you don't get any plasma in it, at all. But, at a very low frequency, you do get some plasmas in and then they get methylated regularly and survive that process. Okay, so that's the restriction/modification system. As I said, it exists only in bacteria. It does not exist in human beings and we don't need such a system because we have an immune system that, in theory, is protecting us from invaders as much as possible. Okay, questions about that? Yeah, back there. Student: Is it possible for the [inaudible] virus genome? Kevin Ahern: Yeah, so, that was the question I was saying. Is it possible that it'll get there before the cutting enzyme can? And the answer is, yes, it can. So if that happens, then the invader will actually be protected. It will replicate and it'll kill the cell. So, as I said, it's not a perfect system, but it's a pretty darn good system. Shannon? Student: Did you just say that we don't have modification? Kevin Ahern: We don't have restriction or modification. So we have neither. And again, we have an immune system that's protecting us extracellularly, not intracellularly. Yeah? Student: What happens to that methyl group after [inaudible] got chopped off? Kevin Ahern: What happens to the methyl group? Student: Yeah, after the enzyme kills the invader. Kevin Ahern: I'm sorry, after what? Student: After the enzyme kills the invader? Kevin Ahern: After the enzyme kills the invader. Oh, I see, so if the enzyme kills the invader, then that means that it didn't get methylated, right? If it gets methylated, then it's just going to stay on there. It's not going to do anything. It's going to be protected, okay? Yes, sir? Student: Doesn't that methylation enzyme also get assistance from the actual, during the reproduction of the DNA itself, when there'll be another reader that comes along and matches methylation states from the parent strand to the new strand? Kevin Ahern: I'm sorry, say it again, now? Student: If you're copying the DNA... Kevin Ahern: Uh-huh? Student: does that methyl get copied, as well? Kevin Ahern: Okay, if you're copying the DNA, does the methyl get copied as well? The methylate still has to come in and do its thing. So is it possible that the restriction enzyme may cut that at a low frequency? Again, the answer is, it's possible, yes. So the methyl doesn't get copied. The methyl has to be put on after the DNA is made. Connie? Student: When you say we don't have a restriction/modification system, does that mean we don't have any restriction enzymes or modification enzymes, at all? Kevin Ahern: We have no restriction enzymes, no modification enzymes, at all. That's correct. I'm not sure if I answered your question properly, back over here, so let me say it one more time. If the methyl group gets put on, then the invader will replicate and kill the cell, basically, because it won't be able to be cut by the enzymes. So if that happens, then you just have plenty of viral DNA. The methyl group doesn't do anything, at all, because as far as the rest of the proteins of the cell are concerned, it's just a GATATC sequence, okay? If we're talking about how we recycle nucleotides, we'll talk about that a little bit when we talk about nucleotide metabolism next term, so I'll save that for that point. I hope that better answers the question for you. Student: Are these all found in, like, the nucleus? Kevin Ahern: Well, bacteria don't have nucleus, yeah. Okay, the one last thing I want to say, and this one doesn't really relate to mechanism so much, but it does remind us of the importance of shape changes in proteins. So the last group of proteins that are of considerable interest with respect to catalysis [inaudible] are known as the myosins, and myosins are part of the actin-myosin pair. Actin is one protein, myosin being another, and these proteins are very, very important for, in fact, they're essential for—muscular contraction. So these proteins, together, produce the contraction that occurs inside of muscles. There's a whole bunch of stuff with respect to mechanism, and I just don't think it really tells us much about mechanism that we haven't already seen. No surprise, you're going to see an activated intermediate that's going to cleave ATP. But the point that I want to make about actin and myosin is this. Contraction happens because myosin literally crawls along the actin, and that crawling requires molecular change in a protein, this protein being myosin. I want you to look and see this protein. Here is myosin, and when ATP, in fact, what this protein does is it hydrolyzes ATP, and that hydrolysis at ATP induces a significant shape change in this molecule. You can see that the unhydrolyzed ATP, and afterATP hydrolysis has happened, this motion has happened inside of this protein. This motion, folks, is what allows you to move. It allows you to have a heart. It allows you to have all kinds of motion necessary to function, and it's happening because of the shape change of a protein. This motion, right here, we can think of as like a claw that is allowing the myosin to crawl its way along an actin filament. A really cool thing, and that happens as a result of this shape change in this protein. The shape changes require the hydrolysis of ATP. So a very cool application of shape changes that we see in proteins happening as a result of catalysis. Most of the rest of the protein, you'll notice, doesn't really change much. It's only this section out here. A picture of myosin is actually on here. You can see these two little heads that are out here, and these heads actually crawl their way along an actin filament. Okay, that's what I want to say about mechanisms of catalysis. With that, I'd like to turn our attention to discussing the allostery and regulation. So allostery and regulation, or control, as I often times describe it, is very, very important with respect to the needs of a cell get harnest on enzymes. Earlier in the term, I mentioned that enzymes can be extraordinarily efficient, extraordinarily fast and I gave the analogy of driving a Maserati to Fred Meyer at 110 miles an hour. You could imagine there's going to be some problems with that if you don't regulate in some way. We regulate with speed limits. Cells regulate enzymes by a variety of mechanisms. One of these mechanisms is known as allostery. I've mentioned it briefly before, but I will say it again and also give you the definition again. Allostery, or allosterism, is the mechanism by which the binding of a small molecule to an enzyme affects the enzyme's activity. So it's the binding of a small molecule to an enzyme that affects the enzyme's activity. When I mentioned this before, I pointed out that not all enzymes are regulated. Cells are very efficient in regulating things. They regulate the most important, or I'll say the first enzyme, in a metabolic pathway, and by controlling the first enzyme, they control all of the things that flow through it. Just like if I put a tollbooth out in front of I-5 in Albany and I stop all the cars there, there aren't going to be many cars getting through, only the ones that the tollbooth allows through. So it's the same thing that happens in metabolic pathways. If we control that first enzyme that catalyzes the first reaction in the pathway, we can essentially control the whole pathway very easily. It's very efficient, and so that's a very useful thing. There are, in the cell, three main mechanisms that cells use to control enzymes. One of them is allosterism, what I've already described to you. I'm going to show you some details of allosterism later today. A second control means that cells have over enzymes is covalent modification. They can covalently modify enzymes. We'll see some examples later which involve the addition or removal of phosphates from enzymes. These covalent modifications can activate or inactivate enzymes, depending upon the enzyme. Student: Allostery [inaudible] also? Kevin Ahern: Allostery can be positive or negative. That's correct. The third mechanism that cells use to control enzymes is controlling whether or not they're synthesized, that is, whether or not the protein is even made. And that seems like, well, duh! It turns out that's one of the most important considerations for many control systems. Is the protein being made by the cell or not? And that control is exhibited in several ways. It could be transcriptional. It could be translational. We'll talk about some of those next term. What I want to do now is talk for a bit about allostery and this very interesting enzyme called ATCase. So let me show you a little bit of this. ATCase catalyzes a reaction that, to be honest with you, we're not going to pay much attention to the reaction itself until next term, but it catalyzes a reaction that is a very, very important reaction, because it's the first step in making pyrimidine nucleotides, the first step. That is, the very first reaction in making a pyrimidine nucleotide is catalyzed by the enzyme ATCase. Now, the enzyme ATCase has a much longer name. It's known as aspartate transcarbamoylase. I'm not going to spell that here. You can get it out of the book if you want to get the word. We commonly abbreviate it ATCase. But please note that when you use an abbreviation, you have to get it right. You can't call it ACTase, for example. ATCase is the proper name. Now, here's the reaction that it catalyzes. You can see it on the screen. It involves an aspartic acid. One of the things that we'll see that's of interest next term is that all of the nucleotides that are made by cells have building blocks that start with amino acids. So we can start with very simple things and make fairly complex molecules, like nucleotides, using various enzymatic systems. The pyrimidine nucleotides include CTP, UTP, and TTP, if we're talking about DNA. To make these nucleotides, it's more than one reaction and this is why I refer to these things as "pathways." To go from these simple molecules here all the way down to the final product, which in this case here is CTP, takes about ten steps. About ten different reactions are necessary... Student: What are the pyrimidine nucleotides, again? Kevin Ahern: What are the pyrimidine nucleotides? That would be CTP, UTP and TTP. It takes about ten steps to get to this final product here. Well, as I said, cells have efficient means of controlling pathways, and cells really don't want to make too much of any given nucleotide. We know from the study of cells that cells that have aberrations in them that cause them to have too much or too little of a given nucleotide cause those cells to have higher rates of mutation. Mutation is generally not a good career move for cells, certainly not in the short term. Over evolutionary history, yes, but over the short term, most mutations are very detrimental to cells. So cells are very much what I describe as control freaks. They put a lot of energy and a lot of controls in the way of preventing the nucleotides from getting too high or too low. Well, what does this all mean? Let's imagine that I'm a cell. I'm sitting around and I'm making pyrimidine nucleotides. I produce CTP and the CTP concentration starts to increase. Well, if the CTP concentration starts to increase too much, I don't want to make any more CTP, so I want to turn off the synthesis. I want to turn off that pathway. It turns out that the enzyme ATCase is an allosteric enzyme and it will bind to the end product of this pathway. So ATCase will bind to CTP, and that's what this little red thing is showing here, and when it binds to CTP the enzyme is turned off. So when the enzyme binds to the end product of the pathway, the CTP accumulation starts to get high, the enzyme gets turned off, then that shuts off essentially all the reactions leading up to CTP, and, until that CTP starts getting used, that pathway will essentially be turned off. Notice I said "essentially." If you remember, I said we think of these mechanisms in terms of on and off, but in reality, they're more like turning the volume down or turning the volume higher, but we still have some things coming through. So allosteric mechanisms do not have on/off switches, but they have turn-down/turn-up situations. Well, the beauty of this is, the end product helps control itself. It can control its own synthesis through this enzyme. That's a very, very useful thing. It's a very simple mechanism. As a result, CTP concentrations inside of cells don't get too high. You say, "What about UTP? "What about TTP?" Well, it turns out UTP and TTP are both ultimately made from CTP. So by controlling CTP, you're controlling all of the pyrimidine nucleotides. One step, one enzyme, one molecule, it doesn't get any more efficient than that. So that's really cool how cells are able to do that. This mechanism I've just described to you has a name. It's called "feedback inhibition." So feedback inhibition occurs when the end product of a pathway inhibits the first enzyme in the pathway. Feedback inhibition occurs when the end product of a pathway inhibits the first enzyme in that pathway. We'll see other examples of feedback inhibition, primarily next term, but there are many examples that cells use of feedback inhibition because it is so simple and easy to control things. Well, that's one interesting aspect of ATCase. There's the actual reaction that is catalyzed, and, no, you don't need to know all this stuff that's on here. I'm just showing you. There's the carbamoyl phosphate. There's aspartic acid. There's the intermediate that it makes. There's a whole bunch of steps, and there's the final product of CTP, right there. I find it really cool and remarkable that the nucleotides can be made from amino acids. We know that amino acids exist in space We know that amino acids can combine in space and people have actually found nucleotide precursors in meteorites floating out there. So this idea of the chemical evolution of life is pretty cool, and we use the materials that are available to us to make us and to make life possible. If I study this enzyme, ATCase, and I study the reaction that it catalyzed, and, remember, ATCase does not make CTP. It's making this aspartyl carbamic molecule. CTP is only made way down the line. That's not made by the ATCase enzyme. So if I take and I study the reaction that ATCase is catalyzing and I study it in the presence of increasing concentrations of CTP, what I see is that the rate of formation, this is the product, we're seeing that the rate of formation is falling, and this is a log scale, so it's a fairly significant drop. This rate is falling as the CTP concentration increases. This graph is showing you visually what I've told you in words. The more CTP there is in the cell, the more the enzyme will be inhibited, but, as I noted, we don't have a complete off switch. We're turning the volume way down, but we haven't turned it off. Now, interestingly, if we examine the catalytic activity, and this is not a very good representation of this, of this enzyme we discover something else very interesting. The last figure I showed you showed the effect of increasing concentration of CTP. CTP is not a substrate. Remember, CTP is a product of that ten-steps-away pathway. It's simply a molecule that binds to ATCase. So it's not a substrate for ATCase. If I take one of the substrates of ATCase and I measure that same rate of formation of product, I see a sigmoidal plot. This is the thing that we referred to on the last exam. That sigmoidal plot is happening because something else is going on in this enzyme. You saw the effect that CTP had. Now you see that a substrate is also having an effect. How do I know it's having an effect? Well, I see a sigmoidal plot. That's not a very good "S" but that's actually a sigmoidal plot. It tells us that, not only can CTP affect the enzyme's activity, but so too can aspartic acid, one of the substrates. So a substrate can affect this enzyme's activity. We see, at this point, two regulators of the enzyme. This regulator is doing to ATCase a very similar thing to what oxygen was doing to hemoglobin. The more of it there was, the more activity we see. That means that aspartate is activating the enzyme. So the substrate, in this case, is activating the enzyme. Now, this actually has biological significance. I want you to think. I'm going to tell you a little bit about that biological significance right now. Cells have to make nucleotides in order to make nucleic acid. They need to make RNA. They need to make protein. They need to make nucleotides. If a cell is getting ready to divide, the cell darn sure better be able to make enough nucleotides. If it doesn't have enough raw materials to make nucleotides, then that cell should not be preparing to divide. Just like if your bank account is broken, you should not be buying beer for a party on Friday. It's not a good career move. You may really regret it on Saturday, and I can guarantee you the cell will really regret it if it tries to go through the division process without having the resources it needs to make the nucleotides for the RNA and DNA necessary to do replication. Well, how does the cell tell if it's got enough? One of the ways is right here. If the cell has a lot of aspartate, what happens to the activity of this enzyme? It goes up. And what is aspartate? Aspartate is an essential building block, not only for nucleotides but also for proteins, and cells need proteins to divide, as well. When the aspartate concentration is high, this is one of the signals to the cell that, "Hey, we can go and have some whoopie! "We can divide!" That's cellular whoopie, by the way. [laughing] "We can divide!" You guys are slow today. So by actually having a little barometer, which is what this is, on the cellular nutrients, the cell is able to make an intelligent decision about to divide or not to divide. That's really cool. So not only are we regulating an enzyme, we're also testing the water. "Do I have enough materials to go ahead and divide?" Student: Is it only making this decision based on the aspartate concentration? Kevin Ahern: Is it only making this decision based on aspartate concentration? The answer is, no, it's not. But this is one very useful piece of information. If cells didn't have enough aspartate, then you could see what would happen to this enzyme. The enzyme wouldn't go and it would stop everything else. So it's a good "no" switch, it's not the only "yes" switch. But, a very good question. Student: If it didn't have enough aspartate...? How is aspartate made? Kevin Ahern: How is aspartate made? Well, aspartate can be made by several mechanisms. There are metabolic pathways that can produce it. I'll mention a couple briefly next term. Also, cells can ingest it. So if they're floating around in a medium that's rich in amino acids or rich in proteins, they have sources of aspartate. So that's really cool. Now, what did I want to say here? I don't want to say anything there. I want to say a little bit of interesting things about the enzyme, itself. I'm focusing now on the protein structure of this enzyme. When we study the protein structure of this enzyme, something interesting happens. Under certain conditions, and what we're looking at here is a centrifugal analysis of this ATCase enzyme, it turns out ATCase has 12 subunits to it. So it's even more complicated than hemoglobin. It has 12 subunits, and if I am careful in how I manipulate those subunits, I can take that 12-unit piece and discover that it's composed of three pieces of an r2 dimer and two pieces of a c3 dimer. I'll show you what those are in a second. Now, in terms of the appearance of this protein, this is what it looks like. We can exclude all the ribbons. I want you to focus on here. You see that the enzyme consists of six units called "c" or "catalytic." These are subunits where reactions are catalyzed. And it has six subunits that are called "r" or "regulatory." Now, you can't see all six of the catalytic because they're underneath. So here's the top three and then there's three underneath there, as well. So we look at it from the side, there's two of the three and there's two of the three, there. So this is like a double decker of trimers right here. These are the catalytic subunits. And the regulatory subunits, you see three pairs of them here. Here's a pair, Here's a pair, Here's a pair. Three sets of pairs of the regulatory subunits, and they are sort of hugging those catalytic subunits. What we're going to see is that the allosteric effectors are going to change how these guys are all arranged. The allosteric effectors, in this case, aspartate or CTP, are going to change the way that the regulatory subunits arrange themselves around the catalytic subunits, and these changes in structure will affect the catalytic activity. Just as we've talked about R and T state with hemoglobin, so, too, do we talk about R and T state with an enzyme. When the enzyme is in the very low activity state, it's in what we call the "T state" or the "tight state." When it's in the high activity state, it's in what we call the "R state" or "relaxed." I'll show you some more figures of that in just a second. Now, before I do that, I need to introduce what will seem to you, at first, like a sort of a curve ball. The curve ball is, I want to spend a few minutes talking about an artificial substrate. An artificial substrate. It's, in fact, a suicide substrate, and though your book doesn't call it that, that's what it is. It's a suicide substrate. So what's a suicide molecule? What was the definition of that, before? It resembles a natural substrate, the enzyme binds it, and it becomes covalently linked to it, right? So this molecule I'm getting ready to describe to you is a man-made molecule. It's not a natural substrate. It's something that we've made to study an enzyme. It resembles the natural substrate. It's not unlike aspartic acid, not unlike aspartic acid. Here's what it looks like. And, we can see that when this guys comes into the enzyme it gets bound covalently to it. This is the synthesis of the molecule. Here is the artificial substrate. This artificial substrate will covalently link to the enzyme itself. It's called PALA, P-A-L-A, and I don't even know the name of it myself, so I always think of it as PALA. Well, what's the significance of that? Why do I tell you about that? Well, it turns out that if I take the enzyme, ATCase, and I add PALA to it, PALA binds, as I expected before, but something unexpected happens. Study of the enzyme linked to PALA first indicated that this enzyme could have two states. It could have a T state and an R state. When you take the enzyme all by itself, and you study it in a centrifuge, you basically see about one form. You don't see two forms. But when you treat it with PALA, you discover that the ones that haven't bound to PALA have one form, but the forms that have bound to PALA have a very different form. These correspond to the T state, on the left, and the R state, on the right. Now you're sitting here very confused. "You said the R state was high activity, "but this is a suicide inhibitor. "If this is suicide inhibited, it has no activity." And that's correct. It turns out that what PALA does, it's catching the enzyme in that high activity state and freezing it there. Why didn't people see this before? The reason people didn't see this before is that when the enzyme binds to the normal substrate, it catalyzes the reaction, it flips into R, it catalyzes the reaction, it flips back out, and you don't see it. Didn't see that state. There's a factor of, I think it's about a couple of hundred to one that the T state is favored. So you don't even see that very tiny percentage that's the R state unless you lock things in it. And when they locked things in it with PALA, they discovered, "Wow, the R state exists for this enzyme." If you look at what this R state looks like, you can see it's very different than the T state. In the R state, the enzyme is relaxed, it's opened up. Access to the catalytic units for the normal substrate is high. The normal substrate, if PALA weren't here, could get in here very easily and cause the enzyme to be able to bind it very readily. On the other hand, the T state, you see everything is up tight, it's up close. It's much more difficult for substrate to get into the catalytic subunit, and that's why we see the T state having less activity than the R state. So the T state and R state are very, very different states of these two enzymes. PALA allows us to see that. You'll notice the arrows here, indicating that the T state is way to the left, the R state is way to the right. You'll also notice something else on this screen, and that is the effect of CTP. Based on what you know about activities of enzymes and the allosteric inhibitors I've described to you, it's not surprising to think, then, that CTP, which reduces the enzymatic activity, is going to favor the T state, whereas aspartic acid is going to favor the R state. Now what's of interest here is that they bind to different places on the protein. R stands for "regulatory subunit." The regulatory subunit is where a regulator would bind, like CTP. You might say, "Oh, well, then that means "aspartate must bind the regulator, as well," and it turns out it doesn't. Why? Because aspartate is a substrate and substrates bind at the catalytic subunit, where the catalysis will occur. So aspartic acid binds to the catalytic subunit and CTP binds to the regulatory subunit. There's CTP locking in the T state. and we basically see that there. And there's the kinetic effect. If we measure the reaction in the presence or absence of CTP, you've seen this earlier, that you see a reduced activity. There it is in the presence of CTP. There it is in the absence of CTP. So, not surprisingly, CTP is turning down that activity. The last thing I want to say about this is something that is even more interesting. This enzyme is playing a critical role in regulating CTP and in measuring the barometer of do we have enough nutrients, in this case, aspartic acid, to go through and do replication. It turns out the enzyme responds to something else, yet. Well, let's think about this before I describe it to you. We think about that when we go to make RNA or we go to make DNA, if we have a lot of pyrimidines, then we need purines, right? Because C pairs with G. G's a purine. C's a pyrimidine. T pairs with A. T's a pyrimidine. A's a purine. We want to have the right balance of nucleotides present in the cell. I told you if we make too much CTP we've got trouble. But what if we have too many purines? What if we have a high level of ATP and GTP? What's going to happen? Well, we may have the same problem with mutation, and if we have that happen, one of the things we would like to do would be to increase the amount of pyrimidines, right? Because if we have a high level of purines and we raise pyrimidines, we're going to have roughly the same balance. So it turns out that ATCase has the ability to sense this, as well. ATCase is allosterically activated by ATP. It's allosterically activated by ATP. Here's the same curve you saw before, now in the presence of ATP, and you see we get increased activity in the presence of ATP. This serves as yet another barometer for the cell. So remember that I said ATCase is telling the cell, "Do you have enough amino acids in the form "of aspartic acid to go through with division?" ATP is an indication of a cell's energy. High ATP, high energy. If the cell is full of energy, the cell is full of purine nucleotides, and the cell has plenty of aspartic acid, this is a sign it's prime time to divide, let's start making some pyrimidines. And that's what this is doing. So this enzyme is performing some very, very important functions in terms of helping cells to make intelligent decisions about dividing. ATP is an allosteric activator of the enzyme and, like CTP, it binds to the regulatory subunits. It binds to the regulatory subunits. That causes the enzyme to shift from T to R. So ATP is going to favor the R state. Aspartic acid is going to favor the R state. CTP is going to favor the T state. This really interesting control system that we see here is one of the reasons why ATCase has been one of the most studied enzymes in biochemistry. It's a classic enzyme for understanding allosteric regulation, and it's not the only enzyme that responds to more than one thing. This ability to respond to different molecules in different ways is really key to having elaborate controls over metabolic pathways, very, very important. Now I'm doing a lot of talking here. Let me ask for questions, and then I'll finish with a couple of things. Shannon? Student: How is it that ATP indicates purine concentration? Kevin Ahern: ATP is a purine. ATP is a purine, so when ATP is high, purine concentration is high. Student: So CTP binding at an R site... Kevin Ahern: Yes? Student: [inaudible] causes the molecular [inaudible]. Kevin Ahern: His question is actually leading to my very next topic. It's a very good question. His question is, do CTP and ATP cause these changes? Or is there something else that's involved? I know it's not exactly what you asked, but that's what the implications of the question are. It turns out that neither one causes this to happen. We thought of cause and effect with hemoglobin. We thought of the first oxygen caused the second one to be favored, caused the third one to be favored, caused the fourth one to be favored. So we saw cooperativity. And we talked about that as a cause and effect. Oxygen caused that to happen. Well, there are other models that are consistent with changes that are not cause and effect, and that's what I'm getting ready to tell you about. Actually, maybe we'll save that for next time, but then I have two things for you. So let me save the answer to that question for next time. I'm running out of time. One, I thought we would sing a song, and then, two, I'll make an announcement about the exam. That way, you'll sing loud. This is a song about taking exams. [laughing] [singing "The Mellow Woes of Testing"] Lyrics: The term is almost at an end, ten weeks since it began. I worried how my grade was 'cause I did not have a plan. The first exam went not so well, I got a 63. 'Twas just about the average score in biochemistry. I buckled down the second time, did not sow my wild oats. I downloaded the videos and took a ton of notes. I learned about free energy and Delta Gee Naught Prime. My score increased by seven points, a C-plus grade was mine. I sang the songs I memorized, I played the mp3s. I learned the citrate cycle and I counted ATPs. I had electron transport down and all of complex vee. I gasped when I saw my exam, it was a 93. So heading to the final stretch, I crammed my memory, and came to class on sunny days for quizzing comedy. I packed a card with info and my brain almost burned out. 'Twas much to my delight I got the "A" I'd dreamed about. So here's the moral of the song, it doesn't pay to stew, if scores are not quite what you want and you don't have a clue. The answers get into your head when you know what to do. Watch videos, read highlights and review, review, review. Ahern: Now, the exams are graded. The average on the exam was 65.5, not a bad one for the first exam. They're available for pickup in the BB office, ALS 2011. I just literally got them done just before I came to class. There's a key posted outside my office. You can look at the key outside my office and I will post a grade distribution later this evening. Student: What was the high score? Kevin Ahern: High score on the exam. This is really interesting. The high score on the exam was 107, perfect, and the low was in the 20's. I want to say about 21. [END]
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Medical_Lectures
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07_Biochemistry_Protein_Characterization_Lecture_for_Kevin_Aherns_BB_450550.txt
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Another Monday beckons, another week beckons. One day closer to an exam. Student: Whoo! Kevin Ahern: Yay, huh? One day closer to your opportunity to show me how much you know. That's good. I hope you had a good weekend. Student: Fantastic. Kevin Ahern: Fantastic? Student: We won. Kevin Ahern: Are we talking about football here? Student: Yeah. Kevin Ahern: Okay. So the football team won. So last time, I threw out the topic to you of the 2D gel electrophoresis, and I think that's a really phenomenal technology. I think it allows not "I think", I know it allows us to do amazingly complex analyses of cells. And if we have cells that have different experiences one being a tumor cell, one not being a tumor cell, one being treated with a drug, one not being treated with a drug, one being starved, the other not being starved, et cetera, et cetera we we can use this technology to see very clearly at the protein level how these changes occur inside of the cells. Several students after the class asked me if there were libraries of gels that were out there that are cells of known treatments. The answer is, there are. But many laboratories will actually do their own side-by-side comparison because one of the things that you see is the reproducibility is not 100% the same, so if you've done both of them in your laboratory at the same time, you're a little bit more able to compare them. So that's something that happens. But, yes, there are libraries of such things out there. And I just realized, I haven't checked the camera to make sure it's properly on the screen. So give me just a second to check that. Doo-do-doo-doo. And the answer is, it was perfect. Alright. There's nothing worse than looking at your video afterwards and you see you had it about halfway on screen and about halfway off the screen. And you guys like that about as much as I do, so, yeah, maybe less than I do. One of the things I skipped over in getting to tell you about 2D gel electrophoresis was to tell you about gel electrophoresis itself. So that's how I'm going to start the lecture today, telling you how gel electrophoresis works and I'm going to talk about two different types of gel electrophoresis. The first type I will talk about is actually the simpler of the two, and it is what we refer to as DNA, separating DNA by agarose gel electrophoresis. Agarose is, and there's the word right there agarose gel electrophoresis, I keep popping out here agarose gel electrophoresis is a technique. I don't have a figure for it anymore. Your book used to have a figure and then they took that away from me, so I don't have the figure out for it. But I can tell you it's, in principle, very much the same as polyacrylamide gel electrophoresis. So let me just show you what that looks like. Agarose gel electrophoresis is what we use to separate fragments of DNA. We can also separate fragments of RNA with it. We do not use agarose gel electrophoresis to separate proteins, and you'll see why that's the case in just a little bit. The first reason, though, that we don't use it to separate proteins is that nucleic acids are way bigger than proteins. The biggest molecules in the cell are DNA molecules, by far. Proteins don't even come close in terms of size. What the agarose provides, in the case of DNA separations, or what the polyacrylamide provides, in the case of protein separations, are a matrix. And we can think of this matrix sort of like it's schematically shown here. The matrix is a series of strands or connected things that provide a support. The support is to support the liquid of the buffer. So just like we could take a mix of Jello and put it into water and boil it, when it cools down, it forms a solid support based on what was in there, so, too, can we do with materials for the gel, the difference being, in the case of a gel, that these strands that provide the support will provide little channels or little holes through which the macromolecules can elute. And I'll show you how that happens, okay? Agarose has bigger holes than polyacrylamide does. So we need those bigger holes to separate DNA molecules. So how do I separate using gel electrophoresis for DNA? Well, first of all, I take my DNA molecules that would be a mixture of different sizes. And I would apply them to the top of my gel, as you can see here. So I make these little indentations that are what are called "wells." And into these wells, we pour our mixture of DNA fragments. DNA fragments are negatively charged. They're polyanionic, meaning that they have many, many negative charges. For every base that we add, we get another negative charge. So the charge is proportional to the length, and the length is proportional to the length. Now, you'll see why that sort of makes sense, in a second. The charge is proportional to the length, and the length is proportional to the length. And what we do in separating these guys is we use an electric field. The electric field we use places a negative charge at the top. You can see that little negative ion right there. And it places a positive charge at the bottom. The DNA molecules, being negatively charged, are repelled by the negative at the top and attracted toward the positive at the bottom. Well since the ratio of the charge to size is constant, that is the longer molecules have more charge, but they also have more size the separation that happens between these molecules is solely on the basis of their size... solely on the basis of their size. The smallest guys can move the fastest through these channels and they go racing through the gel. The largest molecules don't have that same mobility and it takes them longer to get through the gel. So at the end of a stint of gel electrophoresis, what we see is the gel products. So this is a protein gel, but a DNA gel would look very much like this, where we have fragments that have been separated by size. So this would be the largest molecules up here. These would be the smallest molecules down here. And these are specific fragments, in this case, that have been purified of a protein that have a given size that's there. So, in principle, DNA electrophoresis and protein electrophoresis are the same after we have to do some manipulations to proteins to make that happen, and I'll show you how that occurs. So DNA electrophoresis makes sense? Yes, sir? Student: So if the charge on the bottom isn't great enough that it's, it's not just going to tear through the gel? Kevin Ahern: So his question is the charge on the bottom great enough that it's just going to not tear through the gel? In fact the molecules will, if you leave it long enough, go all the way through the gel. Yes, they will. So they will go all the way through, this is cutting out. They will go all the way through the gel. So there's several variables that we have. We don't need to consider them really here, but I will tell you we can change the percentage of agarose, which will actually change the size of those holes that the DNA molecules are passing through. So we can optimize that for different things that we're trying to separate. And I'm getting some noise. Maybe that took care of it. So that's DNA electrophoresis. It's pretty straightforward. With protein electrophoresis, we've got a different consideration. And the reason we've got a different consideration is, first of all, proteins are globs. And second of all, proteins don't have a uniform mass-to-charge ratio. Some proteins are going to be positively charged. Some are going to be negatively charged. Some are going to be neutral. And that charge is really unrelated to the size of the protein. So if we try to separate proteins without some other things to give an artificial size-to-charge ratio that's constant, then we're going to have trouble. Because if I take my mixture of proteins and I've got some positive ones on top and some negative ones in there, the positive ones aren't even going to enter the gel. They're not even going to go in. Boy, this is really misbehaving today. Alright. So I have to do something, then, to make the, I have to do something to make the proteins have a reasonably constant charge, or size-to-charge ratio. So the trick that's used is a very clever one and it works very, very well. It may seem a little odd, at first, but it's actually a very, very good way to give proteins an artificial size-to-charge ratio that's constant. What we do is take the mixture of proteins that we want to separate, and we add excess detergent, called SDS. That stands for "sodium dodecyl sulfate." So it's a long carbon chain molecule that has at one end a sulfate. Now, that sulfate is negatively charged. When these proteins encounter the SDS, if you recall when I talked about what detergents can do to protein, what did I say would happen? They denature, they unfold. So this protein that starts out as a glob, first of all, elongates out into a nice long chain. So, visually, we could imagine this guy is going to look something like a straight DNA molecule, not as big, but a straight DNA molecule. The second thing that happens is these sodium dodecyl sulfates completely envelope the chain. Alright? They just completely go all the way around the thing, making like a Twinkie or something, okay? A Twinkie's got the little chewy center, right? The chewy center being the protein, and it's got this coat of stuff all the way around it. Well, that coat, of course, is proportional to the length of the polypeptide chain. Longer polypeptide chains will have more of those sodium dodecyl sulfates than smaller ones will. So the size-to-charge ratio is relatively constant. It's not absolutely constant, but it's relatively constant. And, in fact, for most purposes, it's constant enough that we can get very, very good separations based on size. So once we've done that, we take our mixture of proteins, that are now all coated with this SDS, and we separate them on a polyacrylamide gel. And as I said earlier, the only difference between agarose and polyacrylamide is that polyacrylamide simply makes smaller pores, smaller holes, for those proteins to go through. We apply an electrical current, just as we did before, negative at the top, positive at the bottom, and we separate solely on the basis of size, how fast they can move through that chamber. Now, and so when we do that, we actually end up getting a gel. This actually is a protein gel. We can see these are marker proteins that have different sizes, the largest ones being up here, the smallest ones being down here. And if we know the sizes of these known proteins over here, we can actually determine the size of an unknown protein by seeing where does it line up with. Is it 50,000 in molecular weight? My protein must be about 50,000 in molecular weight. So this technique has an acronym. First of all, polyacrylamide gel electrophoresis has the acronym PAGE, P-A-G-E. When I use SDS, which I almost always do with proteins, we call it SDS-PAGE. SDS-PAGE. So SDS-PAGE allows me to separate proteins on the basis of their size, very much like I separate DNA molecules on the basis of their size. So when I'm doing that 2D gel electrophoresis that I talked about on Friday, that second dimension, or the first dimension, we had isoelectric focusing, and I said we cut it open, and we laid it on top of this gel? Well, this gel is a polyacrylamide gel. We have to make sure we get some SDS in there so the proteins all make the Twinkie shape, right? And then we run them through in that second dimension as SDS-PAGE. So the first dimension of a 2D gel is isoelectric focusing. The second dimension is SDS-PAGE. And thanks to that, we can actually separate these molecules and determine, literally, the amount and presence or absence of virtually every protein that's made in a given cell. Student: So that SDS part, you just call that isoelectric focusing? Kevin Ahern: What's that? Student: Is that SDS part, you just call that isoelectric focusing? Kevin Ahern: No. Isoelectric focusing is a different technique. That's the one where we used the charge to separate the molecules in the tube? Student: For DNA. Kevin Ahern: No. That's for protein. That's what I talked about last time. Isoelectric focusing, I put the stuff in the tube, and I had the things that had minus 50 all the over to plus 50? Right? So that first dimension, I separate on the basis of what was essentially the pI. Okay? And the second dimension I separate on the basis of size. That second dimension is known as SDS-PAGE. Yes, sir? Student: So the SDS coat doesn't affect so much with the isoelectric focusing? Kevin Ahern: Okay. That's a good question, a common question. So people will frequently say, "Does it screw up the isoelectric focusing?" Well, no, because they've already been separated by the isoelectric focusing. And so we're just covering what's already been separated on the basis of pI with SDS. It doesn't screw anything up, at all. If it did, we would have a problem. Student: Okay. So they're not actually run at the same time? Kevin Ahern: They're not run at the same time, because we have to separate on the basis of pI first. That would be a good exam question if we tried to put the SDS in with the isoelectric focusing, what would happen is, everything would be negatively charged. It would all go to one end. Good question. Yes, Shannon? Student: So how is it physically transferred to the PAGE? Kevin Ahern: It's just laid on top of the gel. It's just, so instead of having individual wells, I would just have a long thing. I'd lay my little tiny tube up there. And I would just lay it on there and run electrical current through it. Yeah. The people who run 2D gels, it's an art. Believe me, it's an art. So getting that little tube on there and not breaking it and fracturing it and everything, so that it lays evenly across the gel surface, is very important, and there really is an art to it. Yes, sir? Student: I've seen agarose gels for DNA that are relatively small. Is there a standardized size for these? Or is there a variation in size? Kevin Ahern: His question is, do we have variations in polyacrylamide that we use for protein gels, because we do see variations that we use for agarose. And the answer is, yes, we do. So we can run polyacrylamide gels varying the percentage of polyacrylamide that's there and also varying the number of links between the individual strands. And that is based on the chemistry that's involved in making a polyacrylamide gel. And all we're doing, in either of those, is really determining the size of those holes that the proteins are moving through. So we can adjust that. If we have a bunch of small proteins, we would run a different kind of a percentage of a gel than we would if we had very large proteins. Student: Do they vary the actual size of the plate itself, though? Kevin Ahern: Do they vary the actual size of the plate itself? Yeah, you can do that. If you're looking at something that is quick and dirty, you can run a very tiny little gel. If you want to go and do 2D gel electrophoresis, you would typically run a fairly large one because you want to get as much separation as possible for those. So, yeah, they do vary the size, as well. Okay. So, good questions. That was pretty much what I wanted to say. I didn't show this last time, but that was from a 2D gel electrophoresis, the difference between a normal, proteins from a normal human colon cell and those from a colorectal tumor. And you could find many differences between these, and those are, as you might imagine, of a considerable amount of interest. Well, one of the things that we do, we're doing all these techniques for, is so that we can purify it, that is, so that we can make plenty of a pure amount of compound of proteins. And, as I mentioned earlier, one of the things we see in the purification process is that that purification, we never really know what's going to work before we do it. So we have to really be very careful to monitor during the purification process where is my protein. Is it in the pellet? Is it in the liquid? Is it in the first part that comes off the anion exchange or the last part that comes off of the anion exchange? Because the worst thing that you can do is assume it's in one fraction and throw that fraction away. And I can tell you hundreds of stories of people who've done exactly that. So people get very careful. If you've been working on this, this is your PhD project, or whatever, you're going to be checking every component of that purification process to make sure that you're not throwing the baby out with the bath water, as it were. Yes, sir? Student: So I assume some of your stories are like, okay, the protein should be here... Kevin Ahern: Should be here, yeah. Student:...but it's actually in this part. Kevin Ahern: Right. Right. So that can be a real problem if that protein is life and death for you, in terms of a thesis or something. So what you see here on the screen is a sort of a depiction of protein separation of a purification process. Here is the unpurified. Here was the stuff after we did one fractionation. I haven't talked about salt fractionation, but one way of separating things. Another separation. Another separation. At each step, it's getting purer and purer, until we finally see at the end, hopefully, something that is almost absolutely pure for us to work with. Well, there are some considerations for that, relevant to the numbers, and I need to sort of step you through this and tell you a little bit about this. So one of the things we want to know in doing a purification is, first of all, where is our protein? But, second, how efficiently am I purifying this protein? Because I'm going to be publishing this result, and I want to report to others, "Hey, this method really is good. "It works very well," et cetera, so that others will have an idea about how much material they're going to get out of it. And so what you see on the screen is a table following the purification of a protein. And, yes, I think that you should be able to do calculations like I'm going to describe to you in a second here. So what we see are the several steps in this purification that you saw in the last figure. You had a homogenization just to bust open the cells. You fractionated it. You did ion exchange chromatography. You did gel filtration. And finally, you did affinity chromatography. And so what this table is showing you is, really, how much of the protein that you're getting, apart from everything else in the process. So, in this case, we started with a protein and we had 15,000 milligrams of protein. And it's very easy to determine the protein concentration of a sample. I bust open a bunch of cells. Maybe I drew up a liter of cells. I bust open the cells, and I get 15,000 milligrams, which is quite a bit of protein, out of my cells, that's here. I haven't done any purification. All I've done is just bust open the cells. So I need to know how much material I have to start with. Well, that's a good starting point, but of more importance to me is, well, how much of my protein is in there? And I don't know in terms of weight but I can measure the activity of the protein. Let's imagine my protein converts one molecule into another. I can take that crude mix and take a very tiny aliquot of it and treat it with this compound that it normally acts on. And I can ask the question, well, how many molecules of this compound that it works on got converted? So I have a definition there of what's called a "unit." Let's say a unit might be a conversion of one nanomole of this molecule into something else. So I would say, well, I measure how many units I have in my total mix that's there. In this case, I had 150,000 units of my desired protein. The specific activity of that is the total number of units that I have, divided by the total amount of protein that I had. So specific activity would give me units per milligram. This would be 150,000 divided by 15,000, or 10. The yield is 100%. And the yield is 100% because this is my starting material. I haven't done anything to it. I haven't lost anything. I haven't gained anything. And my purification level is 1 because, again, I haven't purified, I haven't done anything with it. This is just where I started at. After the salt fractionation, I go back and I look and say, "Whoa! I lost a lot of protein here." I've only got 4,600 milligrams of protein, but謡hoa!悠 still have most of my protein activity. That's really good. I've retained a good deal of my original. I only lost 12,000 units, but I got rid of about 2/3 or 3/4 of the total protein. So this tells me that I have purified my protein to some extent, because now my specific activity has changed from 10 units per milligram to 30 units per milligram, meaning I got rid of a bunch of junk I didn't want. I threw out a little bit of the baby with the bath water, maybe the leg or something like that. [scattered laughter] But... that's bad. [scattered laughter] Bad professor, okay? But I have 30 units per milligram. My yield is the number of units that I have now, compared to how many units I started with. So this, divided by this, times 100 gives me, I've got 92% yield. That's pretty darn good. Ninety-two percent of my protein is there, and three-quarters of the junk I don't want is gone. The purification level is 3 because the specific activity improved by a factor of 3, 30 divided by 10 gave me 3. Well, I can continue this process, and you can go through the numbers and see each of these as you go through. And the bottom line will be that as I get further and further to the bottom, I keep losing more and more of my protein. There's no method that's going to be absolute in terms of keeping all my protein. But what we see is that the specific activity goes up enormously, which means that this stuff right here is really relatively full of my protein and there's very little other stuff that's there. So this has a total activity, 52,500 units, and it only has 1.75 milligrams of protein. That's 30,000 units per milligram. I've got a yield of 35%, meaning I did throw away a lot of the protein. But, by golly, I purified that sucker by a factor of 3,000. Alright? So I got rid of an awful lot of junk in the methods that I used to purify my protein. You might say, "Well, how do you know when you get it absolutely pure?" And the answer is, you never really know. But you can analyze it on a gel and you can see, are there a bunch of other bands on this gel? Or is it just my protein that I'm seeing on that gel? And that can be a really useful thing. If we look at the gel that I just showed you, Oh, wrong one, okay, we can see this has basically gone, they don't really show other bands here unfortunately, but you might imagine that if you had a protein that wasn't very pure you would see some other molecular sizes that would be in this particular lane. And you would see those disappear the further I get along with my purification. Questions on what I've just told you? Yes, sir? Student: So the purification level was the current specific activity level divided by the previous? Kevin Ahern: The purification level is the current specific activity level divided by the starting level. So the starting had 10, and I'm looking at how much I've purified it compared to what it was I started with. So in this case, I had 10 units per milligram to start. Down here, I had 30,000 units per milligram. So my purification level is 30,000 divided by 10, or 3,000. Make sense? Yes? Student: So the gel filtration is the SDS-PAGE? Kevin Ahern: No. Gel filtration is the method I talked about the other day where you separate on the basis of size. Student: I thought SDS-PAGE separated by size. Kevin Ahern: It does. But there are other things that separate by size. So gel electro-, that's where you had the beads with the little holes in them? Yeah. That's gel filtration. Yes, sir? Student: If you continued purifying this, would you start seeing diminishing returns? Kevin Ahern: If you kept purifying this, would you see diminishing returns? In fact, at every step of purification you will always see diminishing returns, yes. Yeah. Alright. So that's purification. What I want to do is spend a little bit of time talking about characterization of proteins, and we're going to talk about some techniques of spectroscopy and also some just simple tools to work with proteins. I'm going to skip over amino acid analysis and I'm not going to hold you responsible for it. There are, suffice it to say that there are a variety of chemical and chromatographic tools for analyzing the amino acids in proteins, but the reality is that people don't do that very much anymore because it's much easier to determine the amino acid composition of a protein if you have the DNA sequence, and it's much easier to sequence the DNA. So I'm not going to talk about amino acid analysis, and as I said, you won't be responsible for that. One of the things that we have to do in working with proteins let's say we get our protein very pure and we want to start to characterize it, start to understand better what all is in it, what it's comprised of, one of the things that we have to do is we actually have to take and cut the protein into smaller pieces. Some proteins can be quite large, many have molecular weights of 200,000 or more, and those are really very difficult for us to work with some of the methods I'm going to be showing you about. So it's desirable, then, to be able to cut proteins into smaller pieces. And to do this, we use a series of chemical reagents or enzymes, depending on what we're trying to do, that will specifically break peptide bonds in a protein at specific places. The first one I'll tell you about is actually a chemical reagent. It's called cyanogen bromide. Now, cyanogen bromide has a very interesting and useful property. When you take and you treat a protein with cyanogen bromide, what happens is every place that there's a methionine residue the peptide bond will be broken. Every place there's a methionine, the peptide bond will be broken. Well, since methionines occur in any given protein at specific places, we get a specific set of fragments that arise from treatment of a protein with cyanogen bromide. You can see other reagents that are here. Blah, blah, blah. The only chemical reagent that I expect that you will know about is cyanogen bromide. It's the most commonly used one, and it is very simple, in terms of what it does. In addition to chemical reagents that we use to chop proteins into smaller pieces, we commonly use enzymes. And enzymes come from our digestive system, for the most part, okay? Our body has enzymes that break down proteins in our digestive system so that when we eat food and there are proteins in there, we can break those proteins down into amino acids that we can use for our own purposes. So some of these that we use are really useful. One is trypsin. Trypsin is a very simple one to understand. It cuts on the carboxyl side, and I'll show you a figure for this in a second but it cuts on the carboxyl side of lysine and arginine residues. That's really useful. So again, lysine and arginines are at specific places in a given protein. By treating a protein with trypsin, I get a specific set of fragments arising from that cleavage. We'll talk more about trypsin later in the term. Thrombin cleaves on the carboxyl side of arginine. That's a very useful tool. So if I compared the pattern that arose from cutting a protein with thrombin compared to the cutting with trypsin, I would guess I would probably get more fragments with trypsin because trypsin cuts near two different amino acids whereas thrombin only cuts near arginine right here. Well, later in the term I'll talk about chymotrypsin. I'll just point out that it cuts near all of these. But you'll notice a pattern. This is a benzene ring, a benzene ring, a benzene ring. So these guys all have aromatic amino acids that they cleave next to, right here. In addition, to some extent it will cleave other amino acids. And the one characteristic you could notice of all of these, at least of four of the five, is that they are hydrophobic. Tyrosine's the only one that's not very hydrophobic there. Well, again, I do think that you should know where thrombin cuts. I think you should know where trypsin cuts. And I think you should know where cyanogen bromide cuts. The other ones are just sort of, I don't think it's really necessary for our purposes. But you should know that enzymes are very useful. They're called proteolytic enzymes, or they're called proteases. Proteases cleave peptide bonds in other proteins. And if the question is, will they cleave bonds within themselves, the answer is, one protease can cleave another protease, you bet. And so, if you leave a protease in a tube over a period of time, it will eventually lose all of its activity because it cuts itself to pieces. Student: Well, how do you store it, then? Kevin Ahern: How do you store it? You store it frozen. Student: Oh. Kevin Ahern: Yep. So you isolate it under conditions where it's not very active, and then you ship it and keep it's frozen so that it's not able to act. Shannon? Student: Is it true that you can store it refrigerated if it's dry? Kevin Ahern: Can you refrigerate it if it's dry? It depends on the protease, but yeah, some of them are actually shipped dry, but they're often kept frozen for that very same reason, as well. Let's see. I talked very briefly before about reducing disulfide bonds. I'll just very briefly show you here, again. I mentioned that, when I talked about mercaptoethanol, I said mercaptoethanol would take a disulfide bond and convert it back to sulfhydryl groups. I said, at the time, there's a molecule called dithiothreitol that will do the thing, and now you see dithiothreitol doing the same thing. What's happening is that this guy is donating electrons to this disulfide bond. So the electrons go here. And when this guy loses its electrons, it becomes a disulfide bond. The same thing happens with mercaptoethanol, actually, as it's reducing disulfides in a protein. This can be important because, again, if we want to get the pieces apart, we may want to break the disulfide bonds of a protein. And, let's see. No surprise. You've had in basic biology, the relationship of the genetic code, which shows how the sequence of DNA ultimately can be converted into the sequence of amino acids in a protein. And, as I mentioned earlier, it's actually much easier to determine the sequence of a protein by sequencing its DNA than to try to determine the individual sequence of amino acids solely starting from the protein alone. So DNA sequence is a much easier way to determine your protein sequence. Well, I want to spend some time talking about an immunological technique that allows us to identify specific proteins from an SDS gel. So let's imagine I've got that SDS get that I separated those proteins on before. And you saw there were a series of bands that were there, on the side of the gel. One of the questions you might ask is, "Well, I'm really interested in a particular protein of my own. How could I tell which one of those bands is the one I'm interested in?" So one way of doing that is this technique that's called "western blotting." And let me take a few minutes and describe western blotting to you. To do that, I need to, first of all, tell you a little bit about antibodies. Antibodies are proteins of the immune system that provide protection for us against outside invaders. And they work by binding to specific structures. So this is a schematic diagram of an antibody. It has one end that binds. It actually has two ends that bind to specific structures. And I'll describe those structures in a second. And the immune system, this allows the immune system to fight off invaders. We'll talk about the immune system in 451 next term. But this allows the immune system to fight off invaders. The reason that we use antibodies in this technique is because of their ability to bind to only very specific structures. So let's imagine that I'm interested in studying a protein that's a protein from HIV. And I've got this mixture of proteins on the side of my gel, and I really wanted to know which protein there was the one that was mine. To do this, I would have had to made an antibody against my protein of interest. So let's say I've got my purified protein. I'm interested in studying it, using a western blot technique. I would actually inject this protein into, say, a bunny rabbit or something like that. And the bunny rabbit's immune system would see this protein coming in as a foreign invader. It doesn't hurt the bunny rabbit in any way. The bunny rabbit's immune system makes antibodies against that. And then I can collect some blood from the bunny rabbit and isolate those antibodies that bind to my protein. And that can take a few weeks to get set up. Just a second, Shannon. I've got my antibody that's there. And it's binding specifically to my protein. Did you have a question? Student: Oh, yeah. Is it possible to carry this out in vitro? Kevin Ahern: To carry out the antibody generation in vitro? Student: Yeah. Kevin Ahern: No. The immune system has to recognize it and then the antibody has to be synthesized. So it can't be made in vitro, no. Alright. Now, so I've got an antibody that binds to my protein, alright? That's the most important component of what I'm going to be showing you. And the beauty of an antibody is it's specific. It will bind to my protein but it won't bind to all the other proteins I might find in blood, or all the other proteins I might find in a plant cell or whatever it is that I'm putting on that gel. It will only bind to the protein that I'm interested in, ideally. So, I've got an antibody that's specific for my, it's called the "antigen." That's the thing it binds to. So there's the antibody. There's the antigen. There's the binding. And the binding can be quite tight, and is necessary for us to do our analysis. Now, this antibody is going to be used in this technique called western blotting that I'm going to show you. So I've got my mixture of proteins. I take my mixture of proteins and I apply them to the top of an agarose gel, I'm sorry, a polyacrylamide gel. And I separate them. So I've got an SDS-PAGE. I've separated my proteins. In this case, they've actually cut out the specific band, but you could do the whole gel, if you wanted to. And I take those proteins in that gel. Gels are kind of hard to handle. They fall apart real readily, like working with Jello or something. So I can take and I can actually use an electric current to transfer those proteins onto a membrane, Onto a membrane, like a sheet of, sometimes you can use like a specialized sheet of paper. So you would transfer all the proteins that's on there onto this sheet of paper. And the proteins would then be stuck to that paper. And we can use techniques to make them stick quite strongly. So now I've got my paper that was the exact match of my gel, and it's got those proteins attached to it. I take the paper and I transfer it to a bag that has some buffer in it, and I add my antibody. I add my antibody. The antibody is given some time, a few hours, to bind to what it's going to bind to. Let's say my protein is right there. The other ones aren't proteins of interest. The antibody binds and I take the piece of paper out. I wash it so that all the things that aren't bound specifically to my protein, all the other antibodies, come off. And then I use a reagent that basically tells me, I ask the question, "Where are the antibodies?" So there are reagents that will light up color where there's an antibody. And now, by identifying the color, I can say, "There's my protein." And I can go all the way back here and say, "There's where my protein was." So it's a very useful technique for specifically identifying a protein of interest in a mixture of other proteins. Very, very useful. A very important technique for me to be able to identify a protein after I've done an SDS-PAGE. So I'm going kind of fast there. I'll slow down and take any questions you might have. Yes. Back in the back. Student: So does the protein, like, do all the other proteins come off of that? Kevin Ahern: Do all the other proteins come off? No, they don't. But the antibody doesn't bind to them, so when I treat to find antibodies, only the one that's got antibodies on it will light up. Yes, sir? Student: Could you treat those antibodies beforehand. Kevin Ahern: So his question was, could I treat the antibodies beforehand? And the answer is, yes, I could. Sometimes people will take, and it's actually easy to put a color onto an antibody, so I don't have to do the treatment afterwards. So now I can just look again, and say, "Where's the color?" There's a variety of ways of visualizing this thing right here. Student: Can you assay the amount? Kevin Ahern: Can you assay the amount? Western blotting gives you a rough idea of amount, but it's not real good for overall quantitation. But it gives you a ballpark idea of the amount, yes. Yes, sir? Student: Okay. You've identified which one out of your original SDS-PAGE is a protein of interest. Kevin Ahern: Yep. Student: Is there a way to dissolve away the gel? Or to isolate the protein for experimental use? Kevin Ahern: Oh, very good question. So he says I've identified my protein. Is there a way I can recover that protein and use it? The answer is, you can extract a protein from there, but frequently you won't want to do that. Anybody know why you wouldn't want to do that? Student: You've denatured the protein. Kevin Ahern: You've already denatured the protein. So what's in there is already probably not of any use to you if you're interested in a protein that's active. Yes, you can, and commonly what people will do is take that and analyze it in another way. So they'll cut it out and use a technique of spectroscopy to further identify it. So the answer is, yes, you can, but it depends on what you want to do with it in terms of whether that's going to be practical for you or not. Student: But you could recover it through like crystalize for an X-ray exam? Kevin Ahern: You would not take this and use it for crystallography or for other analysis like that, no. No. It wouldn't be of use to you. But it is of use in other ways. Yes? Student: So after you identify it and know where it is on there if you want to get more of it without denaturing it Kevin Ahern: You probably didn't put your whole sample on here. So you probably took a pretty small fraction. Student: And then you just do it again and you don't denature it the next time? Kevin Ahern: Uh, no. Because it's not denaturing it. You don't know where it's going to go, right? So there's other things that you have to do. I'm just showing you one way of doing that isolation separation. Yes, sir? Student: You talk about developing antibodies in rabbits. Does that work for non-animal proteins? Kevin Ahern: Can I葉he question, I think, is, will a rabbit make antibodies against a non-animal protein? The answer is, yes. They are specific for structures. So they'll work on proteins. You can make 'em against DNA. You can make 'em against RNA. You can make 'em against carbohydrate, So it's structure that's the important thing, not the source of the molecule or even the type of molecule that it is. Good question. Yes? Student: We were talking the other day about prions. Kevin Ahern: Uh-huh. Student: And you said that immune systems generally don't recognize them. What if you did that cross-speciesation and you injected that like into a different species and it recognized it as a non-native... Kevin Ahern: Okay, so, yeah. The question is, that's getting a little involved, but basically his question is, can I make an immune system recognize a prion. The answer is, yes I can. But that doesn't mean it's going to be an effective treatment, and that is what we were talking about with the immune system the other day. So, yes, I can make antibodies against that, but that's not necessarily going to be something that's going to be useful in terms of helping to treat the disease. In other words, my immune system is not going to protect me against that protein. So... yeah. Ah-bah-dah. Let's see here. So let's spend just a couple of minutes I'm not too far from finishing stuff here spend a couple of minutes talking about, you asked the question earlier about, can I pull this protein out of this gel and use it for something? And I can. So let's imagine I've taken and I've identified my band. I have cut out that band. It could either be from a gel, like I showed here, or it could be from a 2D gel. Either way, I could pull out the band of interest and say, "Okay. Here is my protein. I've separated it by gel electrophoresis. I'd like to know, what is it? What's the sequence of it, for example?" Well, I don't know where in the DNA it came from. I have to actually sequence, in this case, the protein itself. So I take this purified protein that I've got and I might treat it with some enzymes to break it into smaller pieces. I might treat it with a chemical like cyanogen bromide to break it into smaller pieces. And breaking it into smaller pieces is going to be important because the next technique I'm going to describe to you works very well on relatively small pieces of polypeptides. This technique is called MALDI-TOF. And I'm going to show you the image first, okay? Oh, blast it. I'll start with this. MALDI-TOF is a, MALDI, M-A-L-D-I, stands for "matrix assisted laser desorption ionization." You don't need to know that. Okay? It's a mouthful. I always have to look it up each time, so I remember what it is. Matrix assisted laser disruption ionization. What does that mean? It means this technique uses a laser to make a sample volatilize. That's the first part of what happens in mass spec. How many people have done mass spec in a chemistry lab? So mass spec tells us mass. And mass specs work in vacuum chambers. And ions in the vacuum chamber get accelerated and they get accelerated to move up to a detector which detects when things hit them. So MALDI-TOF is a specialized form of mass spectrometry that allows me to analyze relatively large molecules, like polypeptides. To do MALDI-TOF, one takes a protein sample that's purified, that little band that I had, and mixes it with some material that makes it form a sort of a crystal. And that would be on the end of a, let's say a pinhead that I could put into a chamber that would be evacuated. So I've got my sample that's on a pinhead. I put it in this evacuated chamber and it's sitting there, waiting for the analytical process to begin. The "L" said "laser," and laser plays an important role in this process. The laser, I thought I had that figure here and I just don't see it. Okay. Well, it's in the book. I'll post the figure later, to show you. The laser is, there's a laser that's pointed, and the laser hits that sample in the evacuated chamber. So the evacuated chamber has my crystallized material. When the laser hits it, that crystallized sample volatilizes, meaning it leaves the pinhead and goes into a gaseous phase. In the process of that happening, my sample becomes ionized. It becomes charged. Okay? Because it's charged, an electric field will attract it to the detector at the other end. Okay? Let's imagine I've got a sample that's got, let's say, two things in it. One that has a molecule with a mass of 500 and one that has a mass of 1,000. And they're both charged +1. Which one's going to make it faster through the chamber? The 500, right? Because it's got the same charge. It's got less mass. It's going to have less inertia, and it's going to be accelerated faster and is going to arrive at the detector faster. It means that, if we measure carefully the time it takes from volatilization over to the time the detector detects it, that time interval actually tells us the size. Because it'll take something twice as long that's 1,000 in molecular weight as it will for something that's 500. That's really useful for us because with that technique we can determine molecular masses with amazing efficiency. Because of that, we can take a sample that has a polypeptide, and that polypeptide, when it ionizes, will break into pieces. The places where it will break are peptide bonds. So here's a full-length polypeptide that's there. When it ionizes, some of them will be broken here, and I'll get one amino acid. Some will be here, I'll get two amino acids. Some will be broken here, I'll get three amino acids. And if I determine the masses of all those the difference between the peaks is the difference of each amino acid that comes off I can actually determine the sequence of the thing I started with. Yes, it's complicated and yes, it takes a computer to do it. I'm not going to sit and stare at it, myself. But the beauty is that, in a single mass spectrometric analysis, I can determine the sequence of amino acids of that polypeptide I put into the chamber. Now that is really powerful. Because of this technology, because of this technology, a scientist in a modern mass spec lab can determine the sequence and, thus, the identity of 4,000 proteins a day. We could take all the proteins that was on a 2D gel, cut out each spot, have each one analyzed, have each one identified, in a single day. That is an incredibly powerful technique. Question? Student: How long have they be able to do that? How old is this technology? Kevin Ahern: How what? Student: How old is this technology? Kevin Ahern: This technology dates to the '90s. Yeah. So it's relatively new. Yes? Student: So why do you, how do you know which end it starts at, so you have the... Kevin Ahern: Okay. So there are at both ends, and you can actually get this one, as well. That's what that computer has to sort out. Okay? Okay. Kevin Ahern: A lot of stuff there. Let's call it a day and I will see you on Wednesday. Student: [inaudible] Kevin Ahern: I'm sorry? Student: Will we get to talk about photo [inaudible]. Kevin Ahern: We're not going to talk about [inaudible]. Sorry. Yeah. [END]
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Medical_Lectures
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02_Biochemistry_Buffers_Lecture_for_Kevin_Aherns_BB_450550.txt
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Kevin Kevin Ahern: Alright! Good deal. Let's see. A couple things. Somebody asked if I would send, or several people asked if I would send matching, problems for the 6th edition of the textbook compared to the 7th and I did that and hopefully that's helpful to you. As I noted in my e-mail, sometimes the problems change a little bit so I can't tell you they're exactly the same problems but at least they're the ones I assigned the students last year so you can see approximately where those are. I have put a couple of books on reserve in the library. They're not yet available I think. The library takes a little while to get stuff out there, so, when I get the word from the library they're available, they'll be available to you. And I've got a bunch of 6th edition textbooks sitting around as well so if there's interest in those and you want to take a look at one of those come see me, I can work with that also. Okay, not yesterday, Wednesday, Monday, whatever it was... okay, Monday, I got through talking, about just sort of an introduction about chemistry and I started talking about solutions and pH and today I'll spend a fair amount of time talking about pH and buffers. It's important that we all get on the same, page with respect to this. Now somebody asked me a question just before class about, is it true I don't let you use calculators on an exam? And that's true... you don't use calculators on exam. But you also won't have to come up with a number on an exam, okay? You're not going to have to do a logarithm in your head for example, okay, it's not something like that. But you will have to know how to use equations and how those equations tell you information. I'll give you some examples of that today. So even though there are problems you may be actually using a calculator to do them, all the problems that I will assign you, you will not need a calculator to do. Yes, ma'am? Student: So we should commit all the equations to memory? Kevin Ahern: Should you commit all the equations to memory? That's a very good question. In fact, I will give you every equation that you need to know on the exam. Ahh [class laughing] right? Okay, so you won't need to memorize any equations. You won't have to crunch a number as such, okay? You will have to know how to use equations and get information from those equations by their very nature. So that's something you will need to know how to do. The TAs will be helping you with that process and of course I'm available to help you with that process as well, so don't sweat the calculator component, okay? Alright, well usually in my lectures what I have are a series of figures I go through and show you various principles and so forth, and this is one place today and today's an unusual lecture in that I don't really have much in the way of figures I'll be going through and showing you, simply because your book actually I think is rather short-sighted. It's one of the few places where the book is not very good in its coverage of the subject. It's not very good in covering pH and buffers. And so it's important to understand these concepts because, as we shall see, understanding how the pH of a solution affects molecules, we understand better how it affects charges of molecules and ultimately how that affects these molecules that, that are in proteins. Proteins, we are going to come back to many times, are essential obviously for cellular life, but their structure is essential for their function. And one of the things that we will see when we start talking about proteins, and I'll actually... [buzzing] you know, when they redesigned this room, I was told that was going to go away. Space aliens are still here I guess. Okay, when they, proteins get different charges, they adopt different shapes. And so we change the ability of a protein to function as we change the pH of a solution. So, what I'm going to do today... [buzzing] oh, don't do that... [rumbling continues] okay. What I'm going to go through today is talking about pH and buffers, okay, so it's very important to understand that. Alright, well last time I introduced the topic I said pH of course we know is a negative log with a hydrogen ion concentration and pOH is a negative log with a hydroxide ion concentration. Freshman chemistry, okay? pH plus pOH is equal to 14, okay? And by the way, I think you should be able to add 14 and subtract 14 without a calculator so there are, I can't tell you, you won't have to do simple math, but you won't have to do any crunching. Another concept I want to introduce here, I talked last time a little about the fact there's a difference between strong acids and weak acids. And we will be mostly concerned in this course with weak acids. That is acids that do not completely dissociate in water, okay, at a certain pH, alright? Now a prime example of that, acedic acid. As I said, acedic acid ionizes to a very limited extent. If I put it into water, I might get one molecule in a thousand that are ionizing, okay? I put HCl in water and I get every one of them ionizing, meaning they come apart. So there is a fundamental difference between weak acids and strong acids. If I know how much HCl I start with, I know how many protons there are there. I don't necessarily know that if I have acetic acid. The number of protons that I will have free in a solution where there's acetic acid will depend on the pH of that solution, okay? The ionization of acetic acid varies with pH, that's number one. The ionization of a weak acid varies with pH. Okay. Now we'll see that mathematically and I hope you'll think in mathematical terms instead of memorization terms. I can tell you for example if the pH is higher, we're going to have more ionization than if we have the pH being low, okay? When I say ionized with respect to a weak acid, I'm talking about a proton coming off. That's what ionizing is all about. For HAc I can write HAc goes to H+ plus Ac-. Okay. That is an ionization right there. I'm making two ions. Now, this equation you're going to see over and over and over because it's important that we understand what's happening in that ionization. Alright. Well how do we determine, are all weak acids the same? They're not all the same. In fact, we see enormous variety in the strength of weak acids. If I were to define strength of weak acid to you, I would say that at a given pH, a relatively strong weak acid will ionize more than a relatively weak, weak acid. Now we've got strong weak acids and weak, weak acids, alright. Nothing like confusing the picture on the second day of class. Okay. That's important, okay. Now, how do we compare those? Well we compare those where there's a measure of a strength of an acid it's a constant known as Ka. We won't even concern ourselves with that. We're going to be concerned with the negative log of Ka which is the pKa. So just like ph is the negative log of a hydrogen ion concentration, pOH is the negative log of the hydroxide ion concentration, pKa is the negative log of the Ka. We probably won't talk about Ka again after today. We will talk a lot about pKa. So what is pKa? pKa is the measure of the strength of an acid. A strong acid like HCl has no pKa, it completely comes apart, okay. A weak acid like, acetic acid has a pKa of 4.76, okay. What does that mean? Okay, I'll tell you what that means in a second. But I'm going to compare the pKa of acetic acid with that of formic acid, okay. Formic acid is also a weak acid but its pKa is 3.75. And no, you don't need to memorize these numbers, you'll be given them if you need them. Okay. If I compare acetic acid to formic acid, formic acid has a pka of 3.75, acetic acid has a pKa of 4.76, formic acid is stronger of a weak acid than acetic acid is. Okay? Are we clear on that? So when comparing the two, I compare the two pKas, the one that has the lower pKa is the stronger acid. Now we can go through, we can derive all the math, as necessary to do that, but we don't really need to do that in this class, okay. Simple concepts about what Ka is. Well, what pKa is, I'm sorry. Okay. I said HCl, strong acid, doesn't have pKa. Okay. We don't, we don't even consider it. Because it just completely comes apart. We can't mess with that. Alright. So, the lower the pKa, the stronger the acid. Alright? [buzzing] I don't know why it does that. Male student: Do you have your phone in your pocket? Kevin Ahern: I do have my phone in my pocket, but I don't think that should be doing that. Should I, should I mess with it? Male student #1: Yeah Male student #2: Yeah, that will get feedback across. Kevin Kevin Ahern: I'll put it on airplane mode. I've tried this before but it didn't seem to make go away. [Buzzing continues] Female student: [inaudible] Kevin Kevin Ahern: I'll just jump. Okay. Now, I apologize for that noise. I think it's something that this room is haunted or something, it just doesn't, it's stuck there. Alright, let's think about that, that weak acid, let's think about acetic acid for a moment and let's think about what happens with it in a solution. I said if I just dump it into water, okay, a pretty small percentage of that HAc becomes H+ and Ac-. When I'm concerned about pH, the only thing I'm concerned about is the H+, not the Ac-, right? If I have it sitting there and let's say I put a million molecules of acetic acid into that aqueous solution of water and one thousand of those, protons come off, I'm going to have one thousand protons from there come off, I'm going to have one thousand Ac minuses, right? And I'm going to have 999 thousand HAcs left behind, right? Everybody with me? Let's say I add some sodium hydroxide to that solution. Sodium hydroxide of course is a strong base, and like a strong acid, a strong base completely dissociates in water, so if I put a million molecules of NaOH into a solution of water, I get a million molecules of Na+ and a million molecules of OH-. Completely dissociates. So strong acid is equivalent to strong base in terms of its strength. They completely come apart. Let's imagine, if you will, that I put a thousand of molecules, let's make it fun, let's put 449,000 molecules of OH- into that solution. Okay? What's going to happen to that solution? I'm going to have an excess of OH-, right? Is the pH going to go way up? It's gonna go up. But it's not going to go up as much as you might think. Why not? Go ahead. Female student: The buffer's to the limit? Kevin Ahern: Well, you're getting ahead of me in terms of definitions of what a buffer is, but yes, buffers, there are buffers, this is acting as a buffer, but I don't even want to use that term yet. Why doesn't the pH just go through the roof? Female student: The acedic acid dissociates? Kevin Ahern: Acetic acid can dissociate. So I had 499, I'm sorry, I had 999 molecules of that Hac that was sitting there, right? Some of those could give out protons, right? And what's acedic, what's OH going to do when it hits a proton, well it's gonna hit with a proton and make water, thereby neutralizing it, right? So I had a thousand molecules of H+ and I had a thousand molecules of Ac-, and I add 449,000 molecules, why am I think 449, 499 thousand molecules of NaOH, 499 thousand molecules of that Hac is going to give up protons. It's going to give up protons. And at that point, I'm gonna have 500,000 molecules of Ac- and I'm going to have 500,000 of HAc. Right? 1,000 plus 499,000, right? With me? You can say, "yes, Kevin." No you can't obviously. Okay. At that point I have 500,000 molecules of HAc, I have 500,000 molecules of Ac-, and I've got a higher pH. That higher pH turns out to have an important name. When I have equal numbers of Ac- and HAc, I've reached the pKa. pH equals pKa when salt equals acid. With me? pH equals pKa when salt equals acid. So pKa is simply a pH. It's a special pH. It's the pH at which salt equals acid. Now, I told you that acetic acid had a pH of 4.76, right? That tells you something, it tells you that pKa, I said pH, has a pKa of 4.76. That tells you that pKa is a constant. It's a constant for a given acid. Whenever I have acedic acid, it will always be 4.76. It will never change. I can change the protons, I can change the hydroxide, but the pKa will be a constant. The pH will change, but the pKa won't. Alright? Well it turns out there's an equation that relates these things I'm telling you just conceptually at the moment, okay? The equation is an equation you're going to hear a lot about, it's called the Henderson Hasselbalch equation. The Hendrson Hasselbach equation states that, here we go, pH equals pKa plus log of Ac- over HAc. More commonly we will say that pH equals pKa plus log, the concentration of salt, divided by the concentration of acid. I called the thing that has lost the proton the salt. And I think you'll find it much easier to understand if you call it the salt instead of the base. Okay? So, pH equals pKa plus log, the salt, over acid. Acid is the thing that has the proton. Acid is not the proton. The acid is the thing that has the proton. The difference between the salt and the acid is a single proton. There it is right there. Okay? Now, everybody understand the terms I'm talking about? Alright? Salt, acid, Henderson Hasselbalch equation. We're going to use the equation in just a second. How do I make something into a salt? I take protons, I take protons off of an acid, right? I can convert acid into salt by taking protons off and in the example I just gave you, how did I take protons off the acid? I added a strong base. Okay? If I wanted to put protons onto that salt, how do you suppose I would do it? I would add a strong acid. Okay? How would I protons on? Well, if I start dumping protons into the reaction right here, what's going to happen to this equation on the basis of the principle of Levoisier? Male student: It's going to push it left. Kevin Ahern: It's going to move to the left, which means I'm going to make this and I'm going to lose this. Right? You already saw I went to the right when I started taking protons away, the solution starts trying to make them up. Starts trying to replace them, starts making more Ac-. There's a one to one relationship. For every molecule of strong base I added, I lost one of these and I made one of these. The same holds true if I go the other way. If I go the other way and I add protons to this solution, if I add 500 molecules of ACl, I'm going to lose 500 of these, I'm going to make 500 of these. Understanding that is the most thing students screw up on buffer problems. The single most common thing they screw up on. There's a one to one relationship between adding and subtracting acids and bases. That's all there is to it. A very simple concept, okay? Now, let's go back to our equation. Our equation said pH equals pKa plus the log, the concentration of salt, divided by the concentration of the acid, alright? In the example I gave you, we had 500,000 molecules of salt and we had 500,000 molecules of acid. Let's plug in to that equation. They're in the same volume so the volume cancels out, we don't have to worry about concentration, we can actually use numbers, okay? you have a hand up. Female student: Is that a dash or a negative? Kevin Ahern: Uh, where? Female student: [inaudible] Kevin Ahern: This? That's a dash. Yeah, that's not a negative, that's a dash. That's a good question, I didn't notice there was a dash there. Maybe I'll remove that dash so you don't get confused. That's a dash, not a negative pH, that's just a dash. pH equals pKa plus log of salt over acid. Now, Let's think about that. Let's plug in our terms. I want to find the pH of the solution I've just, I've just defined for you. I just made a solution of acetic acid. I have a pKa of 4.76 because that's a constant for acetic acid. And I've got 500,000 molecules of Ac - and 500,000 molecules of HAC. What's the pH? 4.76! You can tell by either the fact I just told you when the two are equal, if that makes sense, but you can also tell it more importantly from the equation. The log of 500,000 over 500,000 is the same as the log of 1, and the log of 1 is equal to zero. Yes, you have to know that, but I'll even put that on the exam. Okay? That's cool. Okay. That's cool. So now I see mathematically why the pKa is the pH at which the salt equals the acid because when the salt equals the acid, this log term becomes zero and pH equals pKa. Questions? I'm kinda going through this kinda blig-da-bleh. Am I that clear? I know I'm not. Am I that fiersome? I probably am. Yeah? [laughing] Oh, is there a question? I'm sorry. Yeah? Male Student: You said if you want to increase the [inaudible], you add a strong base? Kevin Ahern: If I want to increase the amount of salt, okay, I would have to pull protons off of the acid using a strong base. Okay. Important concept and thank you for asking the question. If I want to make acid from salt, I've got to put protons in which means I've got to add a strong acid. Yes? Male Student: So to make acid, you have to use acid? Kevin Ahern: In order to make acid in this system, I have to use protons from a strong acid, that's correct. Okay? So if I guess there's no questions, you guys are ready for a pop quiz. That means you've already understood it, right? Or are there questions? Because I'll find out really quickly if you understood it or not. I guess if there are no questions, you must understand it, therefore the pop quiz is irrelevant, right? Nobody has a question? Male student: I have a question. Kevin Ahern: Okay, good. Male student: What's the Na... Kevin Ahern: You just saved the whole class right there, they should thank you. [class laughing] Male student: What's the Na doing while it dissasociates from the OH in a strong base? Kevin Ahern: So I add NaOH, what's happening to the Na? Nothing, just sits there. Male student: Just hangs? Kevin Ahern: Just sits there. Female student: And that goes to say [inaudible] strong acid [inaudible]. Kevin Ahern: Mhm, the Cl's just going to sit there. And if you keep adding strong acid and strong base, strong acid, strong base, you're making NaCl, and NaCl, and you've got a very salty solution, and that's all that happens. Other good questions. So, you're saved from a pop quiz. How about that? I won't be so nice next time. Alright, now, what we're starting to understand, or what I hope to introduce next is the concept of what is something called a buffer. Okay? You used the term up here of what a buffer was, we need to talk about what a buffer is. So I'm gonna find a buffer for you and then we're going to go through some examples. Alright? A buffer. Buffers are absolutely essential. You'll see why the further we go along. Alright? Definition of a buffer, a buffer is a system that that resists change in pH. It resists them. It doesn't prevent them. It resists them. Okay? In the example I gave you, we dumped a whole bunch of OH in there, but the pH didn't go up very much and I'll show you a graph of that in a bit, okay? The system is acting like a buffer. It's preventing the pH from going up as much as it would if the buffer weren't there. There's a couple of problems that I've assigned in your book that will illustrate to you what a buffer is and how a buffer works. And you'll compare those, actually, it's not in the book, it's actually one of the ones I made up for you on the system, okay? Where there are the ones that Kevin made up on the site, alright? If you click on those, you'll see a couple of them that will illustrate to you how a system that has a buffer differs from a system that doesn't have a buffer if you have the same amount of protons. And you'll see there's a very big difference between the two. Now, so I've defined a buffer for you. A buffer is a system that resists change in pH. Yes, ma'am. Female student: Does it work the same whether the buffer starts in the system or whether you add the buffer later? Kevin Ahern: Does it matter if you add the buffer first or if you add the other stuff later. Turns out it technically doesn't, no. Good question. Yes, sir? Male Student: Approximately, what's the effective range pH-wise of the average buffer? Kevin Ahern: Oh this is a very good question also and thank you for asking that. Very much. What's the range of a buffer? is a buffer an infinite thing? Can a buffer resist pH change forever? No, okay. Buffers have what we call capacity. We'll see some examples, in fact one of the problems in your book actually illustrates capacity to you. It's going to confuse you when you first do it 'cause it's not gonna make any sense. And that should be a clue that something isn't what you think it is. But to answer your question. What's the range of the effectiveness of a buffer? It's mostly a definition thing. But effectively, most buffers are good within one pH unit above or below their pKa. So for acetic acid, the effective buffering range, that is where it's best at resisting change in pH, is from about 3.76 up to about 5.76. One pH unit of its pKa. Either way. Okay? And I haven't given you an example of how a buffer works yet, so I'm gonna do that in a second, but other questions just in general about buffers? Okay. Alright. I keep needing to look up also, make sure there are people up there. Yes? Male student: Will a buffer work outside of its effective range? Kevin Ahern: Will a buffer work outside its effective range? Yes, I mean, will a buffer participate in accepting and donating protons outside that range? Yes, but its effect on controlling the pH will be minimized. So we won't see as strong a protective effect if we get outside of that range. Good question. Alright. So how does a buffer work? Well, let's think back to this buffer that this system that I just described to you. I've got a solution that has 500,000 molecules of Ac- and it has 500,000 molecules of HAc. Equal numbers of salt and acid. Let's imagine that I add 10,000 protons to this system. You can do the math pretty quickly and say, "well, you're gonna have 501,000 molecules "of HAc because we're making Hac "and I have 499,000 of Ac- because I lost Ac-", right? How much would the pH change? Well, it turns out it's not going to change very much. How would I know? I plug it into the Henderson Hasselbalch equation and what I would see is that I would have pH equals 4.76 plus the log of 499,000 divided by 501,000, right? That's very close to one. That means that log term is very close to zero. Right? The buffer is protecting this. Where there's excess protons, the buffer grabs them. Where's there's excess OH, the buffer makes protons. Very, very important concept. Female student: The buffer, in that example you just gave, did you give us what your buffer was, or did you just say you added this much buffer? Kevin Ahern: So, acetic acid is the buffer system here. Thank you for asking that also. Alright. Weak acid systems make great buffers. Anything that has a pKa makes a great buffer in a certain range. Okay? Anything that has a pKa makes a great buffer in a certain range. Now, in this example I just gave you, we don't see the pH change very much. It's a very miniscule change that happens and that logirithm of it actually makes it an even smaller change. If I did this, I would ask you a reasonable question for me to ask you on an exam would be, "is the pH higher than the pKa? "Or is the pH lower than the pKa?" Well I don't have a calculator! You don't have to have a calculator to answer that question. How would you answer that question? Male student: It's lower. Kevin Ahern: It's lower? Why would you say it's lower? Male student: It's been acidified and the lower pH. Kevin Ahern: Okay. So he says it's been acidified and it's true that's done that. But one of the places where students confuse themselves is which acidification. I want you thinking mathematically. Mathematically, why is the pH lower than the pKa. And it is lower than the pKa, you're right. Female student: [inaudible] Kevin Ahern: You're taking the log of a number that's less than one. Exactly! 499,000 divided by 501,000 is a number that's less than one. Just like 499 over 501 is less than one. The logirithm of a number less than one is a negative number. So if I have 4.76 plus a negative number, I have to have the pH lower than 4.76. Right? What if it were higher? What would I have to do to add to the solution to make it be, for example, 501,000 molecules of Ac- and 499,000 molecules of HAc? What would I have to add to make that happen? I wouldn't add salt. I would have to add HCl. I could add salt, I could add salt. But not based on what I just told you. I didn't change the total amount. I said 499 and 501. If I add salt, I'm gonna have 501 and 501, right? I'm going to have a different total amount. When I add a strong acid, I turn salt into acid and my total stays the same. One million molecules. So I add a strong, base to that to make that happen. Pull those protons off to switch the HAc into Ac-. Everybody clear on that? Okay. If I told you I had a solution that had a pH of 4.3 and the system had a pKa of 4.9, more salt or more acid? pH 4.3, pKa 4.9. What does that say about the log term? What does the log term have to be? Students: It has to be negative. Kevin Ahern: It has to be negative. Negative log term, what do I have to have? I have to have more acid, right? Has to be less than one, which means I have to have more, and by the way, HAc only holds for acetic acid, so to make it general, we call it A - and HA, okay. So I have to have more A - than I have HA. Now these are the kinds of things you can work completely without a calculator. You can manipulate that log term in your head. Bang. You got it right there. You don't have to have a calculator. It's important for students to learn how to work problems, okay, to understand the math of what's there, not to see the confusion of the numbers. That's why I don't want you using a calculator. I want you to think about these things. I want you to understand them at a real level and not a "a-duh-duh-duh-duh-duh-duh-duh-duh" level. Because if those numbers that you "duh-duh-duh-duh-duh" in don't have meaning, then you get garbage out as well. You have to put the right stuff in. So what I'm trying to get you to do is to understand how to get that right stuff in. On my exam, all you will have to do is get a solution to the point where it would go into a calculator. If you get down to this is the logirithm of 3, then the logirithm of 3 is the answer and that's it. You don't have to calculate that, okay? Make sense? Okay, um, let's see. Other questions about that? I'm going to show you some examples after if you don't have any questions. Nobody? Okay, alright. Let's see a buffer in action, right? Yes? Female student: [inaudible]. to assume that the concentration of Ac- plus HAc is .1 more? Kevin Ahern: Not for a buffer, no. A buffer can have any concentration. Female student: Okay. That was the concentration of that specific problem. Kevin Ahern: Okay. Okay. Female Student: Okay. So just don't always assume. Kevin Ahern: No, no. So buffers can have any concentration. And that's important because buffers, I can make a buffer as concentrated as I want to. If I say the word "buffer," here's something I want you guys to pop in your heads. Alright? When I say the word buffer, I want you to think of two terms. Two terms I've been using over and over, alright? Salt and acid. That should immediately pop those two up in your head. When I say buffer, if I told you I have a buffer that is .1 molar, the first thing that should pop into your head is salt plus acid equals .1 molar. Because when I have a buffer, I have to have both of those. I have to have salt and I have to have acid. Now the actual amounts of those you may have to calculate, alright, but when I say buffer, salt plus acid equals the total amount of a buffer. Alright? In this case it was .1 but it could be a variety, it could be anything I make up. Yes? Connie. Connie: Just to verify, did you say if the buffer's .1, then salt plus acid equals 1? Kevin Ahern: If I say a buffer is .1, say .1 moles of buffer, if I said I had .1 moles of buffer, then salt plus acid would equal .1 moles, that's correct. Okay? Makes sense? Okay, now. The concentration of a buffer is important. Let's think about a .1 molar buffer. Let's say I have a .1 buffer of acetic acid. My favorite acid, right? And it's at its pKa value. Its pH is at its pKa value. What does that tell me about the concentration of salt and acid in that? They're equal and they're equal to what? .05 each, right? Half the total. What if I add, let's say .1 molar Acl to that in an equal volume. What's going to happen to that buffer? Based on what I just told you, what are you going to see happen? I add HCl, I'm going to lose salt, right? I'm going to make acid. How much salt can I lose? .05, but I just added twice that amount of HCl. Uh oh. When I start doing my subtracting, I discover I have a negative amount of Ac-. Can I have a negative amount of Ac-? No. Something's wrong here, right? Something's wrong, Houston. Okay? Female student: How much [inaudible]? Kevin Ahern: What I've done. I added .1 molar, okay. What I've done is I've just given you an example where I've exceeded the capacity of the buffer. Buffers have limited capacities. Alright? They're not infinite. So one of the reasons we change the concentration of a buffer, okay, is so we can change its capacity. Alright? Capacity's important. Alright, so you'll see one of the problems in the book will actually exceed the capacity of a buffer. When you get to it, I think you'll discover, you'll figure out what it is. Yes? Male student: So regardless of the concentration of the buffer, the capacity won't change [inaudible]. Kevin Ahern: I'm not sure [inaudible] the question. Male student: So you have 500,000 or you have .1 molar versus .05 molar, the capacity's going to stay the same? Kevin Ahern: The capacity is defined by the salt and acid that is there. So if I more than that of strong acid or strong base, I'm going to exceed its capacity. Connie? Connie: So it's not dependant on the concentration at all? Kevin Ahern: It is dependent on the concentration. Absolutely. Connie: Oh, well, because you said it's usually within one pH unit of the pKa. Kevin Ahern: Yep, yep. Connie: What if you have a really concentrated [inaudible]. Kevin Ahern: That's a mathematical question that you're asking me to define non-mathematically. So come see me, I'll show you mathematically what we're talking about. With one pH unit, you're not exceeding capacity. The one pH unit is going to define how much I can add to it to to get to that one pH unit. Right? So I'm limited by that. Alright, okay. So buffers have capacity. I can't exceed those capacities because if I do, the pH is going to go boing! because I no longer have a buffer. The buffer was providing me that, that protection against massive change in pH. When I exceed the buffering capacity, I don't have a buffer anymore. It's gone. The pH is going to go sproing. Yes, sir? Male student: Could it be generalized that the capacity is going to be defined both by concentration and volume combined of the buffer? Kevin Ahern: So, the capacity of a buffer is, so the question that you're asking is if I know the number of moles total a buffer, that defines ultimately the capacity because concentration times volume gives me that and the answer is yes. Okay. Now, I know you're going to have some struggles with this and I understand that. Please come see me, please come see the TAs, work through the problems, okay? I can guarantee you there will be help in getting and understanding of this bigger picture, okay? It's important to get that bigger picture. Gotta keep an eye on the time... Okay, I promised to show you a buffer plot. And this very high quality graphic was drawn by yours truly. [class laughing] This was for you guys in freshman chemistry, did a titration curve, right? Did you like titration curves in freshman chemistry? You did! Okay, good. Most people don't like them. Alright, so this shows the relationship between the pH and the amount of OH I added. In this case, I started with a solution that had essentially all acid to start with. How do I know that? Well, I had a low pH, and right down here the low pH. How would I know a low pH would have mostly acid to start with? How would I determine that? Nobody? If I said extra credit, then everybody would jump right? Male student: What was the question? [laughing] Kevin Ahern: I didn't say I was giving extra credit, I just said if I said I was giving extra credit, okay? The question is how would I know down here I got most things in the acid form? Female student: [inaudible] Kevin Ahern: But I'm not talking about protons. Protons are not acid. I'm talking about HA. How would I know I have most everything in the HA form? You have a friend. What is the friend? The friend is Henderson Hasselbalch equation! Alright? If the pH is low, below the pKa, what happens to that ratio? More salt, more acid? Class: More acid. Kevin Ahern: More acid, right? Bingo. Henderson Hasselbalch tells me and it tells me very quickly. The answers to virtually every question I'm going to ask you will be rooted in that equation. They're gonna be rooted there, okay? You wanna get familiar with that equation. Now, let's see what happens to the solution. I start out with the solution, it's got protons on, I start adding sodium hydroxide. What happens to the pH? The pH rises. It rises relatively rapidly at first. Why does it rise relatively rapidly at first? Male student: It's outside the buffer zone. Kevin Ahern: It's outside the buffering region. The buffering region being plus of minus 1. In this case, the pKa I've got for this is about 2.5. So I'm below 2.5, it rises relatively rapidly and then it starts to level off. And it's leveling off because it's acting as a buffer. It's resisting the change. I'm adding a lot more hydroxide, but the pH is not going up very much. Once I started getting away from that region by more than 1 unit, I all of a sudden see the pH start to go boing. Every buffer plot is going to look like this. This is a visual image of what a buffer is doing. It's resisting a change in pH at a certain range. The maximum resistance is right here where pH equals pKa, and that's another important concept. Not only is pKa the pH at which the buffer has equal salt and acid, it's also the place where there's maximum resistance to change in pH. We have maximum buffering capacity. There's a question here? Yeah. Male student: Molecularly, why does it do that? Why doesn't it just continue on the more you add, the more it disassociates? Kevin Ahern: It is associating. Male student: I know, but what does it plateau? What doesn't it just continue the breaking of the bonds? I don't know why does it pause. Kevin Ahern: Why does it go up and then flatten? Male student: Yeah Kevin Ahern: Okay, well two reasons. Alright? One, plug it in mathematically. Plug in a whole bunch of different values of salt and acid in that ratio and you'll see that mathematically, that's exactly what it does, so mathematically, that's the answer to your question. Male student: I'm talking more molecularly. Like bond-wise. You know what I mean? Kevin Ahern: Well you're talking about ionization. Right? Okay. So ionization is happening because the absence of presence of protons. It's Lavoisier's equation with the HAc going to H+ and Ac-. We put pressure on that equation one way or another. That favors the ionization. Okay? Question? Male student: Wasn't it Le Chatelier's Principle? Kevin Ahern: What did I say? Lavoisier's... Le Chatelier's Principle! God, I do that all the time. It is Le Chatelier's principle. Sorry. Lavoisier was the practical inventor of chemistry, not Le Chatelier. Yes, but thank you, it is Le Chatelier's principle. Male student: Are there ever systems where multiple buffers that have overlapping ranges of effect are used where you could have one say 1.5, 2.5, 3.5? Kevin Ahern: Yeah. So can you have multiple buffers in there? The answer is you can. We're going to do buffers one at a time. I figure you've got enough to think about. But yes, you can. And multiple buffers do complicate the picture a lot. We'll see starting on Friday, I'll be talking about amino acids. And amino acids have multiple buffering regions. And so you'll see a flattening, a flattening, a flattening. That can happen. And so that is a simply example of what you're talking about. Okay, now I promised you guys, any questions about this? We've gotten through a good number of things. I want you to, the most important message I want you to take across, get across from this, is that the ionization is going to be related to pH and its relationship to the pKa. pH, the ionization of a substance is related to the pH of the solution it's in and its pKa value. The more the pH is above the pKa, the more the protons will be gone. The more it's below there, the more the protons will be on. And you don't need to memorize that. Henderson Hasselbalch tells you that. What I'm saying in words to you is what Henderson Hasselbalch is telling you. That's a lot of stuff, why don't we finish with some fun? I thought we might celebrate Henderson Hasselbalch with a song to the tune of "My Country Tis of Thee." I've never sung in front of a class before so let's do this. [professor and class singing] Lyrics: Henderson Hasselbalch You put my brain in shock Oh woe is me The pKa's can make Me lie in bed awake They give me really bad headaches Oh hear my plea Sale minus acid ratios Help keep the pH froze By buffering They show tenacity Compelte audacity If used within capacity To maintain things I know when H's fly A buffer will defy Them actively Those protons cannot waltz When they get bound to salts With this the change in pH halts All praise to thee Thus now that I've addressed This topic for the test I've got know-how The pH I can say Equals the pKa In sum with log of S o'er A I know it now Kevin Ahern: Okay, good place to stop. Thank you [END]
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Medical_Lectures
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Introducing_MRI_Introduction_to_NMR_Nuclear_Magnetism_3_of_56.txt
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the next place we're going to go or really the place where we need to start is that when we take our patient right and we're going to put them inside this thing that we call an MRI scanner and we're going to do a bunch of things that we haven't even talked about yet to generate something that we will call a signal from the body now most of you especially the Radiologists are used to these nice cross-sectional or maybe 3D representations of Mr images we're not actually going to talk about images per se probably until late this afternoon or tomorrow before we ever get to the point of making an image the first thing we need to discuss is where do we get this thing that we call signal intensity or the NMR or MRI signal from where does it come from and we're going to talk a lot about how we generate that signal uh as well as how we manipulate that signal and how some of the parameters in measuring that signal can influence what the image is ultimately going to look like we have to do all of that before we can ever get to the point of actually looking at or generating anything that's close to an image so our starting point is going to be looking at some homogeneous signal what I mean by that is we have a patient in the scanner the signal we're going to be talking about is at this point not coming from any specific location it's whatever we whatever we get from the entire subject now in Mr or in nuclear magnetic resonance there are some choices that we can make as to what exactly it is that we want to measure so typically in MRI the signal intensity derives from hydrogen protons right that are predominantly almost exclusively coming from water molecules in tissue that's the signal that we typically are examining with Mr now there are some people actually here today that do imaging or spectroscopy at least looking at things other than hydrogen so in other words there are other elements in the body for example right phosphorus carbon at least certain isotopes of these uh we can do imaging of sodium there are a whole bunch of different elements that can be used to generate an NMR or an MR signal and perhaps even an image so so the first thing I want to address and I realize that you know this principally doesn't really confront people who are doing clinical Imaging because in clinical Imaging the only thing we're interested in it seems is proton uh that's actually something that's changing by the time you guys are finished with your training and out in practice or doing academics or whatever you might be doing uh you may find that looking at other nuclei is something that is becoming extremely relevant uh you can now actually you know a commercial Mr system that you buy off the shelf can be purchased with the ability to do multinuclear type of Imaging and there are some specific applications like in prostate cancer and in uh metabolism like in diabetes we're looking at some other nuclei looking at sodium and brain tumors where these things are looking like they may actually have clinical application so may be a little bit out there but it's not totally uh it's really not totally off the table so I want to just address for a minute what our options are in terms of where we can derive this Mr signal from uh it can't be anything right for example helium is right is not an option for NMR so there are only a sort of a short list of elements which we can use to generate this nuclear magnetic resonance effect that we're going to be speaking about and the requirements in brief are that whatever the element is that we're looking at cannot have both right an equal number of protons and an equal number of neutrons in its nucleus and one way to think about this qualitatively is that if your nucleus is totally balanced in that way it simply will not undergo this nuclear magnetic resonance effect so there as a result there are specific isotopes of a of a variety of elements that we can use to generate this nuclear magnet magnetic resonance effect and that is because they have what we call nonzero spin okay and this nonzero spin is a quantum mechanical feature right of the nucleus uh the details of which I'm I'm not going to go into right now we can talk about it it's it may be in the notes it's in the book uh but basically in choosing a nucleus it has to be something that does not have this complete balance between protons and neutrons in the in the nucleus now once we have whatever it is that we're going to look at so let's say say it's a hydrogen nucleus right and the reason why hydrogen protons are so relevant to clinical Imaging is two things anyone I have an idea why water Ms so I mean the main reason is that if you look at the human body there's a tremendous amount of water so the biologic abundance of this isotope is very large so it's really most relevant to looking at tissue simply because of its abundance uh the other is that it has a relatively strong NMR effect so of all of these Isotopes they can all undergo this nuclear magnetic resonance process which we're going to speak about shortly but the strength of the response and therefore the strength of the signal which they generate is not the same so carbon 13 for example generates a much weaker NMR response and the signal which you detect is much weaker than that of proton okay so our choice in looking at human tissue is both because of the abundance the fact that water and protons related to water are very prevalent as well as as the strength of the NMR response so what is this nuclear magneet itic resonance phenomenon that we're talking about so first if we look at the nucleus of a proton we peel it off of that water molecule it's a pretty simple structure right it's a single positively charged particle just a proton now it turns out that the presence of charge in this nucleus and yes there is an electron that is orbiting around that nucleus which for now we're going to ignore and we can talk a little bit later about what the effects of those electrons are how we deal with them at this point suffice it to say that the Salient difference between the proton and the electron is that the proton has dramatically higher mass than that electron and we we'll come back to that later so the nuclear proton and its positive charge leads to a phenomenon called nuclear magnetism so in NMR nuclear magnetic resonance the NM stands for nuclear magnetism nuclear magnetism tells us that this nucleus with its charge will behave as a little magnet a little bar magnet which will have mag magnetic field lines just like a little bar magnet that you might place on your refrigerator okay now the reason that this occurs that this charge particle ends up conferring magnetism on the nucleus is a quantum mechanical phenomenon which is not directly analogous to classical electroma magnetism so just to be clear about something we tend to think well we have this proton and it's spinning around right and so if this proton is spinning around right and we take our fingers and we put them in the direction of that spin that we talk about and we can look at the orientation of our thumb the right hand rule everyone had some Physics at some point so the right hand rule tells us that the orientation represented by a vector of the magnetic field generated by this proton looks like that well it is true that the magnetic field generated by this proton looks like that it has this orientation okay the thing that's not true is it's not because there's actually physical movement or spin of that charge that's simply not true so something that you'll see in you know in many books and people like to explain things this way is that it's the actual movement of charge in the nucleus is that by Faraday's law of induction classical electromagnetism generates this magnetic field just to be clear that's not the case uh we could say it doesn't really matter because the bottom line is that it does behave exactly like that type of a magnetic field so nuclear magnetism tells us that if we have a single proton that we can describe its magnetic field by some simple linear vector Vector where the length of the vector is the magnitude of the magnetic field generated by this proton and the orientation is the orientation of that magnetic field okay any questions about this all right so this is the n and the m in NMR now if we look at this a little bit further so we have our proton and its nuclear magnetism the magnetic field generated by this proton looks like this Vector well do we have any way of predicting what the orientation of that magnetic field will be now the first thing we should say is that it's kind of ridiculous to talk about observing the behavior of an individual hydrogen nucleus that's not something that we're really able to do but if on the other hand we have let's say a population of hydrogen nuclei each one of these is identical they contain a single proton and they generate a single unit of nuclear magnetism which can be described by some vector of the length that I've indicated here right the length of those vectors should all be the same so if I have a sample of let's say five nuclei is this what it's going to look like no okay why not so why should the orientations all be exactly the same right and in fact right the reality is that the orientations will not be exactly the same we undo this if we actually look at what's going on we will find that the orientations will be essentially right more or less random now if they were all the same if we had a bunch of units of nuclear magnetism like this that were all the same magnitude and in the same orientation we could actually take these and compute the vector sum and what would it look like right the vector sum would look something like this would have the same orientation and five times the magnitude so if they all had the same orientation our sample of protons would give us some measurable amount of magnetism and with a large enough sample there might actually be a macroscopically measurable amount of protons that would be the equivalent of saying that if I had a glass of water on the table and I gave you a gaus meter right which measures magnetic field strength and you brought it over to that of water which contains zillions of hydrogen protons if they all had the same orientation you should be able to measure some detectable magnetism right which is not the case right this does not pertain the reality is that if we take the vector sum of a large sample it is going to be right zero okay so the sort of basic resting state of a sample of protons whether it's in the liver or it's urine or it's water or whatever it happens to be is going to be a bunch of randomly oriented units of nuclear magnetism that all sum to zero okay now that being said everyone here probably knows if I say I take two little bar magnets off of my refrigerator the things that are holding someone's you know first grade Pro project up to look at so I have these two bar magnets and they have an orientation right they each have a north and a South Pole so I take these two and I bring them close together what's going to happen as I bring those two magnets close together okay if we say again I can't hear depends how you it dep exactly it depends how you hold them so what Smith is getting at is if that if I bring them together like this with North Poles and South poles together I'm going to feel a tremendous amount of resistance as I try and put those things together on the other hand if I don't hold them under too much control chances are they will rotate around and what will happen is I will as I bring this close together I will have the opposite poles attached to each other so my point here is that or on the other hand if I had one magnet on the table and I brought another one in on top of it they would rotate to align with each other so my point is that in the presence of some externally applied magnetic field we know in and this is in classical electromagnetism right macroscopic things that we could observe on the tabletop here that there is going to be an orientation of One MA magnetic field with the other the same thing is true when you deal with this sample of spins okay so if we change things here and we take our sample of protons I left out the little circles I'm just showing you the vectors that represent the amount of nuclear magnetism and its orientation and so if I take all of these things in the absence of any externally applied magnetic field what do I get it sums to zero right this sums to but on the other hand if I apply to this area some magnetic field that let's say looks like this something that is stronger than all of these and has its own orientation that is going to give me some net magnetization that looks like what anyone right it's going to look like this okay at least this is what we would predict based on our little experiment with the bar magnets now it turns out that the reality of the behavior of these units of nuclear magnetism in the presence of some externally applied magnetic field is that there are two potential orientations that each of the these units of nuclear magnetism may take they may Orient themselves right as we would predict in the same orientation with the opposite polarity or they may Orient themselves actually parallel and pointing in the same direction and it turns out that if we look at a large population of these spins that what we end up with is a relative XS in the direction opposite to the applied magnetic field so if we would take a look at what each of these individuals look like we would have several that point in this direction and a slight excess that I have it backwards no slight excess that point in the opposite direction so that the sum right is something that is what is sometimes called anti-parallel to the externally applied magnetic field so if we take a really large sample let's say A you know a phantom a bucket of distilled water when we place it into an externally applied magnetic field all of the billions of protons in that sample are going to assume this Orient ation so that ultimately we will get some net detectable magnetization that is in the opposite orientation to the externally applied magnetic field and the amount at a field strength of about 1.5 Tesla is that this excess right is going to be approximately six in 10,000 units okay so the amount of detectable magnetization from this sample in the externally applied magnetic field is extremely small and ultimately this is what it is that we are going to be detecting as the Mr signal so it's a tiny it's a tiny tiny amount of signal and to make it detectable at all we need to have the presence of this externally applied magnetic field so to just review this very quickly is if we have our sample right of protons and we place it right inside of some right externally applied magnetic field which I'll call B the result is going to be some right detectable amount of magnetization that has an or orientation opposite to the externally applied static magnetic field
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_24_Entanglement_QComputing_EPR_and_Bells_Theorem.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So today I want to finish up a couple of loose ends in the class. The first is by the end of the day, we'll talk about EPR that we've picked up on the last couple of lectures. We're going to close up Bell's theorem from the very beginning. So we talked about Bell's Inequality, and we'll re-prove Bell's Inequality, and then we'll see what quantum mechanics has to say about it. But first, I want to show you something neat. So a couple of times, we've talked about entanglement, and what it can do. And I want to spend the first good chunk of the class talking about what entanglement can do, and also what it can't. So before we get started, I want to remind you of a couple things from the last lecture. So the first is particles can have spin. Half-integer intrisic angular momentum that has nothing to do with being a rigid object that rotates, but is just an intrinsic property of angular momentum. And this spin is observable-- it's represented by an operator. In fact, a vector of operators, Sx, Sy, and Sz, that satisfy the same [? computational ?] relations as the orbital angular momentum. Now last time, one thing we showed is that a state which is up-- or, for example, down-- at an angle theta, so if I measure the spin along this axis, I will always find it's other up or down, spin-1/2, either plus h-bar upon 2 or minus h-bar upon 2. But I can express this state in terms of the states with spin, which are up or down along the z-axis. And we gave an expression-- and I challenge you to follow the logic given to derive this expression. An expression for the state, which is, in this case, down along the angle theta-- it went down when measuring spin along the angle theta-- in terms of up and down of spin along the z-axis, or long angle 0, with theta here being the declination from the vertical. OK. Yep. AUDIENCE: [INAUDIBLE] angle. PROFESSOR: Yeah. There's a general thing for this, that says for a general angle. And it's actually in the notes, which I guess weren't posted from last time. But it's in the notes. And it's also easy to derive right? So what's the point here? The point is you have a spin vector, and it's a vector, Sx, Sy, Sz. And you can compute the operator representing spin along a particular direction by taking a unit vector in that direction and taking the dot product. So that gives you a particular linear combination of Sx, Sy, and Sz. But, from last time, you know that's just some stupid matrix. And you could express a matrix in any basis you want, for example, the basis of up and down in the z-direction, which is what we used last time. Then you can find the eigenvectors of that, and that's how you find these states. So there's a very simple generalization. It just involves an extra angle theta, which is the angle around the equator. OK? AUDIENCE: [INAUDIBLE]. PROFESSOR: Sorry, phi, extra angle phi. So we then did the following thing. I talked about the Stern-Gerlach experiment and, here in an abstract way from the Stern Gerlach experiment, the core of the nugget. And the core of it is this. Suppose I have some spin, and I put it in a particular state. Let's say, for simplicity, up in the z-direction plus down in the z-direction with equal amplitudes. So they're equally likely to be measured. And then, these guys, these two states, should be degenerate in energy, because the system is rotationally invariant. It's just a spin, sitting there. There's rotational invariance, so we know the energy can't depend on the z-component of the angular momentum. But we can break that symmetry and break the degeneracy by turning on a magnetic field. So for a Stern-Gerlach experiment, we turn on a magnetic field that had a gradient. But I just want to look at a constant magnetic field for the moment and see what it does. So we turn a constant magnetic field. That's a contribution of the energy, which is minus the magnetic field dot the magnetic moment, which is a constant times the spin. But if the magnetic field is in the z-direction, then this is just Bz Sz. So we then found yesterday that this led to a splitting-- that the up state had energy plus h-bar omega, and the down state had energy minus h-bar omega, where omega is given by mu naught Bz upon two. Yesterday, we had an additional term which involved the z dependence, the beta z term, but here I'm just looking at a constant magnetic field. But if we know the energies are, these are still the eigenstates of the energy operator. If we know the energies are, we know how the system evolves in time. Here is the initial state. These are the eigenstates. The energies are these. So we can let the system evolve in the time, and we find that all we do, is we evolve the system with phases. This is as usual in quantum evolution. If you have an expansion of the state in energy eigenstates, time evolution is just phases for each term. But note that there's a simple thing. This omega, the thing that determines the rate of evolution of the phase, is controlled by magnetic moment, which is something intrinsic to spin, and the magnetic field, which is something I can dial in an experiment. I can really tune it. Meanwhile, we decide how long to leave the magnetic field on. I can turn the magnetic field on for a while, and I could turn it off while later. So suppose I leave the field on for time t, such that omega t is equal to pi upon 2. OK. So you turn on the magnetic field for a time t, such that omega t is pi upon 2, or t is equal to pi upon 2 omega. What is the state then, afterwards? The state subsequently psi after, is equal to this guy-- omega t is equal to pi upon 2. But what's e to the i pi upon 2? i. Right? And e to the minus i pi over 2, is minus i. So this is an i and a minus i. I can pull out the i so that the phi after is i, a phase, upon root 2 times up z, minus down z. Yeah. That's cool. And this is a state that we saw last time. It's actually up to normalization, which is an i, up or down x in the x-direction. So what this lets us do is something really quite cool. What this tells us is that all these calculations are just the Stern-Gerlach calculation, but even easier, because we don't have a gradient. If we have the spin in some superposition, which, incidentally, was initially up and down in the z-direction, this is equal to up in the x-direction. So in x, we have it up. We turn on a magnetic field in z, and what happens is the spin precesses around the z-axis. And we get down. We get precession. Here's the important point. The important point for everything that's going to follow is this. If I have some spin state, and I want to take it to some other spin state, how do I do so? Well, I can build a machine that does it, by turning on the appropriate magnetic field and having the magnetic field precess the spin. OK. Here I found exactly what magnetic field I need to turn on, with what amplitude, and for how long, such that I think the state up x to down x, with a known phase i. Everyone cool with that? So any unit-- here's the thing I want to convince you of-- any unitary operation that takes you from from one spinner to another spinner-- up z, down y, linear combination of up and down z, some particular linear combination of up and down x. Any such pair of vectors can be related by unitary transformation. And any such unitary transformation can be built by some series of magnetic fields. That cool? You can just can prove this yourself quite quickly, but let me just state it. So now this gives us the ability to do the following. One, we have spins, one, spins that can be put in a system that can be put in states like up z and down z. And two, the ability to rotate states. The ability to evolve states from state to state. From spin state 1 to psi 2 by turning on some magnetic fields, so by suitably choosing our magnetic fields. Everyone cool with that? So here's a question I want to ask, what do we do with this? Imagine we really had this equipment in front of us. What power, what awesomeness could we realize? And this is the entry point to quantum computing. The answer to what can you do with this cool machinery is you build a quantum computer. So let's talk about what that means. So quick question, do quantum computers exist? Yeah. Are they very big? No, OK. The biggest quantum computer that's been built my knowledge is one that has factored the number fifteen Or they might have done 21 at this point, I'm not exactly sure. 21? Done? Yeah, OK. 21's done. So that's the upper end. Well, they're large, physically. It's true. They take up a room. Or at least a large table top, an optics table. But they're not very big in a useful sense. OK. And there's actually reasonable people, not just reasonable, intelligent people-- and reasonable. That's a nontrivial-- Ok. The people who live at the intersection of intelligent and reasonable, which, admittedly as my colleague is pointing out, is not a large overlap. There are people in that overlap who suspect that, for some fundamental reason, it's impossible to build a useful quantum computer, in some deep, deep sense. I don't know whether that's true or not. But we're about to engage in a discussion of what happens when you build quantum computers for n bits were n gets large, like millions and billions as you need for codes or for studying images. And, of course, this is a little strange, because such a compter doesn't exist, which leads to my colleague and friend Scott Aaronson's lovely comment that his job, as someone who studies quantum computers, is to prove things that can't be done with computers that don't exist. Which is what we're about to do, so. So let's talk about quantum computing. So what do I mean by quantum computing? Here's the basic idea. Let's talk about classical computing. Classical computing says, I have these things called bits, which are binary objects. They're either 0, or they're 1. And we realize them in a very particular way. We realize them with a little magnet, which points north or south. Now, it's not always exactly a magnet. It isn't like an old-style hard disk in your computer. It's something a little more fancy than just a magnet. But it's still basically a magnet, and you have some north, south. And whether it's pointing north, or whether it's pointing south down here, which I'll call 0. Or whether it's pointing north down here, which I'll call a 1, determines whether you call this 0 or 1. And classical computation is your data are a set of numbers-- what's pointing down. And the process of doing a classical computation is, build a machine which is governed by classical mechanics, OK, that takes this initial configuration and replaces it with a new one, f of 0, 1. Which is also some number, so maybe it's 0, 0. I don't know-- it's whatever f is. And what you want to do is you want to perform some calculation, which is some function, a known function, of your input variables-- a function, in this case, of two bits, which produces the output that you want, and you build a machine that does that. OK. So let me give you an example. I want to build a classical computer that takes a string of binary integers-- 0, 0, 1, 0, 0. And performs a logical [? NOT ?] on each bit independently. So I need to build that computer out of objects available to me. OK. And I need to use nothing other than the laws of classical mechanics. That has to suffice to describe the system. So let me give an example of such a computer. Why, Allen, why don't you do this calculation? OK. So, that's the input, and I am now the classical computer 0, 1, 1. Right? Now that's not actually what you do in your cell phone or on your laptop. That uses transistors, but it's the same basic idea. You build a set of machines. You build a set of objects, you know, magnetic fields, electric currents that effect this calculation. And so there's some relationship between what electric fields you turn on and what currents you induce to flow and the calculation you want to perform-- f of the input. OK, this is a basic idea of a classical computer. Oh, and one last thing. Suppose we have n bits. Suppose we have n classical bits. Then we have 0, 1, the end. And how many numbers do you have to specify to specify the configuration of n bits? This is sort of [INAUDIBLE]. No, you just need to specify the number of each bit, right? So we need n bits. So n binary numbers. Everyone cool with that? I specify what each register is, and I'm done. Imagine this is quantum mechanical. Instead of having bits, let's take quantum mechanical bits. By which I'm going to mean a system that has two possible observable states, 0 and 1. OK. So these are the possible observable states. And to specify a general configuration, I see that psi is equal to some linear combination, alpha 0 plus beta 1. Now, if I measure, we know that we'll find either 0 or 1. If we measure the spin in the z-direction, we will measure either up or down. However, at a given moment in time, the system need not be in a state corresponding to a definite value. It could be in a general superposition. Agreed? So now here's the question. How do you how many numbers does it take to specify the state of a single quantum bit? Two complex numbers. Right? It takes two complex numbers to specify the state of a bit. And if we have n qubits, and there are n of these guys, well then, how many numbers does it take? Well, I have to specify the state of every possible superposition for every possible configuration the system. So, for example, it could be all 0, 0, 0, dot dot dot 0, plus beta. The first one is 1, and the rest are 0, dot dot dot plus. And how it terms are there? There are 2 to the n terms. Right. So how many numbers do I need to specify this state of n qubits? I need 2 to the n complex numbers. Yeah? Everyone see that? So this immediately leads to a really frustrating reality we have to face. Suppose you want to simulate, using a classical computer, such as sits on my desktop, I wanted to simulate a quantum mechanical system of n bits, or n spin-1/2 states, evolving according to some energy function-- evolving according to some Hamiltonian [INAUDIBLE]. How many variables, and how much memory will it take? 2 to the n, right? If I've got n bits that I want to describe, it's going to take 2 to the n complex numbers. So that's how many variables I have. So if I have 10 bits, literally, 10 little spin-1/2 objects, to accurately specify the quantum configuration system, I need 2 to the 10 complex numbers. And that's for 10 spins. How many things make this piece of chalk? Right? So could you ever, in an honest way, simulate on a classical computer, like sits on my desktop, a quantum mechanical system the size of this chalk. Absolutely not. It's completely and utterly intractable. You have to come up with some sort of approximation scheme. So simulating quantum systems directly, by just taking the spins and representing them with a wave function, is wildly inefficient. Incredibly difficult. Interestingly, the converse is almost certainly not so difficult. It's almost certainly possible for a quantum system to simulate classical evolution quite easily. And how do you know that? Yeah, here we are. Exactly, right? Nature apparently has no trouble simulating classical mechanics with underlying quantum mechanics. Quantum mechanics is the operating system, and classical mechanics is the simulation it's running. Yeah, in a very useful sense, a very real sense. So, at this point, Feynman, I think, was the first person who really pointed this out. I don't know the details of the history, but the lore is that this was really from his observation. Look, if that's true, this opens up a real possibility for computation. If things like spins-- hard problems like calculating how spins evolve or even the motion of chalk in the room-- can be done pretty efficiently by nature, a comput if we can build a computer that used quantum mechanical bits, whose variables were quantum mechanical, and involved all the superpositions and interference effects of quantum mechanics, perhaps we could build computers that run exponentially faster and with exponentially less memory and less resources than classical computer would. Because apparently it works, right? We're here, as was previously said. So this is the question of quantum computing. Can you use quantum mechanics? Can use, in particular, the quantum evolution of the system to perform calculations that you care about, rather than classical computation? And if you do so, do you gain anything? Do you get any sort of speed ups? Is there an enhancement? Is anything better? So, here's the basic gig. The basic gig is that we're going to take our system, considering the following kinds of systems. Our systems are going to be n qubits or n spins. But because I want to emphasize that the substrate doesn't matter--- the underlying material doesn't matter-- I'm going to refer to them purely, and this is totally typical in the field, as qubits, rather than spin systems. And the reason is it might work use little isolated spinning particles with intrinsic spin. Or that might turn out to be technologically infeasible. It shouldn't matter at the end of the day, in the same way that if I ask you, how does you computer work? Or if I ask you to write a code, you write some code in C or Python or whatever-- what the hip kids are using these days. So you write some little Scheme code-- I still love Scheme. You write some little Scheme code, and it performs some calculation as to defined in just logic, in lambda calculus, in abstract logic. Do you need to know the substrate? So you need to know whether your transistors are built out of silicon or germanium? Or indeed, whether it's running on vacuum tubes? No, that's the whole point of abstracting away computation. The substrate doesn't matter. So we can use spin-1/2. And that's going to be useful for us at various moments. But I want to emphasize the important thing is the actual logic of the process, not the underlying substrate. Here's what I need. My systems are going to be n copies, or n qubits, where each bit is specified by either 1, represented by up, or 0, represented by down. I'll use these interchangeably. So that the full system, psi, is a superposition of sum over all possible-- sorry, I didn't mean to write that. So this is my system. It's going to evolve according to some energy operation. And so my input is going to be some wave function, psi n, for these n qubits. Computation is going to be, evolve with some energy operator-- which I've chosen-- to implement an algorithm, in the same way that the precise series of magnetic fields that we turn on in a classical computer, or currents that we allow to flow in a classical computer, implement the algorithm that we want to effect. We evolve with some energy operator, implementing our linear algorithm, and we get some output wave function, psi-- again-- for our n bits, n quantum bits, out. So this is the basic gig with the quantum computation. We're just replacing strings of binary numbers and functions-- evolution according to classical mechanics in Maxwell-- to other strings of numbers with superpositions of states. Superpositions of strings, as it were, evolving according to the Schrodinger equation into again general superposition. So how does this differ? What are the key differences from a classical computer? So the key things are first off, the input and output are not definite values. They could, in general, be superpositions. They do not have to correspond to a definite state in the 1, 0 basis. These, in general, are superpositions of 0, 0, 0, 1, 0, 0, dot dot dot. So the input and output are superpositions of the values we'll measure. OK. Second, when we do measure, at the end of the day, the output, we get interference from the various different terms in the superposition. The different terms in the superposition will lead to interference effects. And so our output will be sensitive to that interference between the different terms in the superposition. It may be that we're in some pure state. But it may be, more generally, we'll be in some superposition of the states we want to measure. So we're going to get interference. So naively, this is a disaster because this means we get probabilistic outcomes. That sounds bad. And so, this leads to the key move in quantum computation. Given that we're going to have interference, and given that those interference effects are going to affect the probabilities of various outcomes, what we're going to want is to tune, or orchestrate, the interference to get a definite outcome. We want to get the correct answer out, rather than just probably the correct answer out. Now there's a slight variation of this. It's not obvious at the beginning, how to do that. It's not even clear that you can. So I'm going to show you that you can. I'm going to show you an explicit algorithm that realizes this. But, OK, that's obviously going to be tricky. Here's something else you can do. You can also focus on special problems. Focus on checkable problems. And what I mean by this is imagine we have some algorithm that gives us an output, and that output is probably the right answer. But we're not sure. There's some quantum mechanical probability that this is the correct answer to our computation-- there's some probability that it's not. So, if the calculation was difficult, but it's easy to check whether you have the right answer then that's great. Imagine it's 10% that you get the right answer, but it's trivial to check. So, for example, factoring numbers, right? If you multiply the numbers, that's easy. You check to see whether you got the right thing. So, for example, if you imagine that I build a computer that is supposed to factor numbers-- I almost said factor prime numbers. That would be a boring computer. So imagine that we build a machine that factors numbers. Right? And so imagine its output is probabilistic. So, you say 15, and I say 3 times 5. You say 15, I say 5 times 3. You say 15, I say 7 tiimes 2. At that point, you're like, well, which one is right? Well, you can check by multiplying. So if you have a problem, which is easy to check, but hard to do, then probablistic is already OK. That's just as true classically as it is quantum mechanically. Those are our basic moves. And so, the key thing for us is that when we have n quantum bits, we have these interference effects. And in particular, as we started talking about last time, we get entanglement effects. What we're going to discover is that the key to making a good quantum algorithm is going to go to attune the entanglement appropriately. So going to have to define that for you in just a minute. Yeah. AUDIENCE: [INAUDIBLE]. PROFESSOR: Sorry? AUDIENCE: [INAUDIBLE] process. PROFESSOR: Well, it may or may not be. So the check, for example, , imagine the process I just gave you. So the question is, what do you do if the checking process is probabilistic? But you can use a classical computer to check, if you have an easy checking algorithm-- for example, multiplying numbers together. AUDIENCE: [INAUDIBLE]. PROFESSOR: No. Good. And so the question then becomes, doesn't that defeat the point of using a quantum computer, if you're using a classical computer to check? And so here's the thing. If I ask you to factor a gigantic number, what do you have to do? Well, you have to take all numbers up to its square root, and you have to multiply them together, and that takes forever. Right? But if I tell you, do these two numbers multiply to give you the third number? That's easy, right? So I use a quantum computer for the hard part, and the classical computer for the forward part. For the checking. And that's a special class of problems which are checkable. These all have fancy names in information theory. I'm not going to use the fancy names. AUDIENCE: [INAUDIBLE] the wave function? PROFESSOR: Yeah, exactly. I mean, suppose I give you some output, and let's say the output is 0, 0 plus 1, 1. Right? What are you going to get, if you measure the first bit? Well, you're either going to get 0 or 1, with one probability or another, right? When we compute the probabilities in general, when we have many terms in our superposition, we're going to get interference effects from different terms in the superposition. And those interference terms will tune the probability, so they're not just the naive probability of that one thing. But you get these interference terms. Norm squared of one, norm squared of two, and then the real part twice real part of the cross term. And that twice the real part of the cross term didn't exist classically. Quantum mechanically, it's important, and it can change the final probability. That's what I mean by the interference effects. OK, so let's make all this much more explicit. So far I've just given you philosophy, and you should be deeply suspicious of philosophy in a physics classroom. So, again, to be explicit, a quantum bit, or qubit, is equal to a two state system, and, again, the substrate doesn't matter. I could be talking about spin-1/2 systems, or I could be talking about ropes on a cylinder with winding mod 2. I could be talking about all sorts of things. But it's some system with two quantum states, 0 and 1. I want to emphasize this is not going to be on the final. So this is just for your moral well being. So a quantum bit is a two state system. We have these two states, which I'll call 0 and 1, and a general state, the general wave function, psi, is equal to alpha 0 plus beta 1. OK. And what this means is the probability that I measure 0 is equal to norm alpha squared, et cetera. OK. Now, again many systems can be used-- many different substrates. This is what I'm going to mean by a qubit. So that's one qubit. The much more interesting system is two qubits. So let's study two qubits. So in the case of two qubits, what's a general state? Well a general state is going to be, well, the first particle could be 0 and the second particle could also be 0. Or we could have the first particle 0 and the second particle 1, with some coefficient, plus beta, plus gamma 1, 0, plus delta 0, 1. OK. So this is just a general superposition. Now, you might worry at this point look, are these identical, or these non-identical spins? But here's the thing. I've got a spin here-- it's in a box-- and I've got another spin here-- it's in a box-- I've got another qubit over here. It's in a box. So they're distinguishable because they're in different places in my laboratory. So these are distinguishable particles. The particle in this box is either up or down. OK. So these are distinguishable, and I don't have to worry about symmetrizatoin, Bosonic or Fermionc statistics, or any of that. They're distinguishable. And we need normalization, so norm alpha squared plus dot dot dot is equal to 1. And what does this mean? What this means is that, for example, the probability of the first part, which I'll call a is equal to 0, is equal to-- well, we sum up all the possible ways we could do that. We have norm alpha squared plus norm beta squared. Whoops-- this was 1, 1. Thank you. And if we're more specific, the probability of the first qubit is 0, and the second qubit is 1-- is equal to 1. I need 0, 1. That's this guy, norm beta squared. OK. What I mean by two qubits. But this immediately leads to a funny thing. There are two kinds of states that a pair of qubits could be in, a very special set of states, which are called separable states. And these are special. And these states correspond to the full system being in the state where the first particles is in one state, and the second particle is in the second state. OK. So, for example, this could be eg the state 1 over root 2-- sorry-- 0 plus 1 for the first particle, and the second particle to be in the state 1 over root 2, 0, minus 1. [INAUDIBLE] Let's just call these general coefficients. a plus b and times c plus d. So what does this equal to? Well, this is of the form-- there's going to be a term that's 0, 0, and that's ac. Plus a term that's 0, 1, and that's ad with a minus. Minus ad. Plus a term that's 1, 0. That's bc. And minus a term that's 1, 1, and that's bd. OK. This is clearly not generic, because it implies relationships amongst the alpha, beta, delta, gamma, apart from just normalizability. Everyone see that? So it's a pretty non-trivial thing that you can write a thing as, the first particle is in one state, and the second particles in the second state. And here's physically what that means. Physically, what that means is that, if you're in a state like this, and I ask you, what's the probability that I measure the first particle to be 0, do I need to know anything about the state of b? No. If I want to know what probability of the first particle is 0, I just take norm a squared. Because I'm going to get plus norm c squared plus norm d squared. So that's fine. So, imagine the probability of the first particle being up or down is independent of any information about the second particle, right? There is another thing that's important. Suppose I tell you, ah ha I've measured, and the first particle is in the state 0. OK. Cool? What is the state subsequent to that measurement? So if we measure a is equal to 0, what is the state subsequent? Psi is equal to-- well, that's 0, and we know we've lost this, so this particular subsystem, this particular qubit has been collapsed to the state 0, so we have 0 times c, 0, minus d, 1 for the second particle. Have we learned anything about the state of the second particle? Have we learned anything about the state of the second particle? Absolutely not. Right? Beforehand, what was the probability that the second article is 0? Norm c squared. And the second particle 1? Norm d squared. Now what's the problem? Same. Norm c squared, norm d squared. So when you have a seperable system, measuring one qubit tells you nothing about the other qubit. Cool? On the other hand, consider a state which is not separable. So the generic states are not separable. And let me give you an example of the state which is not seperable. Which one do I want to pick? Yeah, what the hell. Psi is 1 over root 2, 0, 0, plus 1, 1. Can this be written as the product of the first particle in one state and the second particle in the other? No, because they were have to be cross terms, which don't exist here. Right, just compare this to that form. So, we have that the coefficient of 0, 0 is ac. So a and c must both be non-zero, and the coefficient of 1, 1 is bd. So the coefficient bd must be-- both of those must be non-zero. So ac and bd are all non-zero. That means these terms have to exist in order for it to be separable. Yeah? Because a and b [? can't ?] vanish, and b and c [? can't ?] vanish And these orthogonal states are independent. So this is not a separable state. We call these states entangled. And it's funny, because [INAUDIBLE] I think there's an e in there. AUDIENCE: [INAUDIBLE] PROFESSOR: OK. That's better. Look, I'm not a professor of spelling. It's a little bit funny to give these a special name and call them entangled, because the generic state is entangled. It's sort of like calling mice, mice, and calling all the other mammals non-mice. Oh look, well, that was bad example. Oh look at mice right across from [INAUDIBLE]. AUDIENCE: [LAUGHS]. PROFESSOR: Hearkening back to an earlier lecture. So, in any case, we give these a special name, and the reason we give them a special name is that these seperable states do you more or less what you'd expect classically. There are no strange correlations between the state of one and the state of the other. They're independent quantities. But when you have a generic state, something funny happens. They're entangled. And here's the funny thing that happens. Suppose given the state, what's the probability that I measure the first particle be up, or to be 0? 1/2. What's the probability that I measure the first particle to be 1? 1/2. So, before doing any measurements, the first particle is equally likely to be 0 or 1. So suppose I measure the second particle to be up. OK. Then the probability that I measure the first particle is 0 is equal to 0, and the probability that the first particle is 1 is equal to 1, because I've this collapse onto this term in the wave pack, the wave function. So measuring the second qubit alters the probability distribution for the first qubit. These guys aren't independent. They are correllated, in a way that you studied on the problem set. They are correlated. They're in a correlated state. We call this correlation entanglement. And here's the thing that's most spooky about this, and we'll come back to this in a few minutes, but I didn't tell you anything about the geometry of the set up. But, in fact, when I was thinking of that measurement, I built the two little qubits in my lab, and I took one, and I sent it to France-- because France, you know. And I took the other one, and I said it to the planet Zorg. And so off on the planet is some poor experimentalist huddling in the cold. And our French experimentalist makes a measurement, altering the probability distribution instantaneously on Zorg. That should make you a little worried. That sounds a little crazy. We'll come back to why that is or isn't crazy, and the EPR analysis, which puts flesh on it, sounds crazy, in a little bit. But for the moment, let me just emphasize that, while the generic state is entangled, the generic state is also different than what your classical intuition would expect. There are correlations between the particles. So this is something that happens with quantum mechanical particles that doesn't happen with classical particles. And that means it's something that can be used in a quantum computer that can't be used in a classical computer. So let's see what using entanglement gives us for computation. So let's go come back to the quantum computing problem. And how do we compute with qubits? So how to compute. OK. So again, as usual, the way we take our input to our output is we build a machine that implements an algorithm of our choice by arranging the physical evolution of the system under time. So that means picking an electric field. Sorry, an energy operator-- an electric field, good lord. So it means picking an energy operator. OK. So computing is Schroedinger evolution with our chosen, our attuned, energy operator. So, for example, I want to build for you now a couple of basic circuit elements that you might want to use in a quantum computer. So example one. The first example is, I want to build a NOT gate And what NOT means is that it takes the state 0 and gives me 1, and it takes the state 1, and it gives me 0. OK. This is what NOT does. So how do I build a NOT gate? Well, I can realize this in a nice way. Doot do do. Do I want to say it that way? Yeah, good. So if I realize 0 as 1, 0, and 1 as 0, 1 from last lecture and 0, 1 to 1, 0, so we need an operator. We need time evolution to effect multiplication by an operator. That takes this vector to this vector, and this vector to this vector. We know what that unitary operation is. That unitary operation, which I'll call NOT, must equal to 0, 1, 1, 0. And this operation takes this guy to this guy, and it takes this guy to this guy. Yeah? But I can write this as-- I like this-- shoot, there's a phase-- minus i, e to the i pi over 2, 0, 1, 1, 0. I mean, you can't stop me, right? So this expanded out, as we've done before, expanding this out gives me, with the exponential, 1 plus the thing and all the other terms. This becomes minus i times cosine of pi over 2 times the identity, the 2 by 2 identity, up plus i sine of pi over 2 times 0, 1, 1, 0. But what's cosine of pi upon 2? Yeah, 0. Come on. Is everyone that tired? So cosine pi over 2 is 0. Up. Sine of pi upon 2 is 1, because i times minus i, that's 1 times 0, 1, 1, 0. Pretty solid. But what is this? Well this is the form Schroedinger evolution with a magnetic field, and this is just the unitary transformation unitary transformation for a magnetic field. Well, OK. We can say this. This is the x polymatrix from last time. So this is like Sx, unitary transformation for a magnetic field in the x-direction for some time t. For time t times the frequency, omega, which is given by mu B upon 2 is equal to pi upon 2. OK. Just like before, but for a slightly different one. A slightly different magnetic field. So my point here is that we can pick a magnetic field that does this. We turn a magnetic field with a known amplitude with a known amount of time, details here don't matter so much. The point is we can do it. We turn a magnetic field in the x-direction, and it takes 0 to 1 and 1 to 0. Everyone cool with that? So here is a substrate, an actual physical system that effects this particular evolution. I can build a NOT. The crucial thing is that I can build a NOT gate. And I'll represent that not with some unitary transformation U sub NOT. So that's a useful one, but that's not the most useful gate because, if you only ever impose logical NOTs, you just get everyone angry. But you don't actually get anything done. Second example-- let that be a lesson to you, Congress. So, the second example, if we turn on the magnetic field in the y-direction for a particular time t, what we find is that 0 goes to 1 over root 2, 0 plus 1, and 1 goes to 1 over upon root 2 0 minus 1. And this should be familiar. This is the up x state, and this is the down x state. Just as we talked before. So we turn on some B field, and we get this. So this operation has a name because it's going to turn out to be very useful for us. It's taking a system that's definitely in the state 0, for sure, right? And it put us in a superposition of 0, 1. It's a definite superposition, so it's not like we don't know what happened. But it's a superposition, and you've lost certainty that you'll measure up in the z-direction. You've gained certainty that you measure up in the x-direction. But if we do all our measurements in z, we just taking ourselves from definite to superposition. Cool? So that's useful because we know that's something a quantum computer can do, that a classical computer can't do. Something a quantum computer can take advantage of that classical computer can't take advantage of is this process of putting things into superpositions. So here we've got an operation that puts things in superpositions. And I'll call this Hadamard. I don't know the history of why that's called Hadamard, presumably there's some guy with a last name of Hadamard. Anyway, the U Hadamard does this. And as a matrix, it's represented as 1 over root 2, times 1, 1, 1, minus 1. And there's a the last one, which is going to be useful for me, another one is called C C-NOT. Controlled-NOT. Controlled-NOT does a really cool thing. It takes 0, 0, and 0, 1 and 1, 0, and 1, 1. What does is it says, I'm going to apply a NOT to the second qubit if, and only if, the first qubit is 1. So this takes me to-- 0, 0 goes to-- well, do I perform an NOT on this bit? No, so 0, 0. Do we perform a NOT on this bit? No, 0, 1. Now I do perform a NOT on 0, so I get 1, 1. And 1, 1-- I perform a NOT on this bit, which gives me 1, 0. So this is called controlled-NOT. It's a very useful thing. I'm going to represent this in the following way. I should represent all of these. So this NOT gate, first, I take some initial bit, and it's in some state. And then I impose U-NOT, and it gives me an out state. Similarly, with the Hadamard, I take an initial state n, and I've apply U Hadamard. And I get u out. And for controlled-NOT, I now have two qubits. And I take the two qubits, and I do a controlled-NOT, which is represented in this form. Which says, do a NOT on this guy, controlled by this first bit. And so this acts as U C-NOT. OK. And the key thing here, is that while it's always possible to find a physical real representation of some particular unitary transformation, at the end of the day, all we're going need is some truth table. At the end of the day, all we need is the logic that's being effected. And so, the details of the substrate can be abstracted away. So what we do with this? First, so what can we do? Before we actually talk about what we can do with it, let's briefly talk about what you can't do with it. So what are things you can't do with these sorts of operations? What can't you do? And to me this is among the more surprising things. Remember that what we're doing here is going to be evolving a system for a Schroedinger evolution, and a Schroedinger evolution is linear, it respects superpositions, it's unitary, it preserves probability, and let's just focus on that. It's linear, unitary, and it lots of other properties, [? temporal ?] invariance, unless you turn on a magnetic field, which you do. But in particular, it's linear and unitary. And those facts are going to constrain, powerfully, the kinds of operations we can effect on a quantum system. So, in particular, when we look at just two qubits, there's a beautiful, beautiful theorem says there's no cloning. And here's what that means. The no cloning theorem, pretty high flautin' for what it really is, which is the following. Suppose we have a system, which has input xy. And I want to build a machine that says, look, I've got this first qubit it in the state x, and what I want to do is I want to make a copy. I want to make another quantum system that's in exactly the same state as whatever x is. So you hand me a system where first bit's in state tax, and the second qubit's in state y. And I want to make a copy of x. And y is just, who cares what's in it? So I want this to go to x, x. OK. For all y. So regardless of what data was in here, I want to overwrite that data, and rewrite-- or that datem-- rewrite it with x. Can you do this? No, right. Why? AUDIENCE: [INAUDIBLE] PROFESSOR: Excellent. It would violate linearity and also unitarity, indeed. So to see that quickly, it's easiest to see the unitarity, I think. Well, it violates them both, but for unitarity, you manage to take a linear combination of these guys, where the two states y are orthogonal, and you take the norm squared. So you've normalized it to 1. The linear combination of each one goes to Sx, where the coefficient is the sum of those two terms. So we have, for example, x z, alpha x z plus beta x y goes to alpha plus beta x x. And that's really bad, because if x, z, and y are orthogonal, then normalization is alpha squared plus beta squared is 1. But x x, the normalization is alpha plus beta quantity squared is 1. And in general that's not true. In fact, this could be 0. So this violates linearity and a unitarity rather badly. So you can't clone. This is really disturbing. That means if you have a quantum, and you want make a copy of it, you can't. You can't ever make a copy of your quantum system. One copy. One chance. That's it. No cut and paste. So, as you can imagine that pretty powerfully constrains things that you can do. So, a related thing here is that there's no forgetting. Quantum evolution is, unlike an elephant, it is highly-- well, it's like an elephant, I guess. It remembers very well. It never forgets anything. And you can see that from this. This would be an example of forgetting. You forgot what was in the state y. You can't ever do that. OK. So I leave this as a challenge you to prove this show. It's a simple extension of the same logic. So what can you do? What you can do, is you can entangle two qubits. And that's really the juice of everything. You can entangle. So let me show you entanglement. Good, no e. Sorry, question? So you can entangle, and here's how you do it. Let's take this state 0, 0. So we have two qubits. The first one's in the state 0, and the second one is in the state 0. And now, I'm going to do the following set of operations to it. I'm first going to impose a Hadamard operation on the first qubit, and nothing on the second. And then I'm going to apply controlled-NOT, and we're going to see what I get out. So the initial state is 0, 0. After I Hadamard, well, the first bit is no longer in 0. Hadamard on 0 gives me 0 plus 1. So this is now the state 1 upon root 2. 0 plus 1 times 0, also known as 1 over root 2, 0, 0 plus 1, 0. Now is this the separable state? Yes, there is separated. And now I'm going to perform a controlled-NOT and what the controlled-NOT does, is that it switches the second bit, if the first bit is a 1. So what is the state after we've done this? The state after we've done this is, well, from the first term, 1 upon root 2, from the first term 0, 0-- what happens to that when we controlled-NOT? Well, we NOT this if this is 1. This is not 1, so we don't NOT it. We leave it alone. And the second term-- 1, 0 plus well, we flip this, if this is 1, and not if it's not, so this is 1. We flip it, and we get 1, 1. And now this is the prototypical entangled state-- that I think I just erased. But this is our entitled state. It's not separable. But if I measure the first one, I know what the state of the second one is, which is to say it's entangled. Cool? So by performing this series of operations, which is nothing other than a series of magnetic fields which I'm going to impose to the system, I've taken a state with initial conditions 0, 0, and put it into an entangled state, 0, 0 plus 1, 1. And that's all we need for the first basic algorithm of quantum computation. So this idea the quantum computers might be able to do things faster than classical computers floated around for a while. It took a while for people to make that sharp. And David Deutsch, who is a very entertaining and bombastic speaker, and he wrote-- I guess it's several now-- pretty entertaining books on the topic. And he sounds crazy. You listen to the guy talk, and he sounds nuts. He sounds like he's just way out there. The thing he's just-- gah! As a theorist, you listen to him like, just slow down there, buddy. Right? And so for a long time, I thought the guy-- I only knew his sort of public persona-- I thought, yeah, he's a little bit crazy; I'm not exactly sure-- and this is why everyone thinks he's such a damn genius. Because this is beautiful. So here is-- I don't know how he came up with this, but he's clever. So here is what's now called the Deutsch-- and it's really the one bit version of the Deutsch-Josza algorithm. So there is a first algorithm by Deutsch that didn't quite what it was supposed to do, then it was improved together with Jozsa, and they made an n particle version and everything was awesome. But here's the Deutsch-Jozsa algorithm. And what it is, it's a series of rules for how to make a quantum mechanical system evolve so as to effect the calculation you wanted to calculate. So you have to grant, to begin, you have to let me pose a problem to solve that can be solved in this fashion. And this problem is going to sound a little contrived. And, in fact, it's wildly contrived. It was contrived so that it could be solved in this fashion. But it's actually one that preexists the algorithm itself, so it's not quite as ridiculous So here's the problem. So the statement of the problem is that someone has a function f of x. So, let's say Matt knows a function f of x. Now the thing is, it's extremely expensive to evaluate this function f of x. So the way you evaluate involves putting 20 kilomeres in superposition states with each other. You have to run a whole experiment. And it costs a lot of money to run, so he charges-- $1 million dollars order to-- AUDIENCE: [LAUGHTER] PROFESSOR: Thank you, you guys are not quite old enough to-- so he knows the function f of x and he charges a million dollars in order to evaluate the function. You say, hey, Matt, look, I know this is a function-- which I should tell you f is a function that takes a single bit, 0 or 1, to another single bit, 0 or 1. So it sounds like, how hard could this possibly be? But in fact, it's a very hard function to evaluate. So you say, hey Matt, what's f of 0? And he's like, give me a million bucks. So you give him a million bucks. And he's like, 1. And you're like damn, that cost a lot of money. So now here's the question. So this is not yet the problem. The problem is this. Is f of 0 equal to f of 1 or not? OK. So f of 0 is either 0 or 1. f of 1 is either 0 or 1. Are they equal to each other? So this is easy, right? Classically, this is stupid. You calculate the function f twice. You evaluate f of 0, you get a number. You evaluate f of 1, and you get number. You look at your piece of paper, and you say it's either the same or different. How much does that cost you? Two million bucks. Better have good funding. So this is an expensive-- And here's what Deutsch and Josza have to say. This is really Deutsche at the beginning. It's really quite spectacular. Deutsch says actually, I tell you what, give me a million and a half, and I'll do the computation give you the answer. At which point you think, like I did previously, the guy's clearly raving. But, in fact, he's going to make a profit, And here's how he's going to do it. He's going to build, not a classical computer, but a quantum computer using quantum interference and entanglement to do this calculation. One evaluation. And here's how it's going to work. And the first thing we have to do is a preview, or set up, in order to do this calculation, you need two things. First off, you need Matt to be able to evaluate his function in a way that respects quantum mechanics. So, in particular, Matt had better be able to do his experiment, if I give him an superpositon. So we better be able to effect the calculation in a quantum mechanical way. The same way that we implemented a NOT quantum mechanically, or the controlled-NOT quantum mechanically, or the Hadamard, with some set of magnetic fields. He must be able to implement it quantum mechanically. Otherwise, it's not an interesting function. And let me just point out that any function you can think of can be implemented quantum mechanically, because you are quantum mechanics. OK? You're just not an elegant implementation-- and no offense-- of the quantum mechanical computation. So the set up is that Matt needs to be able to give me-- Matt provides-- a unitary transformation, a unitary operation, use of f that takes two qubits, x and y to x and f of x plus y. Where what this means, f of x plus y, is addition mod two. So what this says is, if y is 0, then this gives me f of x plus 0. If f of x is 0, that's 0 plus 0, so that gives me 0. If f of x is 1, is this gives me 1 plus 0, that's 1. On the other hand, if y is 1, then this gives me-- if f of x is 0, it gives me 1 plus 1, which is 0. And if f of x is 0, it's going to be 0 plus 1, which is 1. Everyone cool with that? Yeah. AUDIENCE: [INAUDIBLE]. --question, but like actually, how do you know that the matrix actually [INAUDIBLE] I mean, how can we know [INAUDIBLE] if matrices actually prove that quantum mechanics [INAUDIBLE] What if the matrix is is just an approximation-- PROFESSOR: You mean what if quantum mechanics is only an approximate description--? Of the-- AUDIENCE: No I'm sorry. [INAUDIBLE] To what if quantum mechanics-- the inelegant reprensentation of [INAUDIBLE] implementation of quantum mechanics PROFESSOR: Is inescapable--? AUDIENCE: [INAUDIBLE] is just an approximation of the problem, or is a really, really good approximation-- PROFESSOR: This is an interesting question. So it's tempting to think that this is a philosophical question, but it turns out not to be in a way that will be made sharp in about 10 minutes with Bell's Inequality. But a complete answer to that question remains open, and, probably, always will. But let me rephrase the question slightly, and tell me if this an accurate statement. Look, at the end of the day, what we're doing is we're going to develop a model where quantum mechanical calculation does something in particular. And that may or may not be a good model the real world. And in particular, whatever the actual thing, the actual system, that we're studying does, may or may not be well described by that quantum mechanical model. So can we check whether or not it is? Is that more or less the question? Yeah, and so the problem is all we can ever do is say that our model is a good or bad model. On the other hand, we can do the following. And this is the really neat thing. You might say, look, underlying quantum mechanics is going to be something more fundamental that's going to lead to slightly different results in exactly the sort of situations where we're going to care about quantum computation of large numbers and bits. And if you tell me just a little tiny bit about what properties that underlying description will have, that becomes an empirical question. So, for example, if you say, look, I suspect that underlying the quantum mechanical probabilities is some classical probability distribution over a hidden variable that you have not actually measured. And what we're going to find out is that we can rule that out experimentally. Just that extra little assumption that there's an underlying hidden variable-- a secret probability distribution on some variable we just haven't observed yet-- that is enough information about the system to rule out that model, amazingly. So I think we'll never have a full answer your question. But all we can do is work and see how well our models fits. And so far, nothing's ever disagreed with the quantum mechanical description. Let me hold off on questions just now. But it's a good and interesting question that's a hard one deal with, by which I mean it's an open question. So Matt provides for us an operator that allows us to calculate f of x. Now you might have said, well look, why not just take x, and why not have Matt build a machine that takes x and gives you f of x. Could you have done that? AUDIENCE: [INAUDIBLE] PROFESSOR: Well, it's not exactly no cloning. But let me leave this to you as a fun thing to think about. Why do we need this carrier bit, as well? OK. So there's our set up. Matt provides this function for us, U. And here's the algorithm. So the algorithm. And it's a series of steps, one by one we do them. We perform these operations are on our qubit. So here's what Deutsch says. Deutsch says start input, a state psi, is equal to 0, 1. First qubit is 0, in the state 0. The second qubit is in the state 1, for sure. We implement that with our boxes, or however we want to implement it. So we find ourselves in a definite state, you know, hard-soft. So we take a hard box, and we take a soft box, and we pull out the hard one and the soft one. One, Hadamard each. Hadamard on both bits. Both qubits. OK. So what does this take us to? It takes us to psi is equal to-- well, the 0 goes to 1 over root 2, times 0 plus 1. Did I erase Hadamard? No, good. There's Hadamard. So it does this-- does this. So it takes the first one to 0 plus 1, and it takes the second one to 1 over root 2, 0 minus 1. Cool? So at this point, this isn't very interesting. What we've done is we take it from one superposition to a different superposition. And doesn't seem to have anything to do with f. In fact, we haven't measured f. Two. Apply f. So we apply our operation U sub f. And well, this is a sort of an entertaining one. If we take this 1 of root 2-- so I'm going to rewrite this in a slightly simpler form-- this is 1/2, 0, times-- 0 times 0 minus 1. That's 1. Plus 1 times 0 minus 1. And the reason I'm doing that is we're going to apply Uf. And Uf, our function f, uses that first bit as a control bit for the second. So here's the control bit for the second. So we apply Uf, and this gives us 1/2. I'm going to actually do this on the next board, because it's going to be gigantic. So this gives us 1/2. 0-- so that first one, this is going to take this and give it 0 plus f of 0, and 1 plus f of 0. So times f of 0, plus 0. Minus f of 0 plus 1. Plus for the 1, this going to be times f of 1, now, plus 0. Minus f of 1, plus 1. OK? Now, here's a crucial step. This is equal to, and note the following, look at this particular guy. So for that particular guy, suppose f of 0 is 0. If f of 0, is 0, then this is 0 plus 0, which is 0. So f of 0 is equal to 0. And this gives me 0, and this gives me 0 plus 1, which is 1. 0 minus 1. On the other hand, if f of 0 is equal to 1, then we get 1 plus 0, which is 1. And here we get 1 plus 1, which is 0 minus 0, which is equal to minus 0 minus 1. Yeah? OK. So I can write this as minus 1 to the f of 0 times 0 minus 1. Everyone cool with that? This is just a little exercise in binary arithmetic. So we can write this first term. This is 1/2 minus 1 to the f of 0, 0, times 0 minus 1. So that's for the first one, and exactly the same logic is going to apply to the second. But now f of 1, instead of f of 0. Plus minus 1 to the f of 1, times 1, times 0, minus 1. Now, I want to point something out to you. If f of 0 is equal to f of 1, than what's true of f of 0 plus f of 1? Well, if they're the same exactly, then either it's 0 plus 0, in which case we get 0, or it's 1 plus 1, in which case we get 0. So this is 0, if it's the same, and 1, if it's not. OK. So we could either know them both, or we can measure f of 0 plus f of 1. So notice what happens here. This is equal to 1/2, and now I'm just going to pull out a factor of f of 0, minus 1 to the f of 0, times-- well, both of these terms have a 0 minus 1 on the second bit, so the second qubit is in the state 0 minus 1. Right, everyone cool with that? So this is equal to, for the first qubit, 0 plus 1, times minus 1 to the f of 0, that I pulled out to square it, plus f of 1, times 0 minus 1. And here's the quantity we wanted to measure. If this is 0, then they're even. Then they're the same. If it's 1, then they're not the same. So at this point we just forget three, forget about the second qubit. Oh, lord. Forget about the second qubit, and so forget about the second qubit just does this, just focus on this guy. And now, if f of 0 plus f of 1 is 0, so that they're the same, this is 0, minus 1 to the 0, 0, so we get 0 plus 1. So same, then our state is 0 plus 1. And if they're different, then we get the state 0 minus 1 Everyone agree with that? But if they're the same we get 0 plus 1, and different we get 0 minus 1. Still doesn't work for us, because if we measure, what's the probability we get 0 here? 1/2. And the probability that we get 1 is 1/2. Similar, if we measure here. It was 0, 1/2. 1, 1/2 On the other hand these states are familiar to us because they're what you get by Hadamarding. So why don't we take these, from these states to these-- by doing the inverse of the Hadamard, which, as it turns out, is Hadamard itself. So four. Hadamard the first bit. And the output is the state, psi out, is equal to 1/2, 1 plus, minus 1 to the f of 0 plus f 1, 0 plus 1/2, 1 minus, minus 1 to the f of 0 plus f 1, 1. And now, if f of 0 and f of 1 are the same, this is a 0. We get 1 plus 1. We get just 0, and this is 0, because this is 1, this is 1. They subtract we get 0. They're same you get this state 0, properly normalized. If they're not the same, you get this state 1, properly normalized. Now if we measure 0, we know they're the same, and if we measure 1, we know they're different. And so with absolute certainty, now five. Measure the first qubit. And we get 0 implies the same, and 1 implies different. And we did all of this with a single evaluation of our function f, right here. This is where we apply our function evaluation. We apply the function evaluation once, and we deterministically get the result, whether they're the same or different. So with one call to Matt, to my Oracle, with one call to Matt, which cost me one million dollars, we get the answer to whether it's the same or different. And that's a factor of 2 faster than the best classical algorithm. But that's not so satisfying. This was supposed to be exponentially better. And so that's where Jozsa comes in, and together with Deutsche, Deutsche and Jozsa the show the following. That there's an exactly analogous problem for n qubits. Wow. There's exactly analogous from for n qubits, the Deutsche-Jozsa problem. And now, how many different strings of integers could you put in? There are now 2 to the n possible states. And if you want to know whether f is the same for all of them, the worst case scenario is you evaluate f on the first possible combination, 0, 0, 0, 0, and you get some number. You measure f on 0, 0, 0, 0, 1, and get the same number, and just keep doing that forever. And you still don't know if they're all the same, until you get to the very last one. So, order 2 to the n is the worst case scenario. But technical scales are of a order 2 to the n. So classically, it takes an enormous number of observations. But in the quantum Deutsche-Jozsa algorithm-- and now in the n qubit Deutsch-Jozsa problem-- one quantum operation, and you get a deterministic result. And all of this, you evaluate it once, and you know. You've solved the problem. So instead of 2 to the n operations, it takes a single one. And now, for a large number n of bits, for example, for large integers-- dealing with very large numbers-- this is dramatically, exponentially more efficient than the classical algorithm. So at this point, people start really seriously thinking about quantum computation, whether you could get it to work. And how much we juice you can get out of actually building such a quantum computer. And this has developed into a whole theory in the whole field of the theory of computation. The thing I want to emphasize is that the crucial move is observing that, in quantum mechanics, you can entangle degrees of freedom. The crucial move is observing that you can entangle degrees of freedom, quantum mechanically. And that's what gave us all of the nice effects. We have these interference effects. And these interference effects lead to the deterministic outcome being correlated with the result of the computation. The interference is crucial. And this brings us to the last point, which is exactly what's so troubling about entanglement. And so here is where EPR come in. And Einstein, Podolsky, and Rosen say the following. They say, look, there are two things that are deeply upsetting about this entanglement story. Let me just give you a precise experiment, they say. They say, let me give you precise experiment that embodies all the weirdness of this. Suppose I take two of these qubits. And I put the qubits in an entangled state, up, up, plus down, down. OK. Let's normalize this with a 1 upon root 2. So there's our state. And then we take the first qubit, so there's our two bits, we take the first qubit, we send it somewhere faraway. And someone named Alice is sitting here and is holding on to that bit. And we take the second bit far away. And someone named Bob, conventionally, is sitting here and holds a second bit. Now given this initial of configuration, what is the probability that Alice will measure the spin to be up? Her spin to be up. 1/2, right? And 1/2 down. Similarly, Bob 1/2 up and down. Once Alice has measured the state to be up, immediately she knows something about Bob's spin. Bob's state will be up, because I chose this one. I could have chosen the other, which is the more popular [INAUDIBLE]. So Bob's will also be up. Now if you look at this list-- you do this over and over and over again-- their list just some random list of ups and downs, ups and downs, but they're exactly correlated amongst each other. So at this point, EPR were really upset. Because they say, look, there are two possibilities. Either there was an answer to the question all the way along of whether Alice's was up and Bob's was up, or Alice's was down and Bob's was down. Or there's some deep non-locality in the universe, such that a distant measurement, causally disconnected, can have an influence on Bob's state, such that they're correlated. This may seem random, but it's certainly not random, because it's correlated with Alice, even though Alice is wildly disconnected, a distant observer. Nothing could have traveled across that distance in the time it took to do the measurement. So they're sort of three responses you could take to this. The first response is, look, there isn't a problem here. It's just saying that quantum mechanics is insufficient. There's secretly a hidden variable, a variable you haven't observed yet, a property of an electron that determines whether it's going to be up or down early on. And the fact that it looks probabilistic just means that there's some classical dynamics for this hidden variable that effectively is probabilistic, like a particle moving in a fluid. Like dust pollen grains in a fluid, it just moves around randomly. Bu it just looks random, and it's not actually random. That's because there's an underlying classical mechanism controlling the probability distribution. The second version is a quantum mechanical version. The second interpretation is to say that, look, this may look upsetting. And I grant you that it looks upsetting, but I'm a quantum mechanic. And quantum mechanics works like a champ. And I'm not about to throw it out, and say that there's some secret, hidden variables. It just works. So just give up on your naive notions of locality, let it go, and just do the quantum mechanical calculation. Practicing physicists look at this, and just yawn. If you're a practicing physicist, you just forget it. Like, obviously, it works, so there's no more conversation to be had. So meanwhile, there's a second version of this, which is slightly more disturbing. Suppose Alice measures up-- and this is all on the z-direction-- but Alice measures up in the z-direction. She thus knows that Bob's particle is up in the z-direction. But simultaneously, Bob could measure spin in the x-direction, and determine that his spin is up in the x-direction as well. At that point, EPR say, look, we measured, empirically, that the particle is both up in the z-direction and up in the x-direction. It's just that we did that measurement using entanglement. But at the beginning of the day, we had that Sx and Sz don't commute. So you can't have a state with definite Sx and definite Sz. You cannot possibly. It doesn't mean anything to say so. Einstein wants to say that this is because quantum mechanics is great, but incomplete. The rest of us want to say that, no it's not, but that sounds like a philosophical question. And that's the way it was treated for a very long time, until you come along to Bell. And Bill made a beautiful observation. Bell said, look, telling me that there's an underlying probabilistic classical description tells me enough to make this an empirical question. Because it's saying that the statistics, the random statistics for Bob and Alice, are correlated by a classical dynamics rather than independent. So here's Bell's version. So remember at the very beginning, we talked about Bell's experiment. We said, consider three binary properties, a, b, and c. The number of some set that are a and not b, plus the number that are b but not c, is always greater than or equal to the number that are a but not c. And the way we proved this was just by noting that, if these are classical deterministic binary properties, then a not b means a not b and c, or a not b and not c. And ditto for each of these other guys. And we ended up with an expression which was, number that are a-- using that logically to, number that are a not b and c plus the number that are not a, b, and not c, is greater than or equal to 0. And that's clearly true. You can't have a number of elements be negative. So this is trivially true, quantum mechanically. But now here's the experiment I want to do. I want to do actually the EPR experiment. And here's the experiment I want to run. Alice is going to measure up or down at 0, and Bob is going to measure up or down at theta. Alice is then going to measure up and down at theta, and Bob is going to measure up and down at 2 theta. And the third experiment is going to be Alice going to measure up and down at 0, and Bob is going to measure up and down at 2 theta. So a is up or down at-- up at 0, b is up at theta, b is up at theta-- and, sorry, this could be down at theta, not b is down a theta. b is going to be up at theta, and c is going to be down at 2 theta. And a, again, up at 0 and not c is down at 2 theta. So we can rephrase this as the probability that given that we are up at 0, what is the probability that we are subsequently down at theta? Plus the probability that we are up at theta, what is the probability that we're up at theta and subsequently down at 2 theta? And then the probability that we are up at 0 and down to theta, using an EPR measurement, where one is performed by Alice and the other is performed by Bob. Exactly as EPR wanted. And we computed this last time-- in fact I just erased, because I'm excited about this, I guess-- I just erased the wave function that we needed, the state we needed. And the state that we needed was that if we are down at the angle theta, then this is equal to cosine of theta upon 2 down at 0, plus i sin theta upon 2, up at 0. And this is enough to answer our question. This is the quantum mechanical prediction. What's the probability that given that we're down at the angle theta, we're up at the angle 0? Well, if we're down at theta, the probably that we're up at 0-- the coefficient is i sine theta upon 2, and the probability is the norm squared of that expansion coefficient. So the probability is sine squared of theta upon 2. Similarly, the probability that we're up at theta and down the 2 theta, well, by rotation by theta, this gives me exactly the same thing. So it's, again, going to be sine squared, sine squared theta upon 2. The probability that we're up at 0 and down at 2 theta, well, just taking a factor of 2 for theta everywhere. And that gives me sine squared of theta. Now, I ask you, is left the left hand side always greater than or equal to the right hand side? And this is easy to check. Let's do this for a very small theta. For a very small theta, sine squared theta. So for small theta, much less than 1, sine theta squared is theta upon 2 the angle squared, which is equal to theta squared upon 4. And the next one is the same thing, plus theta upon 2 squared, which is equal to theta squared upon 4, theta squared upon 4, so theta squared upon two. And the right hand side is sine squared theta, which is theta squared. And is theta squared upon 2 greater than or equal to theta? Certainly not. So quantum mechanics predicts that if we do the EPR experiment, using these observables repeatedly, and built up statistics, what we'll find is an explicit violation of the Bell Inequality. And what that would represent if it were actually true, if we actually observed it, would be a conclusive empirical proof that there are no classical definite configurations underlying the probability of quantum mechanical events. It would say that it's impossible to build a classical theory with hidden variables that are randomly distributed such that you reproduce the predictions of quantum mechanics. We see already that it doesn't agree with quantum mechanics. The question is does it agree with the real world? So someone has to build this experiment and check. And this was done by Alain Aspect. And it violates Bell's Inequality. There is no classical description underlying quantum mechanics. The universe around you is inescapably probabilistic. It evolves in a deterministic fashion through Schrodinger evolution. But when we measure things, we measure results with probabilities. And those probabilities cannot be explained through some underlying classical dynamics. If there's something else underlying quantum mechanics, whatever else we know about it, is it is not classical. And this property of probabilistic evolution, or probabilistic measurement, is an inescapable and empirically verified property of the reality around us. And that's quantum mechanics. Thanks guys.
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_23_More_on_Spin.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Before we get started, let me ask you guys if you have any questions, pragmatic or otherwise, about the course so far. Seriously? To those of you reading newspapers, I encourage you to find a slightly different time to do so. I really encourage you find a slightly different time to do so, thanks. So far we've done basic rules of quantum mechanics. We've done solids. We understand a lot about electrons in atoms, the periodic table, and why diamond is transparent. One thing we did along the way is we talked about spin. We found that when we looked at the angular momentum commutation relations, these guys. We found that the commutators are the same. For these commutators, we can get total angular momentum, l l plus 1 times h bar squared for l squared. And h bar m for a constant and integer m for angular momentum in a particular direction, which we conventionally called z. But we also found that there were half integer values of the total spin and of the spin in a particular direction which, for example, with Spin s as 1/2, 3/2, 5/2, et cetera. And we discovered that riding a wave function on a sphere that interpreted these states as states with definite probability to be at a particular position on a sphere, a function of theta and phi, was inconsistent. In order for the wave function to satisfy those properties that have those eigenvalues and, in particular, half integer eigenvalues, we discovered that the wave function had to be doubly valued. And thus it would equal to minus itself, it equaled 0. So the rest of today and tomorrow is going to be an exploration of spin, or tomorrow-- next lecture, is going to be an exploration of spin and the consequences of these spin 1/2 states. Exactly what they are, we already saw that they're important for understanding the structure of the periodic table. So we know they're there. And they're present from the experiment, the Stern-Gerlach experiment that we've discussed many times. But before I get onto that, I want to improve on the last experiment we did. So in particular, in this last experiment we talked about the effective mass of an object interacting with a fluid or an object interacting with its environment. And why the mass of the object that's moving is not the same as the mass of the object when you put it on a balance. And that's this basic idea of renormalization. And we demonstrated that. I demonstrated that with a beaker of water. So I had a beaker of water and pulled a ping pong ball under water. We calculated that it should accelerate upward when released at 20 times the acceleration of gravity, depending on the numbers you use, very, very rapidly. And in fact it went glug, glug, glug, but it wasn't a terribly satisfying experiment because it's very hard to get the timing right. And the basic issue there is that the time scales involved were very short. How long did it take for a ping pong ball to drop from here to the surface? Not much time. And to rise through the water, not a whole lot of time. So I did that experiment, it was sort of comical. But I wanted to improve on it. So over the weekend, I went down to my basement and tweaked the experiment a little bit. And I called up a couple of my friends, and we did an improved version of this experiment. So this is a diver, I think this one is Kathy. Oh, shoot, we need to turn off the lights. Sorry. I totally forgot. good You'll never see this if we don't. You can do it. There we go. All right. So here you see my friend, I think this one's, actually, is it Kathy? I'm not sure. It's hard to tell when they have their marks on. And we're in the tank at the New England Aquarium, and she's going to perform this experiment for us. And we're going to film it, as you can see, the bubbles moving rather slowly, with a high speed camera filming at 1,200 frames per second with which we'll be able to analyze the data that results. The camera cost as much as a nice house. And it's not mine, but it's important to have friends who trust you. OK, so here we are. You might notice something in the background. Before we get started, I just want to emphasize that one should never take casually the dangers of doing an experiment. When you plan an experiment you must, ahead of time, design the experiment, design the experimental parameters. We designed the lighting. We designed everything. We set it up, but there are always variables you haven't accounted for. And a truly great experimentalist is one who has taken account of all the variables. And I just want to emphasize that I'm not a great experimentalist. So here, for example, is a moment. I probably should have thanked the sharks, it just occurred to me. Anyone who wants to take this experimental data, which, as you can probably guess, filmed for different purposes, I just manage to get the ping pong ball into the tank. Anyone who wants to get this and actually take the data, come to me, I'll give you the raw footage. And you can read off the positions, and hopefully for next lecture we'll have the actual plot of the acceleration as a function of time. So with that said and done, the moral of the story is you have to account for all variables. The other moral of the story is that you saw this very vividly. In the motion of the ping pong ball up, when it was released, there is that very rapid moment of acceleration when it bursts up very, very rapidly. But it quickly slows in its acceleration. Its acceleration slows down. In fact, a slightly funny thing happens. If you look carefully, and again, anyone who wants this can get access to the video, what you'll see is that the ping pong ball accelerates and then it sort of slows down. It literally decreases in velocity, accelerates and slows down. Can anyone think what's going on in that situation? AUDIENCE: Boundary layer formation. PROFESSOR: Sorry? AUDIENCE: Boundary layer formation. PROFESSOR: Good. Say that in slightly more words. AUDIENCE: It's starting to pick up more and more water [INAUDIBLE]. PROFESSOR: Yeah. Exactly. so what's going on is as this guy starts slowly moving along, it's pulling along more and more water, each bit of water around it is starting to drag along the layers of water nearby, and it builds up a sheath of water. Now that water starts accelerating, and the ping pong ball and the bubble of water that it's dragging along need to come to equilibrium with each other. They need to settle down smoothly to a nice uniform velocity. But it takes a while for that equilibrium to happen. And what actually happens is that the ping pong ball drives up. It pulls up the water. Which then drags with the ping pong ball. So you can see that in the acceleration, which is oscillatory with a slight little oscillation. So it's a damped but not overdamped harmonic oscillator motion. Any other questions about the effective mass of an electron and solid before moving-- Yeah? AUDIENCE: Why does it speed up after? That explains why it slows down because it's forming that sheath-- PROFESSOR: Right. Why it speeds back up is it's sort of like a slingshot. As this guy gets going a little ahead of the pack of water, the pack of water has an extra driving force on top of the buoyancy. It has the fact that it's a little bit displaced. So it catches up, but it's going slightly greater velocity than it would be if there were uniform velocity. So this guy catches up with the ping pong ball. OK so this is an of course of course in fluid mechanics, but I guess we don't actually need this anymore. Picking up on spin. So the commutation relations for spin are these. And as we saw last time, we have spin states. We have we can construct towers of states because from the sx and s1 we can build s plus-minus is equal to sx plus-minus sy. Sorry. i, thank you. So we can build towers of states using the raising and lowering operators as plus-minus. And those states need to end, they need to terminate. So we find that the spin can have totalling momentum of s squared h bar squared l l plus 1. And S in some particular direction, which we conventionally called z, is h bar m. I don't want to call this l. I want to call this s. For orbital angular momentum this would be l and this would be an integer. But for spinning angular momentum these are all the states we could build, all the towers we could build, which were 2n plus 1 over 2, which were not expressible in terms of wave functions, functions of a position on a sphere. These are all the 1/2 integer states. So s could be 1/2, 3/2, 5/2, and so on. And then m sub s is going to go from minus s to s in integer steps, just like m for the orbital angular momentum, l. So I want to talk about these states in some detail over the rest of this lecture and the next one. So the first thing to talk about is how we describe spin. In principle, this is easy. What we want, is we want to describe the state of a particle that carries this intrinsic angular momentum spin. So that's easy. The particle sits at some point, but the problem is it could be sitting at some point with angular momentum with spin in the z direction, say, plus h bar over 2. And let's focus on the case s is equal to 1/2. So I'll be focusing, for the lecture, for simplicity on the case, total spin is 1/2, which is the two-state ladder, but all of this generalizes naturally. In fact, that's a very good test of your understanding. So for s as 1/2, we have two states, which I will conventionally call the up in the z direction and the down in the z direction. And I will often omit the subscript. If I omit the subscript it's usually z unless from context you see that it's something else. So the wave function tells us the state of the system. But we need to know now for a spinning particle whether it's in the spin 1/2 up or spin 1/2 down state. And so we could write that as there's some amplitude that it's in the plus 1/2 state of x and at position x. And there's some amplitude that it's at position x and it's in the minus 1/2 state or the down state. And we again need that the total probability is one. Another way to say this is that the probability that we find the particle to be at x with plus or minus h bar upon 2 being the spin in the z direction. So at x, spin in the z direction is h bar upon 2 plus or minus is equal to norm squared of psi plus or minus of x squared. So this is one way you could talk about spin, and you could develop the theory of spin nicely here. But it's a somewhat cumbersome formalism. The formulation I want to introduce is one which involves matrices and which presages the study of matrix mechanics which you'll be using in 805. So instead, I want to take these two components, and what we see already is that we can't use a single wave function to describe a particle at spin 1/2. We need to use two functions. And I want to organize them in a nice way. I'm going to write them as psi is equal to-- and I'll call this capital psi of x-- is a two component vector, or so-called spinner, psi up of x and psi down of x. So it's a two component object. It's got a top and a bottom component. And notice that its conjugate, or its adjoint, psi dagger, is going to be equal to psi up, complex conjugate psi down, a row vector, or a row spinner. And for normalization we'll need that the total probability is 1 which says that psi capital dagger psi, psi capital with psi is equal to 1. But this is going to be equal to the integral dx. And now we have to take the inner product of the two vectors. So integral dx of psi up squared plus psi down squared. Cool? Yep? AUDIENCE: What's the coordinate x representing here? PROFESSOR: The coordinate x is just representing the position. So what I'm saying here is I have, again, we're in one dimension just for simplicity, it's saying, look if I have a particle that carries spin 1/2, it could be anywhere. Let's say it's at position x. So what's the amplitude at position x and spinning up, and I'm not going to indicate spinning down. OK? I like my coffee. So that's what the of x indicates, and I've just been dropping the of x. So there's some probability that it's at any given point and either spin up or spin down. Now, again, it's important, although I'm going to do this, and we conventionally do this spin up and down, this spin is pointing in a vector space that's two dimensional. It's either plus 1/2 or minus 1/2 h bar upon 2 h bar. So it's not like the spin is an arrow in three dimensional space that points. Rather, what it is, it's saying, if I measure the spin along some axis, it can take one of two values. And that was shown in the Stern-Gerlach experiment, where if we have a gradient of the magnetic field, dbz, in the z direction, and this is our Stern-Gerlach box, and we send an electron in, the electron always comes out in one of two positions. OK. Now, this is not saying there is a vector associated with this, that the electron has a angular momentum vector that points in some particular direction. Rather, it's saying that there are two possible values, and we're measuring along the z direction. Cool? So it's important not to make that mistake to think of this as some three dimensional vector. It's very explicitly a vector in a two dimensional vector space. It's not related to regular rotations. Yeah? AUDIENCE: Where you write psi of x equals psi plus 1 plus psi [INAUDIBLE]. PROFESSOR: Yes. AUDIENCE: Do we need a 1 over square root of 2 in front of that thing? PROFESSOR: Yeah, I haven't assumed their normalization, but each one could be independently normalized appropriately. AUDIENCE: So [INAUDIBLE]. PROFESSOR: Right. The whole thing has to be properly normalized, and writing it this way, this is just the [INAUDIBLE]. Good. So there's another nice bit of notation for this which is often used, which is probably the most common notation. Which is to write psi of x is equal to psi up at x times the vector 1, 0 plus psi down of x times the vector 0, 1. So this is what I'm going to refer to as the up vector in the z direction. And this is what I'm going to refer to as the down in the z direction vector. And that's going to allow me to write all the operations we're going to need to study spin in terms of simple two by two matrices. Yeah? AUDIENCE: Will those two psi's not be the same? PROFESSOR: Yeah, in general, they're not. So for example, here's a situation, a configuration, a quantum system could be in. The quantum system could be in the configuration, the particle is here and it's spinning up. And it could be in the configuration the particles over here, and it's spinning down. And given that it could be in those two configurations it could also be in the superposition over here and up and over here and down. Right? So that would be different spatial wave functions multiplying the different spin wave functions, spin part of the wave function. Make sense? OK. So these are not the same function. They could be the same function. It could be that you could be either spin up or spin down at any given point with some funny distribution, but they don't need to be the same. That's the crucial thing. Other questions? Yeah. AUDIENCE: I know that [INAUDIBLE] is a way to call them. So like [INAUDIBLE] something, are they anti-parallel, perpendicular, or are they something? PROFESSOR: Yeah. So here's what we can say. We know that an electron which carries sz is plus h bar upon 2, and a state corresponding to minus h bar upon 2 are orthogonal because those are two different eigenvectors of the same operator, sz. So these guys are orthogonal. Up in the z direction and down to the z directions are orthogonal. And thank you for this question, it's a good way to think about how wrong it is to think of up and down being up and down in the z direction. Are these guys orthogonal? These vectors in space? AUDIENCE: No. PROFESSOR: No, they happen to be parallel with a minus 1 in the overlap, right? So sz as up sc as down are orthogonal vectors, but this is clearly not sz as down, right? So the direction that they're pointing, the up and down, should not be thought of as a direction in three-dimensional space. AUDIENCE: It's just [INAUDIBLE]. PROFESSOR: It's just a different thing. What it does tell you, is if you rotate the system by a given amount, how does the phase of the wave function change. But what it does tell you is how the spin operations act on it. In particular, sz acts with a plus 1/2 or minus 1/2. They're just different states. Yeah? AUDIENCE: Is this similar to what happens when polarizations [INAUDIBLE]? PROFESSOR: It's similar to the story of polarizations except polarizations are vectors not spinners. It's similar in the sense that they look and smell like vectors in three-dimensional space, but they mean slightly-- technically-- slightly different things. In the case of polarizations of light, those really are honest vectors, and there's a sharp relationship between rotations in space. But that's a sort of quirk. They're both spin and vectors. These are not. OK so here's a notation I'm going to use. Just to alert you, a common notation that people use in Dirac notation is to say that the wave function is equal to psi up at x times the state up plus psi down at x times the state down. So for those of you who speak Dirac notation at this point, then this means the same thing as this. For those of you who don't, then this means this. What I want to do is I want to develop a theory of the spin operators, and I want to understand what it means to be a spin 1/2 state. Now, in particular, what I mean by develop a theory of the spin operators, if I was talking about four orbital angular momentum, say the orbital angular momentum in the z direction, I know what operator this is. If you hand me a wave function, I can act on it with lz and tell you what the result is. And that means that I can construct the eigenfunctions. And that means I can construct the allowed eigenvalues, and I can talk about probabilities. Right? But in order to do all that I need to know how the operator acts. And we know how this operator acts. It acts as h bar upon i dd phi where phi is the angular coordinate around the equator. And so given any wave function, a function of x, y, and z or r, theta, and phi, I can act with this operator, know how the operator acts. And it's true, that again, lx with lx is equal to i h bar lz. It satisfies the same time computation relation as the spin. But we know the spin operators cannot be expressed in terms of derivatives along a sphere. I've harped on that many times. So what I want to know is what's the analog of this equation for spin? What is a representation of the spin operators acting on the spinners, acting on states that carry half integer spin? We know it's not going to be derivatives. What is it going to be? Everyone cool with the goal? In order to do that, we need to first decide just some basic definition of spin in the z direction. So what do the angular momentum operators do? Well whatever else is true of the spin in the z direction operator, sz acting on a state up is equal to h bar upon 2 up. And sz actually on a state down is equal to h bar minus upon 2 down. And similarly s squared on up or down-- Oh, by the way, I'm going to relatively casually oscillate between the notations up and plus, up or down, and plus or minus. So sometimes I will write plus for up and minus for down. So I apologize for the sin of this. s squared on plus, this is the plus 1/2 state, is equal to h bar squared ss plus 1, but s is 1/2, so 1/2 times 1/2 plus 1 is 3/4 h bar squared times 3 over 4. And we get the same thing up, ditto down. Because s squared acts the same way on all states in a tower. Going up and down through a tower of angular momentum states, raising and lowering, does not change the total angular momentum because s plus and s minus commute with s squared because they have exactly the same commutation relations as the angular momentum. That's an awesome sound. So I want to know what these look like in terms of this vector space notation, up and down. And for the moment I'm going to dispense entirely with the spatial dependence. I'm going to treat the spatial dependence as an overall constant. So we're equally likely to be in all positions. So we can focus just on the spin part of the state. So again I want to replace up by 1, 0 and down by 0, 1. And I want to think about how this looks. So what this looks like is sz acting on [INAUDIBLE] 1, 0 is equal to h bar upon 2, 1, 0. And sz on 0, 1 is h bar upon 2, 0, 1. And s squared on 1, 0 is equal to 3 h bar squared on 4, 1, 0. And ditto for 0, 1. Yeah? AUDIENCE: That [INAUDIBLE] should have a line over it. PROFESSOR: Oh, thank you. Yes, it really should. AUDIENCE: So how do you get 3/4 there? PROFESSOR: 3/4, good. Where that came from is that remember that when we constructed the eigenfunctions of l squared, l squared acting on a state lm was equal to h bar squared l l plus 1. l plus 1 on [? file m. ?] Now if we do exactly the same logic, which we actually did at the time. We did in full generality whether the total angular momentum was an integer or a half integer. We found that if we took, I'm just going to use for the half integer states the symbol s, but it's exactly the same calculation. s squared on phi and again sm sub s is equal to h bar squared s s plus 1 phi s m sub s. OK and so for s equals 1/2 then s, s plus 1, is equal to 1/2 times 3/2, which is 3/4. Yeah? AUDIENCE: [INAUDIBLE] s equals [INAUDIBLE]. Isn't that [INAUDIBLE]? PROFESSOR: Ah, but remember, does l go negative? Great. Does s go negative? No. s is just labeling the tower. So s is 0, it's 1, it's 2. And so for example, here, these are states where the s is 1/2 and the s in the z direction can be plus 1/2 or minus 1/2. s is 3/2 and then s in the z direction can be m is 3/2, 1/2, minus 1/2, minus 3/2. OK. Other questions? OK, yeah. AUDIENCE: What about the lowering and raising of those? PROFESSOR: Good, we're going to have to construct them, because, what are they? Well, they lower and raise, so we're going to have to build the states that lower and raise. AUDIENCE: And would lowering on the up will give you down, but raising on up--? PROFESSOR: Awesome. So what did it mean that we had towers? Let me do that back here. So the question is, look, we're going to have to use the raising and lowering operators at the end of the day, but what happens if I raise the bottom state-- what if I raise down? I'll get up. And what happens if I raise up? You get 0. That's the statement that the tower ends. On the other hand, if s is 3/2, what happens if I raise 1/2? I get 3/2. And if I raise 3/2, I get nothing, I get 0, identically. And that's the statement that the tower ends. So for every tower labeled by s, we have a set of states labeled by m which goes from minus s to s in integer steps. The raising operator raises us by 1, the lowering operator lowers by 1. The lowering operator annihilates the bottom state, the raising operator kills the top state. Cool? Yeah. AUDIENCE: Do states like 3/2 and 5/2 have anything akin to up and down? PROFESSOR: Yeah, so-- do they have anything akin to up and down. Up and down is just a name. It doesn't really communicate anything other than it's shorthand for spin in the z direction 1/2. So the question could be translated as, are there convenient and illuminating names for the spin 3/2 states? And I don't really know. I don't know. I mean, the states exist. So we can build nuclear particles that have angular momentum 3/2, or 5/2, all sorts of things. But I don't know of a useful name. For the most part, we simplify our life by focusing on the 1/2 state. And as you'll discover in 8.05, the spin 1/2 states, if you know them very, very well, you can use everything you know about them to construct all of the spin 5/2 8/2-- well, 8/2 is stupid, but-- 9/2, all those from the spin 1/2. So it turns out spin 1/2 is sort of Ur-- in a way that can be made very precise, and that's the theory of Lie algebras. Yeah. AUDIENCE: Can you just elaborate on what you meant by, you can't really think of spin as an angular momentum vector? PROFESSOR: Yeah. So OK, good. So the question is, elaborate a little bit on what you mean by, spin can't be thought of as an angular momentum vector. Spin certainly can be thought of as an angular momentum, because the whole point here was that if you have a charged particle and it carries spin, then it has a magnetic moment. And a magnetic moment is the charge times the angular momentum. So if it carries spin, and it carries charge, and thus it carries magnetic moment-- that's pretty much what we mean by angular momentum. That's as good a diagnostic as any. Meanwhile, on top of satisfying that experimental property, this, just as a set of commutation relations, these commutation relations are the commutation relations of angular momentum. It just turns out that we can have states with total angular momentum little s, which is 1/2 integral-- 1/2, 3/2, et cetera. Now, the things that I want to emphasize are twofold. First off, something I've harped on over and over again, so I'll attempt to limit my uses of this phrase. But you cannot think of these states with s is 1/2 as wave functions determining position on a sphere. So that's the first sense in which you can't think of it as equivalent to orbital angular momentum. But there's a second sense, which is that up and down should not be thought of as spin in the z direction being up and spin in the z direction being down meaning a vector in three dimensions pointing up and a vector in three dimensions pointing down, because those states are orthogonal. Whereas these two three-dimensional vectors are not orthogonal-- they're parallel. They have a non-zero inner product. So up and down, the names we give these spin 1/2 states, should not be confused with pointing up in the z direction and down in the z direction. It's just a formal name we give to the plus 1/2 and minus 1/2 angular momentum in z direction states. Does that answer your question? AUDIENCE: Yeah, so, when you make a measurement of the value of spin-- so perhaps you do a Stern-Gerlach experiment-- and you get what the spin is, can you not then say, all right, this is spin plus 1/2, spin minus? It's as z is positive 1/2 as z is minus 1/2? PROFESSOR: Yeah, absolutely. So if you do a Stern-Gerlach experiment, you can identify those electrons that had spin plus 1/2 and those that had spin minus 1/2, and they come out in different places. That's absolutely true. I just want to emphasize that the up vector does not mean that they're somehow attached to the electronic vector that's pointing in the z direction. Good. Yeah. Go ahead, whichever. AUDIENCE: How do we verify that uncharged particles have spin? PROFESSOR: Yeah, that's an interesting question. So the question is, how do we know if an uncharged particle has spin? And there are many ways to answer this question, one of which we're going to come to later which has to do with Bell's inequality, which is a sort of slick way to do it. But a very coarse way is this way. We believe, in a deep and fundamental way, that the total angular momentum of the universe is conserved, in the following sense. There's no preferred axis in the universe. If you're a cosmologist, just stay out of the room for the next few minutes. So there's no preferred axis in the universe and the law of physics should be invariant under rotation. Now, if you take a system that has a bunch of particles with known angular momentum-- let me give you an example. Take a neutron. A neutron has spin 1/2. Wait, how did I know that? We can do that experiment by doing the following thing. We can take a neutron and bind it to a proton and see that the resulting object has spin 1. So let me try to think of a way that doesn't involve a neutron. Grant me for the moment that you know that a neutron has spin 1/2. So let's just imagine that we knew that, by hook or by crook. We then do the following experiment. We wait. Take a neutron, let it sit in empty space. When that neutron decays, it does a very cool thing. It decays relatively quickly into your proton and an electron that you see. You see them go flying away the proton has positive charge, and the electron has negative charge and it goes flying away. But you've got a problem. Because you knew that the neutron had spin 1/2, which is [INAUDIBLE]. And then you decay one into a proton and an electron. And the total angular momentum there is 1/2 plus 1/2 or 1/2 minus 1/2. It's either 1 or 0. And you've got a problem. Angular momentum hasn't been conserved. So what do you immediately deduce? That another particle must have also been emitted that had 1/2 integer angular momentum to conserve angular momentum. And it couldn't carry any charge because the electron and the proton were neutral, and the neutron is neutral. So things like this you can always deduce from conservation of angular momentum one way or the other. But the best way to do it is going to be some version of addition of angular momentum where you have some object like an electron and a proton and you allow them to stick together and you discover it has total spin 1. Yeah. We can talk about that in more detail afterwards. That's a particularly nice way to do the experiment. Yeah. AUDIENCE: Angular momentum [INAUDIBLE] weird vector since if you reflect your system through [INAUDIBLE]. How does that work? PROFESSOR: Yeah, OK, good. I don't want to get into this in too much detail, but it's a really good question, so come to my office hours and ask or go to recitations and ask. It's a really good question. The question is this-- angular momentum has a funny property under parity, under reflection. So if you look in a mirror this way, here's angular momentum and it's got-- right-hand rule, it's got angular momentum up-- if I look in the mirror, it's going this way. So it would appear to have right angular momentum down. That's what it looks like if you reflect in a mirror. Other direction. So that's a funny property of angular momentum. It's also a true property of angular momentum. It's fine. And what about spin, is the question. Does spin also have this funny property under parity, is that basically the question? Yeah, and it does. And working out exactly how to show that is a sort of entertaining exercise. So again, it's beyond the scope of the lecture, so come ask me in office hours and we can talk about that. Yeah, one more. AUDIENCE: For orbital angular momentum, say for l equals 1, we had states like m equals plus 1 and minus 1. PROFESSOR: Yes. AUDIENCE: And we did think of those as angular momentum vectors. PROFESSOR: Absolutely. AUDIENCE: But those states are also orthogonal, are they not? PROFESSOR: Yeah, those states are also orthogonal. AUDIENCE: So even though the angular momentum vectors aren't orthogonal, they're still-- it's just a different sense. PROFESSOR: That's exactly right. So again, even in the case of integer angular momentum, you've got to be careful about talking about the top state and the bottom state corresponding to pointing in some direction, because they're orthogonal states. However, they do correspond to a particular angular momentum vector in three dimensional space. They correspond to a distribution on the sphere. So there's a sense in which they do correspond to real rotations, real eigenfunctions on a sphere, and there's also a sense in which they don't, because they're still orthogonal. That's exactly right. So let me move on. I'm going to stop questions at this point. So good. So these are the properties that need to be satisfied by our operators. And it's pretty easy to see in this basis what these operators must be. Sz has eigenvectors 1, 0 and 0, 1. So Sz should be equal to h bar upon 2 1, 0, 0, minus 1. So let's just check this on 1, 0 gives me 1, 0, so it gives me the same thing back times h bar upon 2. Cool. And acting on 0, 1, or the down state, we get h bar upon 2 times 0 minus 1. That could be minus 1. Oh, sorry, 1, 0 gives me 0 and 0 minus 1 on 1 gives me minus 1, which is 1 with a minus sign. That's a minus sign. So this works out like a champ. And S squared, meanwhile, is equal to-- well, it's got to give me h bar squared times 3/4 for both of these vectors. So h bar squared-- and meanwhile, these are eigenstates-- h bar squared times 3/4 times 1, 0, 0, 1. So we know one other fact, which was brought up just a minute ago, which was that if we take S plus and you act on the state 0, 1, what should you get? If you raise your 1-- 1, 0. Great. So we also worked out the normalization coefficient on the problem set. And that normalization coefficient turns out to be 1. And let's be careful-- we've got an h bar, for dimensional analysis reasons. So meanwhile, S minus, similarly, on 0, 1, is equal to 0. And S plus on 0, 1, is equal to-- oh, sorry, we already did that. We want S minus on 1, 0. Let's see-- S minus on-- we want S plus on 1, 0 is equal to-- right, 0. And S minus on 1, 0 is equal to h bar 0, 1. OK, so putting all this together, you can pretty quickly get that S plus is equal to-- we need an h bar and we need it to raise the lower one and kill the top state. So on 1, 0, what does S plus do? That gives us 0 that gives us 0. Good. And on the lowered state, 0, 1, that gives me a 1 up top and that gives me a 0 downstairs, so it works out like this. So we've got h bar. Similarly, S minus is equal to h bar times 0, 0, 1, 0. So we've got these guys-- so much from just the definitions of raising and lowering. And by taking inner products, you can just derive those two lines from these. But notice that Sx is equal to S plus plus S minus upon 2, and Sy is equal to S plus minus S minus upon 2i. So this tells us that Sx is equal to h bar upon 2 times S plus-- we're going to get a 1 here-- plus S minus-- we're going to get a 1 here-- 0, 1, 1, 0. And Sy is equal to, again, upon 2i times h bar, h bar upon 2i, times S plus, which is going to give me 1 and minus S minus which is going to give me minus 1, 0, 0. But we can pull this i in, so 1/i is like minus i. So minus i times minus 1 is going to give me i and minus i times 1 is going to give me i. So-- AUDIENCE: Shouldn't it be minus i? PROFESSOR: Sorry? Yeah, did I write i? That should've been minus i. Thank you. So now we have a nice representation of these spin operations, of the spin operators. And explicitly we have that Sx is equal to h bar upon 2 0, 1, 1, 0, Sy is equal to h bar upon 2 0, minus i, i, 0. And Sz is equal to h bar upon 2 1, 0, 0, minus 1. So why is Sz the only one that's diagonal? Is it something special about Sz? AUDIENCE: I mean, we've chosen z as the axis along which to project S squared. PROFESSOR: Exactly. So the thing that's special about Sz is that at the very beginning of this, we decided to work in a basis of eigenstates of Sz, with definite values of Sz. So if they have definite values, then acting with Sz is just going to give you a number. That's what it is to be a diagonal matrix. You act on a basis vector, you just get a number out. So we started out by working in the eigenbasis of Sz. And as a consequence, we find that Sz is diagonal. And this is a general truth that you'll discover in matrix mechanics when you work in the eigenbasis of an operator, that operator is represented by a diagonal matrix. And so we often say, rather than to work in an eigenbasis, we often say, to diagonalize. Yeah. AUDIENCE: Are the signs right for Sy? Because if we had h bar over 2i, and as we initially had a 1 in the top right and minus 1 in the bottom left, shouldn't we just multiply by i? PROFESSOR: I'm pretty sure-- so originally, we had a downstairs i, right? So let's think about what this looked like. This was 1 and minus 1, right? Agreed? So this is minus 1 over i. So we pull in the i. So that we go from minus 1 to minus 1 over i. And we go from 1 to 1 over i. And I claim that one over i is minus i. AUDIENCE: Oh, OK. PROFESSOR: OK? And minus 1 over i is i. That cool? Good. OK, so this is i and minus i. I always get that screwy, but it's useful to memorize these matrices. You might think it's a little silly to memorize matrices. But these turn out to be ridiculously useful and they come up all the time. This is called sigma x. This is called sigma y. And this is called sigma z. And different people decide whether to put the 1/2 in there or not, the h bar does not go in there. Some people put in the 2, some people don't put in the 1/2, it's a matter of taste. Just be careful and be consistent, as usual. And these are called the Pauli matrices because A, we really like Pauli and B, Pauli introduced them. Although he didn't actually introduce them in some sense-- this mathematical structure was introduced ages and ages and ages ago. But physicists cite the physicist, not the mathematician. OK I'm not saying that's good. I'm just saying it happens. So notice a consequence of these. An important consequence of these-- the whole point here was to build a representation of the spin operators. Now whatever else the spin operators do, they had better satisfy that computation relation, otherwise they're not really spin operators. That's what we mean by being spin operators. So let's check. Is it true that Sx commutator with Sy is equal to i h bar Sz? So this is a question mark. And let's check. Let's do the commutator. From the Sx, we're going to get an h bar upon 2. From the Sy, from each Sy, we're going to get a factor of h bar upon 2, so I'll just write that as h bar upon 2 squared-- times the commutator of two matrices-- 0, 1, 1, 0 commutator with 0, minus i, i, 0. This is equal to h bar squared upon 4 times-- let's write this out. The first term is going to be this matrix times this matrix. That's going to be, again, a matrix-- 0, 1, 0, 1. So that first one is a 1. 0, 1, minus i-- oh, sorry, that's an i. 0, 1, that's an i, that's a minus i. So 0, 1, 0, i gives me an i. 0, 1, minus i, 0 gives me a 0. Second row-- 1, 0, 0, i gives me 0. And 1, 0, minus i, 0 gives me minus i. And then the second term is the flipped order, right? The commutator term. So we get minus the commutator term, which is going to be 0, minus i, 0, 1. That gives me minus i. 0, minus i, 1, 0, that gives me 0. Bottom row-- i, 0, 0, 1-- 0. And i, 0, 1, 0 give me i. OK, so notice what we get this is equal to h bar squared upon 4. And both of those matrices are the same thing. Those matrices are both i, 0, 0, minus i with minus i, 0, 0, i, giving us i, 0, 0, minus i times 2 from the two terms. The 2's cancel, and this gives me h bar squared upon 2 times i, 0, 0, minus i. But this is also known as-- pulling out an i and an h bar-- times h bar upon 2 1, 0, 0, minus 1. This is equal to i h bar Sz. So these matrices represent the angular momentum commutators quite nicely. And in fact, if you check, all the commutators work out beautifully. So quickly, just as a reminder, what are the possible measurable values of Sz for the spin 1/2 system? What possible values could you get if you measured Sz, spin in the z direction? Yeah, plus or minus h bar upon 2. Now what about the eigenvectors? What are they-- of Sz? In this notation, there are these states. There's one eigenvector, there's the other. But let's ask the same question about Sx. So for Sx, what are the allowed eigenvalues? Well, we can answer this in two ways. The first way we can answer this is by saying look, there's nothing deep about z. It was just the stupid direction we started with. We could have started by working with the eigenbasis of Sx and we would've found exactly the same story. So it must be plus or minus h bar upon 2. But the reason you make that argument is A, it's slick and B, it's the only one you can make without knowing something else about how Sx acts. But now we know what Sx is. Sx is that operator. So now we can ask, what are the eigenvalues of that operator? And if you compute the eigenvalues of that operator, you find that there are two eigenvalues Sx is equal to h bar upon 2 and Sx is equal to minus h bar upon 2. And now I can ask, well, what are the eigenvectors? Now, we know what the eigenvectors are because we can just construct the eigenvectors of Sx plus. And if you construct the eigenvectors of Sx plus-- should I take the time? How many people want me to do the eigenvectors of Sx explicitly? Yeah, that's kind of what I figured. OK, good. So the eigenvectors of Sx are, for example, on 1, 1 is equal to-- well, Sx on 1, 1, the first term, that 0, 1, is going to give me a 1, the 1, 0 is going to give me a 1. So this is h bar upon 2 coefficient of Sx on 1, 1. And Sx on 1, minus 1 is going to give me h bar upon 2 minus 1, minus 1, because all Sx does is swap the first and second components. So it gives me minus 1, takes it to the top, but that's just an overall minus sign. So again, we have the correct eigenvalues, plus and minus h bar upon 2, and now we know the eigenvector. So what does this tell us? What does it tell us that up in the x direction is equal to 1 over root 2 if I normalize things properly. Up in the z direction plus down in the z direction. That's what this is telling me. This vector is equal to up in the z direction-- that's this guy-- plus down in the z direction. But this isn't properly normalized, and properly normalizing it gives us this expression. So what does this tell us? This tells us that if we happen to know that the system is in the state with angular momentum, or spin, angular momentum in the x direction being plus 1/2, then the probability to measure up in the z direction in a subsequent measurement is 1/2. And the probability to measure down is 1/2. If you know it's up in the x direction, the probability of measuring up in the z or down in the z are equal. You're at chance. You're at even odds. Everyone agree with that? That's the meaning of this expression. And similarly, down in the x direction is equal to 1 over root 2, up in the z direction minus down in the z direction. And we get that from here. This state is explicitly, by construction, the eigenstate of Sx as we've constructed Sx. And we have a natural expression in terms of the z eigenvectors up and down. That's what this expression is giving us. It gives us an expression of the Sx eigenvector in the basis of Sz eigenvectors. So for example, this tells you that the probability to measure up in the z direction, given that we measured down in the x direction first-- so this is the conditional probability. Suppose I first measured down in the x direction, what's the probability that I subsequently measure up in the z direction? This is equal to-- well, it's the norm squared of the expansion coefficient. So first down in the x direction and the probability that we're up in the z direction is 1 upon root 2 squared. Usual rules of quantum mechanics-- take the expansion coefficient, take its norm squared, that's the probability-- 1/2. And we can do exactly the same thing for Sy. So let's do the same thing for Sy without actually working out all the details. So doing the same thing for Sy, up in the y direction is equal to 1 upon root 2 times up in the z direction plus i down in the z direction. And down in the y direction is equal to 1 upon root 2 up in the z direction minus i down in the z direction. And I encourage you to check your knowledge by deriving these eigenvectors, which you can do given our representations of Sy. Now here's a nice thing that we're going to use later. Consider the following. Consider the operator S theta, which I'm going to define as cosine theta Sz plus-- oh, sorry. I'm not even going to write it that way. So what I mean by S theta is equal to-- take our spherical directions and consider an angle in the-- this is x, let's say, y, x, z-- consider an angle theta down in the zx plane, and we can ask, what is the spin operator along the direction theta? What's the angular momentum in the direction theta? If theta, for example, is equal to pi/2, this is Sx. So I'm just defining a spin operator, which is the angular momentum along a particular direction theta in the xz plane. Everyone cool with that? This is going to turn out to be very useful for us, and I encourage you to derive the following. And if you don't derive the following, then hopefully it will be done in your recitations. Well, I will chat with your recitation instructors. And if you do this, then what are we going to get for the eigenvalues of S theta? What possible eigenvalues could S theta have? [? AUDIENCE: None. ?] PROFESSOR: Good. Why? AUDIENCE: Because it can't have any other redirection [INAUDIBLE] no matter where you start. PROFESSOR: Fabulous. OK. So the answer that was given is that it's the same, plus or minus h bar upon 2 as Sz, or indeed as Sx or Sy, because it can't possibly matter what direction you chose at the beginning. I could have called this direction theta z. How do you stop me? We could have done that. We didn't. That was our first wave answering. Second wave answering is what? Well, construct the operator S theta and find its eigenvectors and eigenvalues. So I encourage you to do that. And what you find is that, of course, the eigenvalues are plus or minus h bar upon 2 and up at the angle theta is equal to cosine of theta upon 2 up in the z direction plus sine of theta upon 2 down in the z direction. And down theta is equal to cosine theta upon 2 up in the z direction minus sine of theta over 2 down in the z direction. Oops, no. I got that wrong. This is sine, and this is minus cosine. Good. That makes more sense. OK. So let's just sanity check. These guys should be properly normalized. So if we take the norm squared of this guy, the cross terms vanish because up z and down z are orthogonal states. So we're going to get a co squared plus the sine squared. That's 1. So that's properly normalized. Same thing for this guy. The minus doesn't change anything because we norm squared. Now, I'll check that they're orthogonal. If we take this guy dotted into this guy, everything's real. So we get a cosine from up up, we're going to get a cosine sine. And from down down, we're going to get a minus cosine sine. So that gives us 0. So these guys are orthogonal, and they satisfy all the nice properties we want. So this is a good check-- do this-- of your knowledge. And if we had problem sets allowed this week, I would give you this in your problem set, but we don't. Yeah, OK. Suppose that I measure. So I want to use these states for something. Suppose that I measure Sz and find Sz is equal to 1/2 h bar plus h bar upon 2, OK, at some moment in time. First question is easy. What's the state of the system subsequent to that measurement? AUDIENCE: [INAUDIBLE] PROFESSOR: Man, you all are so quiet today. What's the state of the system subsequent to measurement that Sz is plus h bar upon 2? AUDIENCE: Up z. PROFESSOR: Up z. Good. So our state psi is up z upon measurement, OK, after measurement. And I need new chalk. OK. Now, if I measure Sx, so 1, 2, measure Sx, what will I get? AUDIENCE: [INAUDIBLE] PROFESSOR: OK. What values will I observe with what probabilities? Well, first off, what are the possible values that you can measure? AUDIENCE: Plus or minus h bar over 2. PROFESSOR: Right, the possible eigenvalues, so which is plus or minus h bar upon 2, but with what probability? AUDIENCE: [INAUDIBLE] PROFESSOR: Exactly. So we know this from the eigenstate of Sx. If we know that we're in the state up z, we can take linear combinations of this guy to show that up z is equal to 1 over root 2. So let's just check. If we add these two together, up x and down x, if we add them together, we'll get 1/2, 1/2, 2/2, a root 2 times up z, is up x plus down x, up x plus down x, dividing through by the 1/2. So here we've expressed up in the z direction in a basis of x and y, which is what we're supposed to do. So the probability that I measure a plus 1/2 as x is equal to plus 1/2 h bar upon 2 is equal to 1 over root 2 squared, so 1/2. And ditto, Sx is equal to minus h bar upon 2. Same probability. OK? What about Sy? Again, we get one of two values. Oops, plus or minus h bar upon 2. But the probability of measuring plus is equal to 1/2, and the probability that we measure minus 1/2 is 1/2. That's h bar upon 2. OK? So this should look familiar. Going back to the very first lecture, hardness was spin in the z direction, and color was spin in the x direction, I guess. And we added, at one point, a third one, which I think I called whimsy, which is equal to spin in the y direction. OK? And now all of the box operation that we used in that very first lecture, you can understand is nothing other than stringing together chains of Stern-Gerlach experiments doing Sx, Sy, and Sz. So let's be explicit about that. Let's make that concrete. Actually, let's do that here. So for example, suppose we put a random electron into an Sz color box. OK. Some are going to come out up, and some are going to come out down in the z direction. And if we send this into now an Sx color box, this is going to give us either up in the x direction or down in the x direction. And what we'll get out is 50-50, right? OK. So let's take the ones that came out down. And if we send those back into an Sz, what do we get? AUDIENCE: 50-50. PROFESSOR: Yeah, 50-50, because down x is 1 over 2 up z plus 1 over 2 down z. What was going on in that very first experiment, where we did hardness, color, hardness? Superposition. And what superposition was a hard electron? It was 1 over a 2, white plus black. We know precisely which superposition, and here it is. OK? And now, let's do that last experiment, where we take these guys, Sx down, and I'm turning this upside down, beam joiners. We take the up in the x direction and the down in the x direction, and we combine them together, and we put them back into Sz, what do we get? Well, what's the state? 1 over root 2 down x plus 1 over root 2 up x? AUDIENCE: [INAUDIBLE] PROFESSOR: It's up z. Up z into an Sz box, what do we get? Up z with 100%. That's white with 100%, or I'm sorry, hard with 100%. AUDIENCE: [INAUDIBLE] down z with a down z. PROFESSOR: Sorry. AUDIENCE: You put in down. PROFESSOR: Oh, I put in down. Shoot, I'm sorry. Well, yeah, indeed, I meant to put in up. Yes, down z with 100% confidence. If we remove the mirror, 50-50. If we add in the mirror, 100%. And the difference is whether we're taking one component of the wave function, or whether we're superposing them back together. All right? Imagine we know we have a system in this state, and I say, look, this component is also coincidentally very far away, and I'm going to not look at them. So of the ones that I look at, I have 1 over root 2 up x. But if I look at the full system, the superimposed system, that adds together to be an eigenstate of Sz. These are our color boxes. Yeah? AUDIENCE: But how do you know, when you put the two beams to the beam joiner, it serves to add their two states together? PROFESSOR: Yeah, excellent. This is a very subtle point. So here we have to decide what we mean by the beam splitter. And what I'm going to mean by the beam splitter, by the mirrors and beam joiners-- so the question is, how do we know that it does this without changing the superposition? And what I want to do is define this thing as the object that takes the two incident wave functions, and it just adds them together. It should give me the direct superposition with the appropriate phases and coefficients. So if this was plus up x, then it stays plus up x. If it's minus up x, it's going to be minus up x. Whatever the phase is of this state, when it gets here it just adds together the two components. That's my definition of that adding box. AUDIENCE: So realistically, what does that look like? PROFESSOR: Oh, what? You think I'm an experimentalist? [LAUGHTER] Look, every time I try an experiment, I get hit by a shark. OK? [LAUGHTER] Yeah, no. How you actually implement that in real systems is a more complicated story. So you should direct that question to Matt, who's a very good experimentalist. OK. So finally, let's go back to the Stern-Gerlach experiment, and let's actually run the Stern-Gerlach experiment. I guess I'll do that here. So let's think about what the Stern-Gerlach experiment looks like in this notation, and not just in this notation, in the honest language of spin. And I'm going to do a slightly abbreviated version of this because you guys can fill in the details with your knowledge of 802 and 803. OK. So here's the Stern-Gerlach experiment. We have a gradient in the magnetic field. This is the z direction. And we have a gradient where B in the z direction has some B0, a constant plus beta z. OK? So it's got a constant piece and a small gradient. Everyone cool with that? It's just the magnetic field gets stronger and stronger in the z direction, there's a constant, and then there's a rate of increase in the z direction. And I'm going to send my electron through. Now remember, my electron has a wave function, psi electron, is equal to-- well, it's got some amplitude to be up, a up z, plus some amplitude to be down, b down z. And if this is a random electron, then its state is going to be random, and a and b are going to be random numbers whose norm squared add up to 1, proper normalization. Cool? So here's our random initial state, and we send it into this region where we've got a magnetic field gradient. And what happens? Well, we know that the energy of an electron that carries some angular momentum is a constant, mu naught, times its angular momentum dotted into any ambient magnetic field. Whoops, sorry, with a minus sign. This is saying that magnets want to anti-align. Now, in particular, here we've got a magnetic field in the z direction. So this is minus mu 0 Sz Bz. And Bz was a constant, beta 0-- sorry, B0 plus beta z. So the energy has two terms. It has a constant term, which just depends on Sz, and then there's a term that depends on z as well as depending on Sz [INAUDIBLE]. So we can write this as a-- so Sz, remember what Sz is. Sz is equal to h bar upon 2, 1, 0, 0, minus 1. So this is a matrix with some coefficient up a-- do I want to write this out? Yeah, I guess I don't really need to write this out. But this is a matrix, and this is our energy operator. And it acts on any given state to give us another state back. It's an operator. OK. And importantly, I want this to be only-- this is in some region where we're doing the experiment, where we have a magnetic field gradient. Then outside of this region, we have no magnetic field and no magnetic field gradient. So it's 0 to the left and 0 to the right. So as we've talked about before, the electron feels a force due to this gradient to the magnetic field. The energy depends on z, so the derivative of the energy with respect to z, which is the force in the z direction, is non-zero. But for the moment that's a bit of a red herring. Instead of worrying about the center of mass motion, let's just focus on the overall phase. So let's take our initial electron with this initial wave function a up z plus b down z, and let's note that in this time-- so what does this matrix look like? OK. So fine, let's actually look at this matrix. So Sz is the matrix 1, 0, 0, minus 1 times h bar upon 2. So we have h bar upon 2 minus mu 0. And then B0 plus beta z, which I'll just write as B, which is equal to some constant C, 0, 0, some constant minus C. Everyone agree with that? Where a constant, I just mean it's a number, but it does depend on z, because the z is in here. Now, here's the nice thing. When we expand the energy on up z, this is equal to C of z up z, because there's our energy operator. And energy on down z is equal to minus Cz of z down to z. So up and down in the z direction are still eigenfunctions of the energy operator. We've chosen an interaction, we've chosen a potential which is already diagonal, so the energy is diagonal. It's already in its eigenbasis. So as a consequence, this is the energy, energy of the up state is equal to, in the z direction, is equal to plus C. And energy in the down state, energy of the down state, is equal to minus C of z. Everyone agree with that? So what this magnetic field does it splits the degeneracy of the up and down states. The up and down states originally had no energy splitting. They were both zero energy. We turn on this magnetic field, and one state has positive energy, and the other state has negative energy. So that degeneracy has been split. Where did that original degeneracy come from? Why did we have a degeneracy in the first place? AUDIENCE: [INAUDIBLE] PROFESSOR: Spherical symmetry. And we turn on the magnetic field, which picks out a direction, and we break that degeneracy, and we lift the splitting. OK? So here we see that the splitting is lifted. And now we want to ask, how does the wave function evolve in time? So this was our initial wave function. But we know from the Schrodinger equation that psi of t is going to be equal to-- well, those are already eigenvectors-- so a e to the minus i e up t upon h bar up z plus b e to the minus i e down t upon h bar down z. All right? But we can write this as equals a e to the i mu 0 b0 t over 2 plus i mu 0 beta t z upon 2 up z plus b same e to the minus i--oops, that should be, oh, no, that's plus-- that's e to the i mu 0 b0 t over 2. And if we write this as two exponentials, e to the minus i mu 0 beta t z upon 2-- oh no, that's a minus, good, OK-- down z. OK. So what is this telling us? So what this tells us is that we start off in a state, which has some amplitude to be up and some amplitude to be down, a and b. And at a later time t, sine of t, what we find after we run it through this apparatus is that this is the amplitude. What do you notice? Well, we notice two things. The first is that the system evolves with some overall energy. So the phase rotates as usual. These are energy eigenstates, but the amount of phase rotation depends on z. So in particular, this is e to the i some number times z. And what is e to the i some number times z? If you know you have a state that's of the form e to the i some number, which I will, I don't know, call kz times z, what is this telling you about the system? This is an eigenstate of what operator? AUDIENCE: Momentum. PROFESSOR: Momentum in the z direction. It carries what momentum? AUDIENCE: h bar kz. PROFESSOR: h bar kz, exactly. So what about this? If I have a system in this state, what can you say about its momentum in the z direction? AUDIENCE: [INAUDIBLE] PROFESSOR: It's non-zero. Right here's a z. These are a bunch of constants. It's beta, t, mu 0, 2, i. So it's non-zero. In fact, it's got minus some constant times z. Right? So this is a state with negative momentum. Everyone agree? It's got momentum z down in the z direction. What about this guy? Momentum up. So the states that are up in the z direction get a kick up. And the states that are down in the z direction get a kick down. They pick up some momentum down in the z direction. Yeah? AUDIENCE: Where did that big T come from? PROFESSOR: Big T should be little t, sorry. Sorry, just bad handwriting there. That's just t. Yeah? AUDIENCE: So in this case, the eigenvalues-- it's a function of z? PROFESSOR: Mhm. AUDIENCE: Is that allowable? PROFESSOR: Yeah. So that seems bad, but remember what I started out doing. Oop, did I erase it? Shoot, we erased it. So we started out saying, look, the wave function is up in the z direction, or really rather here, up in the z direction times some constant. Now, in general, this shouldn't be a constant. It should be some function. So this should be a function of position times up in the z direction. This is a function position times down in the z direction. So what we've done here is we said, under time evolution, that function changes in time, but it stays some linear combination up and down. So you can reorganize this as-- the time evolution equation here is an equation for the coefficients of up z and the coefficient of down z. AUDIENCE: OK. PROFESSOR: Other questions? OK. So the upshot of all this is that we run this experiment, and what we discover is that this component of the state that was up z gets a kick in the plus z direction. And any electron that came from this term in the superposition will be kicked up up z. And any electron that came from the superposition down z will have down. Now, what we really mean is not that an electron does this or does that, but rather that the initial stage of an electron that's here, with the superposition of z up and down, ends up in the state as a superposition of up z being up here and down z being down here. OK? An electron didn't do one, it didn't do the other. It ends up in a superposition, so the state at the end. So what the Stern-Gerlach experiment has done, apparatus has done, is it's correlated the position of the electron with its spin. So if you find the amplitude, to find it up here and down is very small. The amplitude to find it up here and up is very large. Similarly, the amplitude, to find it down here and up is 0. And the amplitude, to find it down here and down is large. Cool? So this is exactly what we wanted from the boxes. We wanted not to do something funny to the spins, we just wanted to correlate the position with the spin. And so the final state is a superposition of these guys. And which superposition? It's exactly this superposition. So these calculations are gone through in the notes that are going to be posted. OK. So questions at this point? Yeah? AUDIENCE: And so when you put the electron through the Stern-Gerlach device, does that count as a measurement of the particle's angular momentum? PROFESSOR: Excellent. When I put the electrons through the Stern-Gerlach device, do they come out with a definite position? AUDIENCE: No. PROFESSOR: No. At the end of the experiment, they're in a superposition of either being here or being here. And they're in a superposition of either being up spin or down spin. Have we determined the spin through putting it through this apparatus? No. We haven't done any measurement. The measurement comes when we now do the following. We put a detector here that absorbs an electron. We say, ah, yeah, it got hit. And then you've measured the angular momentum by measuring where it came out. If it comes out down here, it will be in the positive. So this is a nice example of something called entanglement. And this is where we're going to pick up next time. Entanglement says the following. Suppose I know one property of a particle, for example, that it's up here, or suppose I'm in the state psi is equal to up, so up here and up in the z direction plus down there and down in the z direction. OK? This means that, at the moment, initially, if I said, look, is it going to be up or down with equal probabilities, 1 over root 2 with equal amplitudes, if I measure spin in the z direction, what value will I get? What values could I get if I measure spin in the z direction? Plus or minus 1/2. And what are the probabilities that I measure plus 1/2 or minus 1/2? AUDIENCE: [INAUDIBLE] PROFESSOR: Even odds, right? We've done that experiment. On the other hand, if I tell you that I've measured it to be up here, what will you then deduce about what its spin is? AUDIENCE: Plus. PROFESSOR: Always plus 1/2. Did I have to do the measurement to determine that it's plus 1/2. No, because I already know the state, so I know exactly what I will get if I do the experiment. I measure up here, and then the wave function is this without the 1 over root 2, upon measurement. But as a consequence, if I subsequently measure up z, the only possible value is up in the z direction. Yeah? And this is called entanglement. And here it's entanglement of the position with the spin. And next time, what we'll do is we'll study the EPR experiment, which says the following thing. Suppose I take two electrons, OK, take two electrons, and I put them in the state one is up and the other is down, or the first is up and the second is down. All right? So up, down plus down, up. OK? I now take my two electrons, and I send them to distant places. Suppose I measure one of them to be up in the x direction. Yeah? Then I know that the other one is down in that direction. Sorry. If I measure up in the z direction, I determine the other one is down in the z direction. But suppose someone over here who's causally disconnected measures not spin in the z direction, but spin in the x direction. They'll measure one of two things, either plus or minus. Now, knowing what we know, that it's in the z eigenstate, it will be either plus 1/2 in the x direction or minus 1/2 in the x direction. Suppose I get both measurements. The distant person over here does the measurement of z and says, aha, mine is up, so the other one must be down. But the person over here doesn't measure Sz, they measure Sx, and they get that it's plus Sx. And what they've done as a result of these two experiments, Einstein, Podolsky, and Rosen say, is this electron has been measured by the distant guy to be spin z down, but by this guy to be measured spin x up. So I know Sx and Sz definitely. But that flies in the face of the uncertainty relation, which tells us we can't have spin z and spin x definitely. Einstein and Podolsky and Rosen say there's something missing in quantum mechanics, because I can do these experiments and determine that this particle is Sz down and Sx up. But what quantum mechanics says is that I may have done the measurement of Sz, but that hasn't determined anything about the state over here. There is no predetermined value. What we need to do is we need to tease out, we need an experimental version of this tension. The experimental version of this tension, fleshed out in the EPR experiment, is called Bell's inequality. We studied it in the very first lecture, and we're going to show it's a violation in the next lecture. See you guys next time. [APPLAUSE]
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_9_Operator_Methods_for_the_Harmonic_Oscillator.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So it's a beautiful recording. OK, so to get started, questions? From last time? Barton covered for me last time. I fled. I was out of town. I was at a math conference. It was pretty surreal. Questions, yes. AUDIENCE: What can you tell us about the exam? PROFESSOR: About--? AUDIENCE: The exam. PROFESSOR: The exam. Yes, absolutely. So the exam is, as you all know, on Thursday, a week hence. So on Tuesday we will have a lecture. The material Tuesday will not be covered on the exam. The exam will be a review of everything through today's lecture, including the problems that, which for some technical reason I don't know why didn't get posted. But it should be up after lecture today. The exam will be a combination of short questions and computations. It will not focus on an enormous number of computations. It will focus more on conceptual things. But there will be a few calculations on the exam. And I will post some practice problems over the next couple of days. AUDIENCE: Do we have a problem set due next week? PROFESSOR: You do have a problem set due Tuesday. And that is part of your preparation for the exam. Here's a basic strategy for exams for this class. Anything that's on a problem set is fair game. Anything that's not covered on a problem set is not going to be fair game. If you haven't seen a new problem on it, broadly construed, then you won't-- I won't test you on a topic you haven't done problems on before. But I will take problems and ideas that you've studied before and spin them slightly differently to make you think through them in real time on the exam. OK? From my point of view, the purpose of these exams is not to give you a grade. I don't care about the grade. The purpose of these exams is to give you feedback on your understanding. It's very easy to slip through quantum mechanics and think, oh yeah, I totally-- I got this. This is fine. But it's not always an accurate read. So that's the point. Did that answer your question? Other questions? Exam or-- yeah. AUDIENCE: About the harmonic oscillator actually. PROFESSOR: Excellent. AUDIENCE: So when we solved it Tuesday using the series method, so there are two solutions technically, the even solution and the odd term solution. So did boundary conditions force the other one to be completely zero, like the coefficient in front of it? So there's like an A0 term which determines all the other ones. But there's an A0 term and an A1 term for the evens and odds. So did the other ones just have to be 0? PROFESSOR: This is a really good question. This is an excellent question. Let me ask the question slightly differently. And tell me if this is the same question. When we wrote down our differential equation-- so last time we did the harmonic oscillator. And Barton did give you the brute force strategy for the harmonic oscillator. We want to find the energy eigenstates, because that's what we do to solve the Schrodinger equation. And we turn that into a differential equation. And we solve this differential equation by doing an asymptotic analysis and then a series expansion. Now, this is a second order differential equation. Everyone agree with that? It's a second order differential equation. However, in our series expansion we ended up with one integration constant, not two. How does that work? How can it be that there was only one integration constant and not two? It's a second order differential equation. Is this he question? AUDIENCE: Yeah. PROFESSOR: OK, and this is an excellent question. Because it must be true, that there are two solutions. It cannot be that there is just one solution. It's a second order differential equation. Their existence in uniqueness theorems, which tell us there are two integration constants. So how can it possibly be that there was only one? Well, we did something rather subtle in that series expansion. For that series expansion there was a critical moment, which I'm not going to go through but you can come to my office hours again, but just look through the notes. There's an important moment in the notes when we say, aha, these terms matter. But what we did is we suppressed a singular solution. There's a solution of that differential equation which is not well-behaved, which is not smooth, and in particular which diverges. And we already did, from the asymptotic analysis, we already fixed that the asymptotic behavior was exponentially falling. But there's a second solution which is exponentially growing. So what we did, remember how we did this story? We took our wave function and we said, OK, look, we're going to pull off-- we're going to first asymptotic analysis. And asymptotic analysis tells us that either we have exponentially growing or exponentially shrinking solutions. Let's pick the exponentially shrinking solutions. So phi e is equal to e to the minus x over 2a squared squared, times some-- I don't remember what Barton called it. I'll call it u of x. So we've extracted, because we know that asymptotically it takes this form. Well, it could also take the other form. It could be e to the plus, which would be bad and not normalizable. We've extracted that, and then we write down the differential equation for u. And then we solve that differential equation by series analysis, yeah? However, if I have a secondary differential equation for phi, this change of variables doesn't change the fact that it's a secondary differential equation for u, right? There's still two solutions for u. One of those solutions will be the solution of the equation that has this asymptotic form. But the other solution will be one that has an e to the plus x squared over a squared so that it cancels off this leading factor and gives me the exponentially growing solution. Everyone cool with that? So in that series analysis there's sort of a subtle moment where you impose that you have the convergent solution. So the answer of, why did we get a first order relation, is that we very carefully, although it may not have been totally obvious, when doing this calculation one carefully chooses the convergent solution that doesn't have this function blowing up so as to overwhelm the envelope. That answer your question? AUDIENCE: Yep. PROFESSOR: Great. It's a very good question. This is an important subtlety that comes up all over the place when you do asymptotic analysis. I speak from my heart. It's an important thing in the research that I'm doing right now, getting these sorts of subtleties right. It can be very confusing. It's important to think carefully through them. So it's a very good question. Other questions before we move on? OK. So I'm going to erase this, because it's not directly germane, but it is great. OK, so one of the lessons of this brute force analysis was that we constructed the spectrum, i.e., the set of energy eigenvalues allowed for the quantum harmonic oscillator, and we constructed the wave functions. We constructed the wave functions by solving the differential equation through asymptotic analysis, which give us the Gaussian envelope, and series expansion, which give us the Hermite polynomials. And then there's some normalization coefficient. And then we got the energy eigenvalues by asking, when does this series expansion converge? When does it, in fact truncate, terminate, so that we can write down an answer? And that was what gave us these discrete values. But fine, we can see that it would be discrete values. We're cool with that. In fact, Barton went through the discussion of the node theorem and the lack of degeneracy in one dimensional quantum mechanics. So it's reasonable that we get a bunch of discrete energy eigenvalues, as we've talked about now for two lectures. However, there's a surprise here, which is that these aren't just discrete, they're evenly spaced. We get a tower, starting with the lowest possible energy corresponding to a-- sorry, E0-- starting with the lowest possible energy, which is greater than 0, and a corresponding ground state wave function. And then we have a whole bunch of other states, phi 1, phi 2, phi 3, phi 4, labeled by their energies where the energies are evenly spaced. They needed to be discrete, because these are bound states. But evenly spaced is a surprise. So why are they evenly spaced? Anyone, based on the last lecture's analysis? Yeah, you don't have a good answer to that from last lecture's analysis. It's one of the mysteries that comes out of the first analysis. When you take a differential equation, you just beat the crap out of it with a stick by solving it. With differential equations strategies like this you don't necessarily get some of the more subtle structure. One of the goals of today's lecture is going to be to explain why we get this structure. Why just from the physics of the problem, the underlying physics, should you know that the system is going to have evenly spaced eigenvalues? What's the structure? And secondly, I want to show you a way of repeating this calculation without doing the brute force analysis that reveals some of that more fine grain structure of the problem. And this is going to turn out to be one of the canonical moves in the analysis of quantum mechanical systems. So from quantum mechanics to quantum field theory this is a basic series of logic moves. What I'm going to do today also has an independent life in mathematics, in algebra. And that will be something you'll studying in more detail in 8.05, but I would encourage you to ask your recitation instructors about it, or me in office hours. So our goal is to understand that even spacing and also to re-derive these results without the sort of brutal direct assault methods we used last time. So what I'm going to tell you about today is something called the operator method. It usually goes under the name of the operator method. To get us started let's go back to look at the energy operator for the harmonic oscillator, which is what, at the end of the day, we want to solve. p squared over 2m plus m omega squared upon 2, x squared. And this is the operator that we want to-- whose eigenvalues we want to construct, whose eigenfunctions we want to construct. Before we do anything else, we should do dimensional analysis. First thing you when you look at a problem is do some dimensional analysis. Identify the salient scales and make things, to the degree possible, dimensionless. Your life will be better. So what are the parameters we have available to us? We have h bar, because it's quantum mechanics. We have m, because we have a particle of mass, m. We have omega, because this potential has a characteristic frequency of omega. What other parameters do we have available to us? Well, we have c. That's available to us. But is it relevant? No. If you get an answer that depends on the speed of light, you made some horrible mistake. So not there. What about the number of students in 8.04? No. There are an infinite number of parameters that don't matter to this problem. What you want to know is, when you do dimensional analysis, what parameters matter for the problem. What parameters could possibly appear during the answer? And that's it. There are no other parameters in this problem. So that's a full set of parameters available to us. This has dimensions of momentum times length. This has dimensions of mass, and this has dimensions of one upon the time. And so what characteristic scales can we build using these three parameters? Well, this is a moment times a length. If we multiply by a mass, that's momentum times mass times x, which is almost momentum again. We need a velocity and not a position, but we have 1 over time. So if we take h bar times and omega, so that's px times m over t, that has units of momentum squared. And similarly, this is momentum which is x, which is length mass over time. I can divide by mass and divide by frequency or multiply by time, so h bar upon m omega. And this is going to have units of length squared. And with a little bit of foresight from factors of two I'm going to use these to define two link scales. x0 is equal to h bar-- I want to be careful to get my coefficients. I always put the two in the wrong place. So 2 h bar upon m omega. Square root. And I'm going to define p0 as equal to square root of 2 h bar times m omega. So here's my claim. My claim is at the end of the day the salient link scales for this problem should be integers or dimensionless numbers times this link scale. And salient momentum scales should be this scale. Just from dimensional analysis. So if someone at this point says, what do you think is the typical scale, what is the typical size of the ground state wave function, the typical link scale over which the wave function is not 0? Well, that can't possibly be the size of Manhattan. It's not the size of a proton. There's only one link scale associated with the system. It should be of order x0. Always start with dimensional analysis. Always. OK, so with that we can rewrite this energy. Sorry, and there's one last energy. We can write an energy, the thing with energy, which is equal to h bar omega. And this times a frequency gives us an energy. So we can rewrite this energy operator as h bar omega times p squared over p0. So this has units of energy. So everything here must be dimensionless. And it turns out to be p squared over p0 squared plus x operator squared over x0 squared. So that's convenient. So this has nothing to do with the operator method. This is just being reasonable. Quick thing to note, x0 times p0 is just-- the m omegas cancel, so we get root 2h bar squared, 2 h bar. Little tricks like that are useful to keep track of as you go. So we're interested in this energy operator. And it has a nice form. It's a sum of squares. And we see the sum of squares, a very tempting thing to do is to factor it. So for example, if I have two classical numbers, c squared plus d squared, the mathematician in me screams out to write c minus id times c plus id. I have factored this. And that's usually a step in the right direction. And is this true? Well yes, it's true. c squared plus d squared and the cross terms cancel. OK, that's great. Four complex numbers, or four C numbers. Now is this true for operators? Can I do this for operators? Here we have the energy operator as a sum of squares. Well, let's try it. I'd like to write that in terms of x and p. So what about writing the quantity x minus ip over x0 over p0, operator, times x over x0 plus ip over p0. We can compute this. This is easy. So the first term gives us the x squared over x0 squared. That last term gives us-- the i's cancel, so we get p squared over p0, squared. But then there are cross terms. We have an xp and a minus px, with an overall i. So plus i times xp over x0 times p0. x0 times p0, however, is 2 h bar. So that's over 2 h bar. And then we have the other term, minus px. Same thing. So I could write that as a commutator, xp minus px. Everyone cool with what? Unfortunately this is not what we wanted. We wanted just p squared plus x squared. And what we got instead was p squared plus x squared close plus a commutator. Happily this commutator is simple. What's the commutator of x with p? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah, exactly. Commit this to memory. This is your friend. So this is just i h bar, so this is equal to ditto plus ditto plus i h bar. Somewhere I got a minus sign. Where did I get my minus sign wrong? x with ip. Oh now, good. This is good. So x with p is i-- so we get an i h bar. No, I really did screw up the sign. How did I screw up the sign? No I didn't. Wait. Oh! Of course. No, good. Sorry, sorry. Trust your calculation, not your memory. So the calculation gave us this. So what does this give us? It gives us i h bar. So plus. But the i h bar times i is going to give me a minus. And the h bar is going to cancel, because I've got an h bar from here and an h bar at the denominator minus 1/2. So this quantity is equal to the quantity we wanted minus 1/2. And what is the quantity we wanted, x0 squared plus p0 squared? This guy. So putting that all together we can write that the energy operator, which was equal to h bar omega times the quantity we wanted, is equal to-- well, the quantity we wanted is this quantity plus 1/2. h bar omega times-- I'll write this as x over x0 plus ip over p0, x over x0 plus ip over p0, hat, hat, hat, plus 1/2. Everyone cool with that? So it almost worked. We can almost factor. So at this point it's tempting to say, well that isn't really much an improvement. You've just made it uglier. But consider the following. And just trust me on this one, that this is not a stupid thing to do. That's a stupid symbol to write, though. So let's define an operator called a, which is equal to x over x0 plus ip over p0, and an operator, which I will call a dagger. "Is that a dagger I see before me?" Sorry. x over x0 minus ip over p0. Hamlet quotes are harder. So this is a dagger. And we can now write the energy operator for the harmonic oscillator is equal to h bar omega times a dagger a plus 1/2. Everyone cool with that? Now, this should look suggestive. You should say, aha, this looks like h bar omega something plus 1/2. That sure looks familiar from our brute force calculation. But, OK, that familiarity is not an answer to the question. Meanwhile you should say something like this. Look, this looks kind of like the complex conjugate of this guy. Because there's an i and you change the sign of the i. But what is the complex conjugate of an operator? What does that mean? An operator is like take a vector and rotate. What is the complex conjugation of that? I don't know. So we have to define that. So I'm now going to start with this quick math aside. And morally, this is about what is the complex conjugate of an operator. But before I move on, questions? OK. So here's a mathematical series of a facts and claims. I claim the following. Given any linear operator we can build-- there's a natural way to build without making any additional assumptions or any additional ingredients. We can build another operator, o dagger, hat, hat, in the following way. Consider the inner product of f with g, or the bracket of f with g. So integral dx of f complex conjugate g. Consider the function we're taking here is actually the operator we have on g. I'm going to define my-- so this is a perfectly good thing. What this expression says is, take your function g. Act on it with the operator o. Multiply by the complex conjugate. Take the integral. This is what we would have done if we had taken the inner product of f with the function we get by taking o and acting on the function g. So here's the thing. What we want-- just an aside-- what we want to do is define a new operator. And here's how I'm going to define it. We can define it by choosing how it acts. I'm going to tell you exactly how it acts, and then we'll define the operator. So this operator, o with a dagger, called the adjoint, is defined in the following way. This is whatever operator you need, such that the integral gives you-- such that the following is true. Integral dx o on f, complex conjugate, g. So this is the definition of this dagger action, the adjoint action. OK so o dagger is the adjoint. And sometimes it's called the Hermetian adjoint. I'll occasionally say Hermetian and occasionally not, with no particular order to it. So what does this mean? This means that whatever o dagger is, it's that operator that when acting on g and then taking the inner product with f gives me same answer as taking my original operator and acting on f and taking the inner product with g. Cool? So we know how-- if we know what our operator o is, the challenge now is going to be to figure out what must this o dagger operator be such that this expression is true. That's going to be my definition of the adjoint. Cool? So I'm going to do a bunch of examples. I'm going to walk through this. So the mathematical definition is that an operator o defined in this fashion is the Hermetian adjoint of o. So that's the mathematical definition. Well, that's our version of the mathematical definition. I just came back from a math conference, so I'm particularly chastened at the moment to be careful. So let's do some quick examples. Example one. Suppose c is a complex number. I claim a number is also an operator. It acts by multiplication. The number 7 is an operator because it takes a vector and it gives you 7 times that vector. So this number is a particularly simple kind of operator. And what's the adjoint? We can do that. That's easy. So c adjoint is going to be defined in the following way. It's integral dx f star of c adjoint g is equal to the integral dx of c on f complex conjugate times g. But what is this? Well, c is just a number. So when we take its complex conjugate we can just pull it out. So this is equal to the integral dx c complex conjugate, f star g. But I'm now going to rewrite this, using the awesome power of reordering multiplication, as c star. And I'm going to put parentheses around this because it seems like fun. So now we have this nice expression. The integral dx of f c adjoint g is equal to the integral dx of f c star g, c complex conjugate g. But notice that this must be true for all f. It's true for all. Because I made no assumption about what f and g are, true for all f and g. And therefore the adjoint of a complex number is its complex conjugate. And this is the basic strategy for determining the adjoint of any operator. We're going to play exactly this sort of game. We'll put the adjoint in here. We'll use the definition of the adjoint. And then we'll do whatever machinations are necessary to rewrite this as some operator acting on the first factor. Cool? Questions? OK, let's do a more interesting operator. By the way, to check at home, and I think this might be on your problem set-- but I don't remember if it's on or not. So if it's not, check this for yourself. Check that the adjoin of the adjoint is equal to the operator itself. It's an easy thing to check. So next example. What is the adjoint of the operator derivative with respect to x? Consider the operator, which is just derivative with respect to x. And I want to know what is the adjoint of this beast. So how do we do this? Same logic as before. Whatever the operator is, it's defined in the following way. Integral dx, f complex conjugate on dx dagger on g. This is equal to the integral dx of-- how we doing on time? Good-- integral dx of dx, f complex conjugate on g. Now, what we want is we want to turn this into an expression where the operator is acting on g, just as our familiar operator ddx. So how do I get the ddx over here? I need to do two things. First, what's the complex conjugate of the derivative with respect to x of a complex function? MIT has indigestion. So this is integral dx, derivative with respect to x of f complex conjugate, g. And now I want this operator. I want derivative acting on g. That's the definition. Because I want to know what is this operator. And so I'm going to do integration by parts. So this is equal to the integral, dx. When I integrate by parts I get an F complex conjugate, and then an overall minus sign from the integration by parts minus f complex conjugate dx g. Was I telling you the truth earlier? Or did I lie to you? OK, keep thinking about that. And this is equal to, well the integral of dx, f complex conjugate if minus the derivative with respect to x acting on g. Everyone cool with that? But if you look at these equalities, dx adjoint acting on g is the same as minus dx acting on g. So this tells me that the adjoint of dx is equal to minus dx. Yeah? AUDIENCE: Are you assuming that your surface terms vanish? PROFESSOR: Thank you! I lied to you. So I assumed in this that my surface terms vanished. I did a variation by parts. And that leaves me with a total derivative. And that total derivative gives me a boundary term. Remember how integration by parts works. Integration by parts says the integral of AdxB is equal to the integral of-- well, AdxB can be written as derivative with respect to x of AB minus B derivative with respect to x of A. Because this is A prime B plus B prime A. Here we have AB prime. So we just subtract off the appropriate term. But this is a total derivative. So it only gives us a boundary term. So this integral is equal to-- can move the integral over here-- the integral and the derivative, because an integral is nothing but an antiderivative. The integral and the derivative cancel, leaving us with the boundary terms. And in this case, it's from our boundaries which are minus infinity plus infinity, minus infinity and plus infinity. Now, this tells us something very important. And I'm not going to speak about this in detail, but I encourage the recitation instructors who might happen to be here to think to mention this in recitation. And I encourage you all to think about it. If I ask you, what is the adjoint of the derivative operator acting on the space of functions which are normalizable, so that they vanish at infinity, what is the adjoint of the derivative operator acting on the space of functions which is normalizable at infinity? We just derive the answer. Because we assume that these surface terms vanish. Because our wave functions, f and g, vanish at infinity. They're normalizable. However, if I had asked you a slightly different question, if I had asked you, what's the adjoint of the derivative operator acting on a different set of functions, the set of functions that don't necessarily vanish at infinity, including sinusoids that go off to infinity and don't vanish. Is this the correct answer? No. This would not be the correct answer, because there are boundary terms. So the point I'm making here, first off, in physics we're always going to be talking about normalizable beasts. At the end of the day, the physical objects we care about are in a room. They're not off infinity. So everything is going to be normalizable. That is just how the world works. However, you've got to be careful in making these sorts of arguments and realize that when I ask you, what is the adjoint of this operator, I need to tell you something more precise. I need to say, what's the adjoint of the derivative acting when this operator's understood as acting on some particular set of functions, acting on normalizable functions? Good. So anyway, I'll leave that aside as something to ponder. But with that technical detail aside, as long as we're talking about normalizable functions so these boundary terms from the integration by parts cancel, the adjoint of the derivative operator is minus the derivative operator. Cool? OK, let's do another example. And where do I want to do this? I'll do it here. So another example. Actually, no. I will do it here. So we have another example, which is three. What's the adjoint of the position operator? OK, take two minutes. Do this on a piece of paper in front of you. I'm not going to call on you. So you can raise you hand if you-- OK, chat with the person next to you. I mean chat about physics, right? Just not-- [LAUGHS]. AUDIENCE: [CHATTING] PROFESSOR: OK, so how do we go about doing this? We go about solving this problem by using the definition of the adjoint. So what is x adjoint? It's that operator such that the following is true, such that the integral dx of f complex conjugate with x dagger acting on g is equal to the integral dx of what I get by taking the complex conjugate of taking x and acting on f and then integrating this against g. But now we can use the action of x and say that this is equal to the integral dx of x f complex conjugate g. But here's the nice thing. What is the complex conjugate of f times the complex function of f? x is real. Positions are real. So that's just x times the complex conjugate of f. So that was essential move there. And now we can rewrite this as equal the integral of f complex conjugate xg. And now, eyeballing this, x dagger is that operator which acts by acting by multiplying with little x. Therefore, the adjoint of the operator x is equal to the same operator. x is equal to its own adjoint. OK? Cool? So we've just learned a couple of really nice things. So the first is-- where we I want to do this? Yeah, good. So we've learned a couple of nice things. And I want to encode them in the following definition. Definition-- an operator, which I will call o, whose adjoint is equal to o, so an operator whose adjoint is equal to itself is called Hermetian. So an operator which is equal to its own adjoint is called Hermetian. And so I want to note a couple of nice examples of that. So note a number which is Hermetian is what? Real. An operator-- we found an operator which is equal to its own adjoint. x dagger is equal to x. And what can you say about the eigenvalues of this operator? They're real. We use that in the proof, actually. So this is real. I will call an operator real if it's Hermetian. And here's a mathematical fact, which is that any operator which is Hermetian has all real eigenvalues. So this is really-- I'll state it as a theorem, but it's just a fact for us. o has all real eigenvalues. AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah? AUDIENCE: Is it if and only [INAUDIBLE]? PROFESSOR: No. Let's see. If you have all real eigenvalues, it does not imply that you're Hermetian. However, if you have all real eigenvalues and you can be diagonalized, it does imply. So let me give you an example. So consider the following operator. We've done this many times, rotation in real three-dimensional space of a vector around the vertical axis. It has one eigenvector, which is the vertical vector. And the eigenvalue is 1, so it's real. But that's not enough to make it Hermetian. Because there's another fact that we haven't got to yet with Hermetian operators, which is going to tell us that a Hermetian operator has as many eigenvectors as there are dimensions in the space, i.e., that the eigenvectors form a basis. But there's only one eigenvector for this guy, even though we're in a three-dimensional vector space. So this operator, rotation by an angle theta, is not Hermetian, even though its only eigenvalue isn't in fact real. So it's not an only if. If you are Hermetian, your eigenvalues are all real. And you'll prove this on a problem set. Yeah? AUDIENCE: If you're Hermetian, are your eigenfunctions normal? PROFESSOR: Not necessarily. But they can be made normal. We'll talk about this in more detail later. OK. Let's do a quick check, last example. And I'm not actually going to go through this in detail, but what about p? What about the momentum operator? First off, do you think the momentum is real? It sure would be nice. Because its eigenvalues are the observable values of momentum. And so its eigenvalues should all be real. Does that make it Hermetian? Not necessarily, but let's check. So what is the adjoint of p? Well, this actually we can do very easily. And I'm not going to go through an elaborate argument. I'm just going to know the following. p is equal to h bar upon i ddx. And this is an operator. This is an operator. So what's the adjoint of this operator? Well, this under an adjoint gets a minus sign, right? It's itself up to a minus sign. So is the derivative Hermetian? No, it's in fact what we anti-Hermetian. Its adjoint is minus itself. What about i? What's its adjoint? Minus i. Sweet. So this has an adjoint, picks up a minus. This has an adjoint, picks up a minus. The minuses cancel. p adjoint is p. So p is in fact Hermetian. And here's a stronger physical fact. So now we've seen that each of the operators we built is x and p, true of the operators we've looked at so far is Hermetian, those that correspond to physical observables. Here's a physical fact. All the observables you measure with sticks are real. And the corresponding statement is that all operators corresponding to observables, all operators must be Hermetian. To the postulate that says, "Observables are represented by operators," should be adjoined the word "Hermetian." Observables are represented in quantum mechanics by Hermetian operators, which are operators that have a number of nice properties, including they have all real eigenvalues. Cool? OK. Questions? Yeah. AUDIENCE: If it has to be Hermetian and not just have real eigenvalues, does that mean the eigenvalues always need to form some kind of basis? PROFESSOR: Yeah, the eigenvectors will. This is connected to the fact we've already seen. If you take an arbitrary wave function you can expand it in states with definite momentum as a superposition. You can also expand it in a set of states of definite energy or of definite position. Anytime you have a Hermetian operator, its eigenvectors suffice to expand any function. They provide a basis for representing any function. So that's the end of the mathematical side. Let's get back to this physical point. So we've defined this operator a and this other operator a dagger. And here's my question first. Is a Hermetian? No. That's Hermetian. That's Hermetian. But there's an i. That i will pick up the minus sign when we do the complex conjugation. Oh, look. Sure was fortuitous that I called this a dagger, since this is equal to a dagger. So this is the adjoint of a. So this immediately tells you something interesting. x and p are both observables. Does a correspond to an observable? Is it Hermetian? Every intervals is associated to a Hermetian operator. This is not Hermetian. So a does not represent an observable operator. And I will post notes on the web page, which give us a somewhat lengthy discussion-- or it might be in one of the solutions-- a somewhat lengthy discussion of what it means for a and a dagger to not be observable. You'll get more discussion of that there. Meanwhile, if a is not observable, it's not Hermetian, does it have real eigenvalues? Well, here's an important thing. I said if you're Hermetian, all the eigenvalues are real. If you're not Hermetian, that doesn't tell you you can't have any real eigenvalues. It just says that I haven't guaranteed for you that all the eigenvalues are real. So what we'll discover towards the end of the course when we talk about something called coherent states is that in fact, a does have a nice set of eigenvectors. They're very nice. They're great. We use them for lasers. They're very useful. And they're called coherent states. But their eigenvalues are not in general real. They're generically complex numbers. Are they things you can measure? Not directly. They're related to things you can measure, though, in some pretty nice ways. So why are we bothering with these guys if they're not observable? Yeah. AUDIENCE: E [INAUDIBLE]. PROFESSOR: Yeah, good. Excellent. That's really good. So two things about it. So one thing is this form for the energy operator is particularly simple. We see the 1/2. This looks suggestive from before. But it makes it obvious that E is Hermetian. And that may not be obvious to you guys. So let's just check. Here's something that you'll show on the problem set. AB adjoint is equal to B adjoint A adjoint. The order matters. These are operators. And so if we take the adjective of this, what's this going to give us? Well we change the order. So it's going to be a dagger, and then we take the dagger of both of the a dagger. So this is self-adjoint, or Hermetian. So that's good. Of course, we already knew that, because we could have written it in terms of x and p. But this is somehow simpler. And it in particular emphasizes the form, or recapitulates the form of the energy eigenvalues. Why else would we care about a and a dagger? OK, now this is a good moment. Here's the second reason. So the first reason you care is this sort of structural similarity and the fact that it's nicely Hermetian in a different way. Here's the key thing. Key. a and a dagger satisfy the simplest commutation relation in the world. Well, the second simplest. The simplest is that it's 0 on the right-hand side. But the simplest not trivial commutation relationship. a with a dagger is equal to-- so what is a dagger equal to? We just take the definition. Let's put this in. So this is x over x0 plus ip over p0, comma, x over x0 minus i, p over p0, hat, hat, hat, hat, bracket, bracket. Good. So here there are going to be four terms. There's x commutator x. What is that? What is the commutator of an operator with itself? 0. Because remember the definition of the commutator A, B is AB minus BA. So A with A is equal to AA minus AA. And you have no options there. That's 0. So x with x is 0. p with p is 0. So the only terms that matter are the cross terms. We have an x with p. And notice that's going to be times a minus i with p0 and x0. And then we have another term which is p with x, which is i, p0 over x0. So you change the order and you change the sign. But if you change the order of a commutator, you change the side. So we can put them both in the same order. Let me just write this out. So this is i over x0 p0. So this guy, minus i over x0p0. But x0p0 is equal to 2h bar, as we checked before. This was x with p. And then the second term was plus i, again over x0p0, which is 2h bar, p with x. This x with p is equal to? i h bar. So the h bar cancels. The i gives me a plus 1. And p with x gives me minus i h bar. So the h bar and the minus i gives me plus 1. Well that's nice. This is equal to 1. So plus 1/2, therefore a with a dagger is equal to 1. As advertised, that is about as simple as it gets. Notice a couple of other commutators that follow from this. a dagger with a is equal to minus 1. We just changed the order. And that's just an overall minus sign. And a with a is what? 0. a dagger with a dagger? Good. OK. So we are now going to use this commutation relation to totally crush the problem into submission. It's going to be weeping before us like the Romans in front of the Visigoths. It's going to be dramatic. OK, so let's check. So let's combine the two things. So we had the first thing is that this form is simple. The second is that the commutator is simple. Let's combine these together and really milk the system for what it's got. And to do that, I need two more commutators. And the lesson of this series of machinations, it's very tempting to look at this and be like, why are you doing this? And the reason is, I want to encourage you to see the power of these commutation relations. They're telling you a tremendous amount about the system. So we're going through and doing some relatively simple calculations. We're just computing commutators. We're following our nose. And we're going to derive something awesome. So don't just bear with it. Learn from this, that there's something very useful and powerful about commutation relations. You'll see that at the end. But I want you to on to the slight awkwardness right now, that it's not totally obvious beforehand where this is going. So what is E with a? That's easy. It's the h bar omega a dagger a plus 1/2. So the 1/2, what's 1/2 commutator with an operator? 0. Because any number commutes with an operator. 1/2 operator is operator 1/2. It's just a constant. That term is gone. So the only thing that's left over is h bar omega, a dagger a with a. The h bar omega's just a constant. It's going to pull out no matter which term we're looking at. So I could just pull that factor out. So this is equal to h bar omega times a dagger a minus a a dagger a. But this is equal to h bar omega-- well, that's a dagger a a, a a dagger a. You can just pull out the a on the right. a dagger a minus a a dagger a. That's equal to h bar omega. Well, a dagger with a is equal to a dagger with a minus 1 is equal to minus h bar omega. And we have this a leftover, a. So E with a is equal to minus a. Well, that's interesting. Now, the second commutator-- I'm not going to do it-- E with a dagger is going to be equal to-- let's just eyeball what's going to happen. They can be a dagger. So we're going to have a dagger a dagger minus a dagger a dagger a. So we're going to have an a dagger in front and then a dagger. So all we're going to get is a sign. And it's going to be a dagger plus a dagger. I shouldn't written that in the center. Everyone cool with that? Yeah. AUDIENCE: The h bar where? PROFESSOR: Oh shoot, thank you! h bar here. Thank you. We would have misruled the galaxy. OK, good. Other questions? You don't notice-- you haven't noticed yet, but we just won. We just totally solved the problem. And here's why. Once you see this, any time you see this, anytime you see this commutator, an operator with an a is equal to plus a times some constant, anytime you see this, cheer. And here's why. Yeah, right. Exactly. Now. Whoo! Here's why. Here's why you should cheer. Because you no longer have to solve any problems. You no longer have to solve any differential equations. You can simply write down the problem. And let's see that you can just write down the answer. Suppose that we already happened to have access-- here in my sleeve I have access to an eigenfunction of the energy operator. E on phi E is equal to E phi E. Suppose I have this guy. Cool? Check this out. Consider a new state, psi, which is equal to a-- which do I want to do first? Doesn't really matter, but let's do a. Consider psi is equal to a on phi E. What can you say about this state? Well, it's the state you get by taking this wave function and acting with a. Not terribly illuminating. However, E on psi is equal to what? Maybe this has some nice property under acting with E. This is equal to E on a with pfi E. Now, this is tantalizing. Because at this point it's very-- look, that E, it really wants to hit this phi. It just really wants to. There's an E it wants to pull out. It'll be great. The problem is it's not there. There's an a in the way. And so at this point we add 0. And this is a very powerful technique. This is equal to Ea minus aE plus aE, phi E. But that has a nice expression. This is equal to Ea minus aE. That's the commutator of E with a. Plus a. What's E acting on phi E? Actually, let me just leave this as aE. So what have we done here before we actually act? What we've done is something called commuting an operator through. So what do I mean by commuting an operator through? If we have an operator A and an operator B and a state f, and I want A to act on f, I can always write this as-- this is equal to the commutator of A would be plus BA acting on f. So this lets me act A on f directly without B. But I have to know what the commutator of these two operators is. So if I know what the commutator is, I can do this. I can simplify. When one does this, when one takes AB and replaces it by the commutator of A with B, plus BA, changing the order, the phrase that one uses is I have commuted A through B. And commuting operators through other is an extraordinarily useful tool, useful technique. Now let's do y. So here what's the commutator of E with a? We just did that. It's minus h bar omega a. And what's aE on phi E? What's E on phi E? E. Exactly. Plus Ea on phi. And now we're cooking with gas. Because this is equal to minus h bar omega a plus Ea, hat. I'm going to pull out this common factor of a. So if I pull out that common factor of a, plus E, a phi E, and now I'm going to just slightly write this instead of minus h bar omega plus E, I'm going to write this as E minus h bar omega. I'm just literally changing the order of the algebra. E minus h bar omega. And what is aE? Psi. That was the original state we started with, psi. Well, that's cool. If I have a state with energy E and I act on it with the operator a, I get a new state, psi, which is also an eigenstate of the energy operator, but with a slightly different energy eigenvalue. The eigenvalue is now decreased by h bar omega. Cool? And that is what we wanted. Let's explore the consequences of this. So if we have a state with eigenvalue E, we have phi E such that E on phi E is equal to E phi E. Then the state a phi E has eigenvalue as energy, eigenvalue E minus h bar omega. So I could call this phi sub E minus h bar omega. It's an eigenfunction of the energy operator, the eigenvalue, E minus h bar omega. Agreed? Do I know that this is in fact properly normalized? No, because 12 times it would also be a perfectly good eigenfunction of the energy operator. So this is proportional to the properly normalized guy, with some, at the moment, unknown constant coefficient normalization. Everyone cool with that? So now let's think about what this tells us. This tells us if we have a state phi E, which I will denote its energy by this level, then if I act on it with a phi E I get another state where the energy, instead of being E, is equal E minus h bar omega. So this distance in energy is h bar omega. Cool? Let me do it again. We'll tack a on phi E. By exactly the same argument, if I make psi as equal to a on a phi E, a squared phi E, I get another state, again separated by h bar omega, E minus 2h bar omega. Turtles all the way down. Everyone cool with that? Let's do a slightly different calculation. But before we do that, I want to give a a name. a does something really cool. When you take the state phi E that has definite energy E, it's an energy eigenfunction, and you act on it with a, what happens? It lowers the energy by h bar omega. So I'm going to call a the lowering operator. Because what it does is it takes a state with phi E, with energy eigenvalue E to state with energy E minus h bar omega. And I can just keep doing this as many times as I like and I build a tower. Yes? AUDIENCE: What happens when you apply a to the ground state? PROFESSOR: Very good question. Hold on to that for a second. We'll come back to that in just a second. So this seems to build for me a ladder downwards. Everyone cool with that? But we could have done the same thing with a dagger. And how does this story change? What happens if we take a dagger instead of a? Well, let's go through every step here. So this is going to be E on a dagger. And now we have E a dagger, a dagger, E, a dagger. What's E with a dagger? E with a dagger is equal to same thing but with a plus. And again, psi. Same thing, because the a dagger factors out. Yeah? So we go down by acting with a. We go up by acting with a dagger. And again, the spacing is h bar omega. And we go up by acting with a dagger again. So a and a dagger are called the raising and lowering operators. a dagger, the raising operator. a dagger phi E plus h bar omega. So what that lets us do is build a tower of states, an infinite number of states where, given a state, we can walk up this ladder with the raising operator, and we can walk down it by the lowering operator. So now I ask you the question, why is this ladder evenly spaced? There's one equation on the board that you can point to-- I guess two, technically-- there are two equations on the board that you could point to that suffice to immediately answer the question, why is the tower of energy eigenstates evenly spaced. What is that equation? AUDIENCE: Commutators? PROFESSOR: Yeah, those commutators. These commutators are all we needed. We didn't need to know anything else. We didn't even need to know what the potential was. If I just told you there's an energy operator E and there's an operator a that you can build out of the observables of the system, such that you have this commutation relation, what do you immediately know? You immediately know that you get a tower of operators. Because you can act with a and raise the energy by a finite amount, which is the coefficient of that a in the commutator. This didn't have to be the quantum mechanics of the harmonic oscillator at this point. We just needed this commutator relation, E with a, E with a dagger. And one of the totally awesome things is how often it shows up. If you take a bunch of electrons and you put them in a magnetic field, bunch of electrons, very strong magnetic field, what you discover is the quantum mechanics of those guys has nothing to do with the harmonic oscillator on the face if it's magnetic fields, Lorentz force law, the whole thing. What you discover is there's an operator, which isn't usually called a, but it depends on which book you use-- it's n or m or l-- there's an operator that commutes with the energy operator in precisely this fashion, which tells you that the energy eigenstates live in a ladder. They're called Landau levels. This turns out to be very useful. Any of you who are doing a UROP in the lab that has graphene or any material, really, with a magnetic field, then this matters. So this commutator encodes an enormous amount of the structure of the energy eigenvalues. And the trick for us was showing that we could write the harmonic oscillator energy operator in terms of operators that commute in this fashion. So we're going to run into this structure over and over again. This operator commutes with this one to the same operator times a constant that tells you have a ladder. We're going to run into that over and over again when we talk about Landau levels, if we get there. When we talk about angular momentum we'll get the same thing. When we talk about the harmonic oscillator we'll get the same thing. Sorry, the hydrogen system. We'll get the same thing. So second question, does this ladder extend infinitely up? Yeah, why not? Can it extend infinitely down? AUDIENCE: Nope. PROFESSOR: Why? AUDIENCE: Ground state. PROFESSOR: Well, people are saying ground state. Well, we know that from the brute force calculation. But without the brute force calculation, can this ladder extend infinitely down? AUDIENCE: [INAUDIBLE] you can't go under the minimum. PROFESSOR: Brilliant. OK, good. And as you'll prove on the problems, that you can't make the energy arbitrarily negative. But let me make that sharp. I don't want to appeal to something we haven't proven. Let me show you that concretely. In some state, in any state, the energy expectation value can be written as the integral of phi complex conjugate-- we'll say in this state phi-- phi complex conjugate E phi. But I can write this as the integral, and let's say dx, integral dx. Let's just put in what the energy operator looks like. So psi tilda, we can take the Fourier transfer and write the psi tilda p, p squared upon 2m-- whoops, dp-- for the kinetic energy term, plus the integral-- and now I'm using the harmonic oscillator-- plus the integral dx of psi of x norm squared, norm squared, m omega squared upon 2x squared. Little bit of a quick move there, doing the Fourier transfer for the momentum term and not doing the Fourier transform but it's OK. They're separate integrals. I can do this. And the crucial thing here is, this is positive definite. This is positive definite, positive definite, positive definite. All these terms are strictly positive. This must be greater than or equal to 0. It can never be negative. Yeah? So what that tells us is there must be a minimum E. There must be a minimum energy. And I will call it minimum E0. We can't lower the tower forever. So how is this possible? How is it possible that, look, on the one hand, if we want, if we have a state, we can always build a lower energy state by acting with lowering operator a. And yet this is telling me that I can't. There must be a last one where I can't lower it anymore. So what reaches out of the chalkboard and stops me from acting with a again? How can it possibly be true that a always lowers the eigenfunction but there's at least one that can't be lowered any further. Normalizable's a good guess. Very good guess. Not the case. Because from this argument we don't even use wave functions. AUDIENCE: P zero less than [INAUDIBLE] PROFESSOR: That would be bad. Yes, exactly. So that would be bad, but that's just saying that there's an inconsistency here. So I'm going to come back to your answer, a non-normalizable. It's correct, but in a sneaky way. Here's the way it's sneaky. Consider a state a on phi-- let's say this is the lowest state, the lowest possible state. It must be true that the resulting statement is not phi minus h bar omega. There can't be any such state. And how can that be? That can be true if it's 0. So if the lowering operator acts on some state and gives me 0, well, OK, that's an eigenstate. But it's a stupid eigenstate. It's not normalizable. It can't be used to describe any real physical object. Because where is it? Well, it's nowhere. The probability density, you'd find it anywhere. It's nowhere, nothing, zero. So the way that this tower terminates is by having a last state, which we'll call phi 0, such that lowering it gives me 0. Not the state called 0, which I would call this, but actually the function called 0, which is not normalizable, which is not a good state. So there's a minimum E0. Associated with that is a lowest energy eigenstate called the ground state. Now, can the energy get arbitrarily large? Sure. That's a positive definite thing, and this could get as large as you like. There's no problem with the energy eigenvalues getting arbitrarily large. We can just keep raising and raising and raising. I mention that because later on in the semester we will find a system with exactly that commutation relation, precisely that commutation relation, where there will be a minimum and a maximum. So the communication relation is a good start, but it doesn't tell you anything. We have to add in some physics like the energy operators bounded below for the harmonic oscillator. Questions at this point? Yeah? AUDIENCE: So you basically have shown this ladder has to exist if I'm a particular energy eigenstate and I can kind of construct a ladder. How do I know that I can't construct other, intersecting ladders? PROFESSOR: Yeah, that's an excellent question. I remember vividly when I saw this lecture in 143A, and that question plagued me. And foolishly I didn't ask it. So here's the question. The question is, look, you found a bunch of states. How do you know that's all of them? How do you know that's all of them? So let's think through that. That's a very good question. I'm not going to worry about normalization. There's a discussion of normalization in the notes. How do we know that's all of them? That's a little bit tricky. So let's think through it. Imagine it's not all of them. In particular, what would that mean? In order for there to be more states than the ones that we've written down, there must be states that are not on that tower. And how can we possi-- wow, this thing is totally falling apart. How do we do that? How is that possible? There are two ways to do it. Here's my tower of states. I'll call this one phi 0. so I raise with a dagger and I lower with a. So how could it be that I missed some states? Well, there are ways to do it. One is there could be extra states that are in between. So let's say that there's one extra state that's in between these two. Just imagine that's true. If there is such a state, by that commutation relation there must be another tower. So there must be this state, and there must be this state, and there must be this state, and there must be this state. Yeah? OK, so that's good so far. But what happens? Well, A on this guy gave me 0. And this is going to be some phi tilde 0. Suppose that this tower ends. And now you have to ask the question, can there be two different states with two different energies with a0? Can there be two different states that are annihilated by a0? Well, let's check. What must be true of any state annihilated by a0? Well, let's write the energy operator acting on that state. What's the energy of that state? Energy on phi 0 is equal to h bar omega. This is a very good question, so let's go through it. So it's equal to h bar omega times a dagger a plus 1/2 on phi 0. But what can you say about this? Well, a annihilates phi 0. It gives us 0. So in addition to a being called the lowering operator it's also called the annihilation operator, because, I don't know, we're a brutal and warlike species. So this is equal to h bar omega-- this term kills phi 0-- again with the kills-- and gives me a 1/2 half leftover. 1/2 h bar omega phi 0. So the ground state, any state-- any state annihilated by a must have the same energy. The only way you can be annihilated by a is if your energy is this. Cool? So what does that tell you about the second ladder of hidden seats that we missed? It's got to be degenerate. It's got to have the same energies. I drew that really badly, didn't I? Those are evenly spaced. So it's got to be degenerate. However, Barton proved for you the node theorem last time, right? He gave you my spread argument for the node theorem? In particular, one of the consequences of that is it in a system with bound states, in a system with potential that goes up, you can never have degeneracies in one dimension. We're not going to prove that carefully in here. But it's relatively easy to prove. In fact, if you come to my office hours I'll prove it for you. It takes three minutes. But I don't want to set up the math right now. So how many people know about the Wronskian? That's awesome. OK, so I leave it to you as an exercise to use the Wronskian to show that there cannot be degeneracies in one dimension, which is cool. Anyway, so the Wronskian for the differential equation, which is the energy eigenvalue equation. There can be no degeneracies in one dimensional potentials with bound states. So what we've just shown is that the only way that there can be extra states that we missed is if there's a tower with exactly identical energies all the way up. But if they have exactly identical energies, that means there's a degenerate. But we can prove that there can't be degeneracies in 1D. So can there be an extra tower of states we missed? No. Can we have missed any states? No. Those are all the states there are. And we've done it without ever solving a differential equation, just by using that commutation relation. Now at this point it's very tempting to say, that was just sort of magical mystery stuff. But what we really did last time was very honest. We wrote down a differential equation. We found the solution. And we got the wave functions. So, Professor Adams, you just monkeyed around at the chalkboard with commutators for a while, but what are the damn wave functions? Right? We already have the answer. This is really quite nice. Last time we solved that differential equation. And we had to solve that differential equation many, many times, different levels. But now we have a very nice thing we can do. What's true of the ground state? Well, the ground state is annihilated by the lowering operator. So that means that a acting on phi 0 of x is equal to 0. But a has a nice expression, which unfortunately I erased. Sorry about that. So a has a nice expression. a is equal to x over x0 plus ip over p0. And so if you write that out and multiply from appropriate constants, this becomes the following differential equation. The x is just multiplied by x. And the p is take a derivative with respect to x, multiply by h bar upon i. And multiplying by i over h bar to get that equation, this gives us dx plus p over h bar x0-- sorry, that shouldn't be an i. That should be p0. x on phi 0 is equal to 0. And you solved this last time when you did the asymptotic analysis. This is actually a ridiculously easy equation. It's a first order differential equation. There's one integration constant. That's going to be the overall normalization. And so the form is completely fixed. First order differential equation. So what's the solution of this guy? It's a Gaussian. And what's the width of that Gaussian? Well, look at p0 over h bar x0. We know that p0 times x0 is twice h bar. So if I multiply by x amount on the top and bottom, you get 2 h bar. The h bars cancel. So this gives me two upon x0 squared. Remember I said it would be useful to remember that p0 times x0 is 2h bar? It's useful. So it gives us this. And so the result is that phi 0 is equal to, up to an overall normalization coefficient, e to the minus x squared over x0 squared. Solid. So there. We've solved that differential equation. Is the easiest, second easiest differential equation. It's our first order differential equation with a linear term rather than a constant. We get a Gaussian. And now that we've got this guy-- look, do you remember the third Hermite polynomial? Because we know the third excited state is given by h3 times this Gaussian. Do you remember it off the top of your head? How do you solve what it is? How do we get phi 3? First off, how do we get phi 1? How do we get the next state in the ladder? How do we get the wave function? Raising operator. But what is the raising operator? Oh, it's the differential operator I take with-- OK, but if I had a dagger, it's just going to change the sign here. So how do I get phi 1? Phi 1 is equal to up to some normalization. dx minus 2 over x0 squared, x0, phi 0. So now do I have to solve the differential equation to get the higher states? No. I take derivatives and multiply by constants. So to get the third Hermite polynomial what do you do? You do this three times. This is actually an extremely efficient way-- it's related to something called the generating function, and an extremely efficient way to write down the Hermite polynomials. They're the things that you get by acting on this with this operator as many times as you want. That is a nice formal definition of the Hermite polynomials. The upshot of all of this is the following. The upshot of all this is that we've derived that without ever solving the differential equation the spectrum just from that commutation relation, just from that commutation relation-- I cannot emphasize this strongly enough-- just from the commutation relation, Ea is minus a times the constant, and Ea dagger is a dagger times the constant. We derive that the energy eigenstates come in a tower. You can move along this tower by raising with the raising operator, lowering with the lowering operator. You can construct the ground state by building that simple wave function, which is annihilated by the lowering operator. You can build all the other states by raising them, which is just taking derivatives instead of solving differential equations, which is hard. And all of this came from this commutation relation. And since we are going to see this over and over again-- and depending on how far you take physics, you will see this in 8.05. You will see this in 8.06. You will see this in quantum field theory. This shows up everywhere. It's absolutely at the core of how we organize the degrees of freedom. This structure is something you should see and declare victory upon seeing. Should see this and immediately say, I know the answer, and I can write it down. OK? In the next lecture we're going to do a review which is going to introduce a slightly more formal presentation of all these ideas. That's not going to be material covered on the exam, but it's going to help you with the exam, which will be on Thursday. See you Tuesday.
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_15_Eigenstates_of_the_Angular_Momentum_Part_1.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right. Hi, everyone. AUDIENCE: Hi. PROFESSOR: We're getting towards the end of the semester. Things are starting to cohere and come together. We have one more midterm exam. So there is an exam next Thursday, the 18th. OK? There will be a problem set due. It'll be posted later today, and it will be due next week on Tuesday as usual. Of course, next week on Tuesday is a holiday technically, so we'll actually make the due date be on Wednesday. So on Wednesday at 10 o'clock. You should think of this problem set as part of the review for the exam. Material that is covered today and on Thursday will be fair game for the exam. So the format of the exam is going to be much more canonical. It's going to be a series of short answer plus a series of computations. They'll be, roughly speaking, at the level of the problem sets, and a practice exam will be posted in the next couple of days. The practice exam is going to be of the same general intellectual difficulty, but it's going to be considerably longer than the actual exam. So as a check for yourself here's what I would recommend. I would recommend sitting down and giving yourself an hour and a half or two hours with the practice exam and see how that goes. OK? Try to give yourself the time constraint. And here's one of the things that-- there's a very strong correlation between having nailed the problem sets, and worked through all the practice exams, and just done a lot of problems, and doing well on exams. You'll be more confident. The only way to study for these things is just do a lot of problems. So on Stellar, for example, are a bunch of previous problem sets, and previous exams and practice exams from previous years. I would encourage you to look at those also on OCW and use those as practice. OK, any practical questions before we get on to the physics. Yeah? AUDIENCE: Will the exam cover all the material, or all the material [INAUDIBLE]? PROFESSOR: This is a cumulative topic. So the question is does it cover everything, or just last couple of weeks. And it's a cumulative topic, so the entire semester is necessary in order to answer the problems. Other questions? OK. Anything else? So quick couple of review questions before we launch into the main material of today. These are going to turn out to be useful reminders later on in today's lecture. So first, suppose I tell you I have a system with energy operator E, which has an operator A such that the commutator of E with A would say plus h bar A. OK? What does that tell you about the spectrum of the energy operator? AUDIENCE: It's a ladder. PROFESSOR: It's a ladder, exactly. So this tells you that the spectrum of E, or the energy eigenvalues En, are evenly spaced-- and we need a dimensional constant here, omega-- evenly spaced by h bar omega. And more precisely, that given a state phi E, we can act on it with the operator A to give us a new state, which is also an energy eigenstate, with energy E plus h bar omega. right? So any time you see that commutation relation, you know this fact to be true. Second statement. Suppose I have an operator B which commutes with the energy operator. OK? So that the commutator vanishes. What does that tell you about the system? AUDIENCE: Simultaneous eigenfunctions [INAUDIBLE]. PROFESSOR: Excellent. So one one consequence is that there exists simultaneous eigenfunctions phi sub E, B, which are simultaneous eigenfunctions of both E and B. What else does it tell you? Well, notice the following. Notice that if we took this computation relation and set omega to 0, we get this commutation relation. So this commutation relation is of the form of this commutation relation with omega equals 0. So what does that tell you about the states? About the energy eigenvalues? What happens if I take a state phi sub E and I act on it with B? What do I get? What can you say about this function? Well, what is its eigenvalue under E? It's E. It's the same thing, because they commute. You can pull the E through, multiply it, you get the constant E, you pull it back out. This is still an eigenfunction of phi of the energy operator with the same eigenvalue. But is it necessarily the same eigenfunction? No, because B may act on phi and just give you a different state. So this is some eigenfunction with the same energy, but it may be a different eigenfunction. OK? So let's think of an example of this. An example of this is consider a free particle. In the case of a free particle, we have E is equal to P squared upon 2m. And as a consequence E and P commute. Everyone agree with that? Everyone happy with that statement? They commute. So plus 0 if you will. And here what I wanted to say is when you have an operator that commutes with the energy, then there can be multiple states with the same energy which are different states, right? Different states entirely. So for example, in this case, what are the energy eigenfunctions? Well, e to the ikx. And this has energy h bar squared k squared upon 2m. But there's another state, which is a different state, which has the same energy. e to the minus ikx. OK? So when you have an operator that commutes with the energy operator, you can have simultaneous eigenfunctions. And you can also have multiple eigenfunctions that have the same energy eigenvalue, but are different functions. For example, this. Everyone cool with that? Now, just to make clear that it's actually the commuting that matters, imagine we took not the free particle, but the harmonic oscillator. E is p squared upon 2m plus m omega squared upon 2 x squared. Is it true that E-- I'll call this harmonic oscillator-- does E harmonic oscillator commute with P? No, you're going to get a term that's m omega squared x from the commutator. The potential is not translationally invariant. It does not commute with momentum. So this is not equal to 0. So that suggests that this degeneracy, two states having the same energy, should not be present. And indeed, are the states degenerate for the harmonic oscillator. No. No degeneracy. Yeah? AUDIENCE: We did something [INAUDIBLE] where I think it said that the eigenfunctions were complete. PROFESSOR: Yes. AUDIENCE: What does that mean? PROFESSOR: What does it mean for the eigenfunctions to be complete? What that means is that they form a basis. AUDIENCE: So the basis doesn't necessarily mean not [INAUDIBLE]. PROFESSOR: Yeah, no one told you the basis had to be degenerate, and in particular, that's a excellent-- so the question here is, wait a minute, I thought a basis had to be a complete set-- if you had an energy operator and you constructed the energy eigen-- this is a very good question. Thank you. If I have the energy operator and I construct it's energy eigenfunctions, then those energy eigenfunctions form a complete basis for any arbitrary function. Any function can be expanded in it, right? So for example, for the free particle, the energy eigenfunctions are e to the ikx, the momentum eigenfunctions for any value of k. But wait, how can there be multiple states with the same energy? Isn't that double counting or something? And the important thing is those guys have the same energy, but they have different momenta. They're different states. One has momentum h bar k. One has momentum minus h bar k. And you know that that has to be true, because the Fourier theorem tells you, in order to form a complete basis, you need all of them, all possible values of k. So there's no problem with being a complete basis and having states have the same energy eigenvalue, OK? It's a good question. Other questions? Yeah? AUDIENCE: So if we have a potential that only admits bound states, we'll never have this commutation happen basically? PROFESSOR: Yeah, exactly. Excellent. So the observation is this. Look, imagine I have a potential that's not trivial. It's not 0, OK? Will the momentum commute with the energy operator. No, because it's got a potential that's going to be acted upon by P, so you'll get a derivative term. But more precisely, if I have a system with bound states, I have to have a potential, right? And then I can't have P commuting with the energy operator, which means I can't have degeneracies. So indeed, if you have bound states, you cannot have degeneracies. That's exactly right. Yeah? AUDIENCE: But doesn't this break down in higher dimensions? PROFESSOR: Excellent, so we're going to come back to higher dimensions later. So the question predicts what's going to happen in the rest of the lecture. What we're going to do in just a minute is we're going to start working in three dimensions for the first time. We're going to leave 1D behind. We're going to take our tripped out tricycle and replace it with a Yamaha. As you'll see, it has the same basic physics driving its, well, self. It's the same dynamics. But I want to emphasize a couple things that are going to show up. So the question is, isn't this story different in three dimensions? And we shall see exactly what happens in higher dimensions. We'll work in two dimensions. We'll work in three dimensions. We'll work in more. Doesn't really matter how many dimensions we work in. You'll see it. OK, third thing. You studied this in some detail in your problem set. Suppose I have an energy operator that commutes with a unitary operator, U, OK? So it commutes to 0. And U is unitary, so U dagger U is one, is the identity. So what does this tell you? Well, first off from these guys it tells us that we can have simultaneous eigenfunctions. It also tells us too that if we take our state phi and we act on it with U, this could give us a new state, phi tilde sub E, which will necessarily have the same energy eigenvalue, because U and E commute. But it may be a different state. We'll come to this in more detail later. But the third thing, and I want to emphasize this, is this tells us, look, we have a unitary operator. We can always write the unitary operator as e to the i of a Hermitian operator. So what is the meaning of the Hermitian operator? What is this guy? So in your problem set, you looked at what unitary operators are. And in the problem set, it's discussed in some detail that there's a relationship between a unitary transformation, or unitary operator, and the symmetry. A symmetry is when you take your system and you do something to it, like a rotation or translation, and it's a symmetry if it doesn't change anything, if the energy remains invariant. So if the energy doesn't change under this transformation, we call that a symmetry. And we also showed that symmetries, or translations, are generated by unitary operators. For example, my favorite examples are the translate by L operator, which is unitary. And also e to the minus dxl. And the boost by q operator, which similarly is e to the minus qdp. And the time translation operator, U sub t, which is equal to e to the minus i t over h bar energy operator. OK? So these are transformation operators. These are symmetry operators, which translate you by L, boost or speed you up by momentum q, evolve you forward in time t. And they can all be expressed as e to the unitary operator. So this in particular is l i over h bar p. And this is similarly x. And earlier we understood the role of momentum having to do translations in the following way. There's a beautiful theorem about this. If you take a system and its translation invariant, the classical statement of Noether's theorem is that there's a conserved quantity associated with that translation. That conserved quantity is the momentum. And quantum mechanically, the generator of that transformation the Hermitian operator that goes upstairs in the unitary is the operator associated to that conserved quantity, associated to that observable. You have translations. There's a conserved quantity, which is momentum. And the thing that generates translations, the operator that generates translations, is the operator representing momentum. So each of these are going to come up later in today, and I just wanted to flag them down before the moment. OK, questions before we move on? Yeah? AUDIENCE: So you made the claim that every unitary operator can be expressed as p to the eigenfunction. PROFESSOR: OK, I should be a little bit careful, but yes. That's right. AUDIENCE: But if I take the [INAUDIBLE] I should be able to figure out what it is, but you can't take the [INAUDIBLE] PROFESSOR: The more precise statement is that any unitary-- any one parameter family of unitary operators can be expressed in that form. And then you can take a derivative. And that's the theory of [INAUDIBLE], which is beyond the scope. Let me make a very specific statement, which is that one parameter of [INAUDIBLE] unitary transformation. So translations by l, where you can vary l, can be expressed in that form. And that's a very general statement. OK, so with all that as prelude, let's go back to 3D. So in 3D, the energy operator-- so what's going to change? Now instead of just having position and its momentum, we now also have-- I'll call this P sub x-- we can also have a y-coordinate and we have a z-coordinate. And each of them has its momentum. P sub z and P sub y. And here's just a quick practical question. We know that x with Px is equal to i h bar. So what do you expect to be true of x with y? AUDIENCE: 0. PROFESSOR: Why? AUDIENCE: [INAUDIBLE]. PROFESSOR: What does this equation tell you? What is its physical content? Well, that they don't commute, good. What does that tell you physically? Yes? AUDIENCE: That there's an uncertainty principle connecting the two. PROFESSOR: Excellent. So that's one statement. So the consequence of this is that there's an uncertainty principle. Delta x delta Px must be greater than or equal to h bar upon 2. What's another way of saying this? Do there exist simultaneous eigenfunctions of x and P? AUDIENCE: No. PROFESSOR: No. No simultaneous eigenfunctions. OK, so you can't have a definite value of x and a definite value of P simultaneously. There's no such state. It's not that you can't know. It's that there's no such state. Do you expect to be able to know the position in x and the position in y simultaneously? Sure. OK, so this turns out to be 0. And in some sense, you can take that as a definition of quantum mechanics. x and y need to be 0. And similarly, Px and Py commute. The momenta are independent. However, Py and y should be equal to minus i h bar. Good. Exactly. So the commutators work out exactly as you'd naively expect. Every pair of position and its momenta commute canonically to i h bar. And every pair of coordinates commute to 0. Every pair of momenta commute to 0. Cool? So what kind of systems are we going to interested in? Well, we're going to be interested in systems where the energy operator is equal to P vector hat squared upon 2m plus U of x and y and z, hat, hat. You can see why dropping the hats becomes almost / So in this language, we can write the Schrodinger equation. This is just a direct extension of the 1D Schrodinger equation. i h bar dt of psi. Now our wave function is a function of x and y and z. There's some finite probability then to find a particle at some position. That position is labeled by the three coordinates. Is equal to-- and of t. Is equal to-- well, I'm actually write this in slightly different form. This is going to be easier if I use vector notation. So I'm going to write this as psi of r and t, where r denotes the position vector, is equal to the energy operator acting on it. And P is just equal to minus i h bar the gradient. So this is minus i h bar squared, or minus h bar squared upon 2m gradient squared plus u of x or now u of r psi of r and t. Quick question, what are the units or what are the dimensions of psi of r in 3D? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah, one over length to the root three halves. And the reason is this norm squared gives us a probability density, something that when we integrated over all positions in a region integral d 3x is going to give us a number, a probability. So its actual magnitude must be-- or its dimension must be 1 over L to the 3/2. Just the cube of what it was in 1D. And as you'll see on the problems set and as we'll do in a couple lectures down the road, it's convenient sometimes to work in Cartesian, but it's also sometimes convenient to work in spherical coordinates. And it does not matter. And here's a really deep statement that goes way beyond quantum mechanics. It does not matter which coordinates you work in. You cannot possibly get a different answer by using different coordinates. So we're going to be ruthless in exploiting coordinates that will simplify our problem throughout the rest of this course. In the notes is it a short discussion of the form of the Laplacian, or the gradient squared in Cartesian spherical and cylindrical coordinates. You should feel free to use any coordinate system you want at any point. You just have to be consistent about it. So let's work out a couple of examples. And here are all we're going to do is apply exactly the same logic that we see over and over in 1D to our 3D problems. So the first example is a free particle in 3D. So before I get started on this, any questions? Just in general 3D questions? OK. So this stuff starts off easy. And I'm going to work in Cartesian coordinates. And a fun problem is to repeat this analysis in spherical coordinates, and we'll do that later on. OK, so free particle in 3D, so what is the energy eigenfunction equation look like? We want to find-- the Schrodinger equation has exactly the same structure as before. It's a linear differential equation. So if we find the eigenfunctions of the energy operator, we can use superposition to construct the general solution, right? So exactly as in 1D, I'm going to construct first the energy eigenfunctions , and then use them in superposition to find a general solution to the Schrodinger equation. OK? So let's construct the energy eigenfunctions. So what is the energy eigenvalue equation look like? Well, E on psi is equal to minus h bar squared upon 2m. And in Cartesian, the Laplacian is derivative respect to x squared plus derivative with respect to y squared plus derivative with respect to z squared. And we have no potential, so this is just psi. So that would be energy operator acting on it. And the eigenvalue equation is at a constant, the energy E on psi satisfies this equation. I'm going to write this phi sub e to continue with our notation of phi being the energy eigenfunctions. It of course, doesn't matter what I call it, but just for consistency I'm going to use the letter phi. So this is a very easy equation to solve. In particular, it has a lovely property, which is that it's separable. OK? So separable means the following. It means, look, I note, I just observed that this differential equation can be written as a sum of terms where there's a derivative with respect to only one variable. There's a differential operator with respect to only one variable and another differential operator with respect to only one variable added together. And when you see that, you can separate. And here's what I mean by separate. I'm going to just construct. I'm not going to say that this is a general solution. I'm just going to try to construct a set of solutions of the following form. Psi E of x, y, and z is equal to psi E-- I will call this psi sub x of x times phi sub y of y times phi sub z of z. And if we take this and we plug it in, let's see what we get. This gives us that E on phi E is equal to minus h bar squared upon 2m. Well the dx squared acting on phi sub e is only going to hit this guy. So I'm going to get phi x prime prime phi y phi z. Plus from the next term phi y prime prime phi x phi z. And from the next term phi z prime prime phi x phi y. But now I can do a sneaky thing and divide the entire equation by phi sub e. Phi sub e is phi x phi y phi z. So if I do so, I lose the phi y phi z, and I divide by phi x. And in this term, when I divide by phi x phi y phi z, I lose the x and z, and I have a left over phi sub y, or phi y. And similarly here, phi sub z upon phi z. Everyone cool with that? Yeah? AUDIENCE: Can we also lose the [INAUDIBLE]? PROFESSOR: I don't think so. Minus h bar squared over 2m just hangs out for the ride. So when I take the derivatives, I get these guys. And I have E times the function. We could certainly write this as 2m over h bar squared and put it over here. That's fine. OK, so this is the form of the equation we have, and what does this give us? What content does this give us? Well, note the following. This is a funny system. This is a function of x. This is a function of y, g of y only, and not of x or z. And this is a function only of z, and not of x or y. Yeah? So we have that E, and let's put this as minus 2m over h bar squared is equal to a function of x plus a function of y plus a function of h. What does this tell you? AUDIENCE: They're all constant. PROFESSOR: They're all constant, right. So the important thing is this equation has to be true for every value of x, y, and z. It's a differential equation. It's true everywhere. It's true here. It's true there. It's true at every point. Yeah? So for any value of x, y, and z, this equation must be true. So now imagine I have a particular solution g at h. I'm going to fix y and z to some particular point. I'm going to look right here. And here that fixes y and z. So these are just some numbers. And suppose we satisfy this equation. Then there's a very x leaving y and z fixed. What must be true of f of x? AUDIENCE: Constant. PROFESSOR: It's got to be constant, exactly. So this tells us that f of x is a constant. I.e. phi x prime prime of over phi x is a constant. I'll call it epsilon x. And phi y prime prime of over phi y and this is a function of y of x is equal to epsilon y. And actually for-- yeah, that's fine. For fun I'm going to put in a minus. It doesn't matter what I call this coefficient. And similarly for phi z prime prime z over phi z is equal to minus epsilon z. So this tells us that minus 2m upon h bar squared e is equal to minus epsilon x minus epsilon y minus epsilon z. And any solutions of these equations with some constant value of epsilon x, epsilon y, and epsilon z is going to give me a solution of my original energy eigenvalue equation, where the value of capital E is equal to the sum. And I can take the minus signs make this plus plus. Yeah? AUDIENCE: Can epsilon x, epsilon y, and epsilon z-- can one of them be negative if the other's are sufficiently positive or vice versa? Or is that [INAUDIBLE]? PROFESSOR: Let's check. Let's check. So what are the solutions of this equation? Yeah. So solutions to this equation phi double-- so let's write this in a slightly more familiar form. This says that phi prime prime plus epsilon x phi is equal to 0, OK? But this just tells you that phi is exponential. Phi is equal to a e to the Ikx kxx plus B e to the minus ikxx, where k squared is equal to kx squared is equal to epsilon x. OK? So this becomes-- and similarly for epsilon y and epsilon z, each with their own value of ky and kz who squares the epsilon accordingly. So what can you say about these epsilons? Well, the epsilons are strictly positive numbers. So to answer your question. So the epsilons have to be positive. OK, so this equation becomes, though, E is equal to-- I'm going to put the h bar squared over 2m back on the other side. h bar squared upon 2m. Epsilon x, epsilon y, epsilon z. But those are just Kx squared plus Ky squared plus Kz squared. Kx squared plus Ky squared plus Kz squared, also known as h bar squared upon 2m k vector squared, where the wave function, phi sub E is equal to some overall normalization constant times, for the first function-- where did the definition go? Right here. So from here, phi x is the exponential with Kx. This is an exponential y with Ky. And then the exponential in z with Kz. e to the ikx times x plus Ky times y plus Kz times z. Let me write this as e to the i. e to the i. Also known as some normalization constant e to the i k vector dot r vector. OK? So the energy, if we have a free particle, the energy eigenfunctions can be put in the form, or at least we can build energy eigenfunctions of the form, plane waves with some 3D momentum with energy E is equal to h bar squared k squared upon 2m, just as we saw for the free particle in one dimension. And the actual wave function is nothing but a product of wave functions in 1D. Yeah? AUDIENCE: What happened to the minus ikx minus iky minus ikz terms? PROFESSOR: Good, so here I just dropped these guys. So I just picked examples where we just picked e to the ikz. That's an excellent question. So I've done something here. In particular, I looked at a special case. And here's an important lesson from the theory of the separable equations, which is that once I separate-- so if I have a separable equation and I find it separated solution, phi E is equal to phi x of x, phi y of y phi z of z, not all functions-- not all solutions of the equation are of this form. They're not. I had to make that assumption. I said suppose it's of this form right here. So this is an assumption. OK? However, these form a good basis. By taking suitable linear combinations of them, suitable super positions, I can build a completely general solution. For example, as was noted, the true solution of this equation, even just focusing on the x, is a superposition of plus and minus waves, waves with plus positive negative momentum. So how do we get that? Well, we could write that as phi is equal to e to the ikx phi y phi z plus e to the minus ikx phi y phi z with the same phi y and phi z. So these are actually in there. Yeah? AUDIENCE: My other question is so I still don't see why any of the epsilons can't have a negative sign. You have an exponential, a real exponential as one of your products. PROFESSOR: OK, so if we had a negative epsilon, is that wave function going to be normalizable? AUDIENCE: Oh, as r goes to-- but can you just keep the minus term? PROFESSOR: In which direction? AUDIENCE: Oh, right. PROFESSOR: So if it's converging in this direction, it's got to be growing in this direction. And that's not going to be normalizable. And so as usual with the plane wave, we can pick the oscillating solutions that are also not normalizable to one, but they're delta function normalizable. And so that's what we've done here. It's exactly the same thing as in 1D. Yeah? AUDIENCE: So does this mean that any superposition of plane waves with wave vector equal in magnitude will also be an eigenfunction of the same energy? PROFESSOR: Absolutely. Awesome. Great observation. So the observation is this. Suppose I take K, I'll call it K1. So this is a vector such that K1 squared times h bar squared over 2m is equal to E. Now there are many vectors k which have the same magnitude, but not the same direction. So we could also make this equal to h bar squared K2 squared vector squared over 2m where K2 is not equal to K1 as a vector, although they share the same magnitude. So that's interesting. So that looks a lot like before. In 1D, we saw that if we have k or minus k, these have the same energy. All right? Now if we have any K, K1-- so this is 1D. In 3D, if we have K1 and K2 with the same magnitude and the same energy, they're degenerate. That's interesting. Why? Why do we have this gigantic degeneracy of the energy eigenfunctions for the free particle in three dimensions? Yeah? AUDIENCE: Well, there are an infinite number of directions it could be going in with the same momentum. PROFESSOR: Awesome. So this is clearly true that there are an infinite number of momenta with the same magnitude. So there are many, many, but why? Why do they have the same energy? Couldn't they have different energy? Couldn't this one have a different energy? AUDIENCE: [INAUDIBLE]. PROFESSOR: Excellent, it's symmetric. The system is invariant under rotation. Who are you to say this is the x direction? I call it y. Right? So the system has a symmetry. The symmetry is rotations. And when we have a symmetry, that means there's an operator, a unitary operator, which affects that rotation, the rotation operator. It's the operator that takes a vector and rotates in some particular way. We have a unitary operator that's a symmetry that means it commutes with the energy operator. But if it commutes with the energy operator, we get can degeneracies. We can get states that are different states mapped to each other under our unitary operator, under our rotation. We get states which are different states manifestly. But which have the same energy, which are shared energy eigenvalues. Cool? And this is a really lovely example, both in 1D and 3D, that when you have a symmetry, you get degeneracies. And when you have a degeneracy, you should be very suspicious that there's a symmetry hanging around, lurking around ensuring it, OK? And this is an important general lesson that goes way beyond the specifics of the free particle. Yeah? AUDIENCE: So that occurs in systems with bound states [INAUDIBLE]? PROFESSOR: Yeah, it occurs in systems with bound states and systems with non bound states. So here we're talking about a free particle. Certainly not bound. And its true. For bound states, we'll also see that there will be a degeneracy associated with symmetry. Now your question is a really, really good one, because what we found-- let me rephrase the question. The question is, look, in 1D when we had bound states, there was no degeneracy. Didn't matter what you did to the system. When you had bound states, bound states were non degenerate. In 3D, we see that when you have a free particle, you again get degeneracy. In fact, you get a heck of a lot more degeneracy. You get a sphere's worth. Although actually, that's a sphere in 0 dimension, right? It's a 0 dimensional sphere, two points. So you get a sphere's worth of degenerate states for the free particle. Well, what about bound states? Are bound states non degenerate still? Fantastic question. Let's find out. So let's do the harmonic oscillator. Let's do the 3D harmonic oscillator to check. So the 3D harmonic oscillator, the potential is h bar squared. And let's pick for fun the rotationally symmetric 3D harmonic oscillator. m omega squared upon 2 x squared plus y squared plus z squared. This could also be written m omega squared upon 2 r squared. So I could write it in spherical coordinates or in Cartesian coordinates. This is really in vector notation. Doesn't matter. It's the same thing. 3D harmonic oscillator. So what you immediately deduce about the form of the energy eigenfunctions? Well, we have that E phi E is equal to P squared over minus h bar squared over 2m-- so here's the energy eigenvalue equation-- times dx squared plus dy squared plus dz squared plus m omega squared upon 2 times x squared plus y squared plus z squared phi E. But I can put this together in a nice way. This is minus h bar squared upon 2m dx squared plus m omega squared upon 2 x squared plus ditto for y plus ditto for z phi E. Yeah? Everyone agree? So this is differential operator that only involves x. Doesn't involve y or z. Ditto y, but no x or z. And ditto z, but no x or y. Aha, this is separable just as before. So now we have a nice separable system where I want to solve the equations 3 times, once for x, y, and z. And I'm just going to write it for x, y, and z epsilon sub x phi x is equal to minus h bar squared over 2m dx squared plus m omega squared upon 2 x squared phi sub x. And ditto for x, y, z. And then phi E is going to be equal to phi x of x, phi y of y, and phi z of z, OK? Where E is equal to epsilon x plus epsilon y plus epsilon z. Note that I've used a slightly different definition of epsilon here as before. Here it's explicitly the energy eigenvalue. So what this is telling is, look, we know what this equation is. This equation is the same equation we ran into in the 1D harmonic oscillator. It's exactly the 1D harmonic oscillator problem. So the solution to the 3D harmonic oscillator problem can be written for energy eigenfunctions, can be constructed by taking a harmonic oscillator in the x direction-- and we know what those are. There's a tower of them. There's a ladder of them created by the raising operator and lowered by the lowering operator. And similarly for y, and similarly for z. So I'm going to write this slightly differently in the same place as phi sub E is equal to phi sub nx of x. The state with energy E sub nx. Phi sub ny of y. Phi sub nz of z. Where these are all the single dimension, one dimensional harmonic oscillator eigenfunctions. And E is equal to Ex E sub nx plus E sub ny plus E sub nz. OK? So let's look at the consequences of this. So first off, does anyone have any questions? I went through the kind of quick. Any questions about that? If you're not comfortable with separable equations, you need to become super comfortable with separable equations. It's an important technique. We're going to use it a lot. So the upshot is that if I write-- in fact, I'm going to write that in slightly different notation. If I write phi E is equal to phi n of x, where it's the-- and phi l of y and phi m of z-- actually that's a stupid ordering. Let's try that again. l, m, n. That is the alphabetical ordering. With the energy is now equal to-- we know the energy is of a state with of the harmonic oscillator with excitation number l. It's h bar omega, the overall omega, times l plus 1/2. But from this guy it's got excitation number m, so energy of that is h bar omega m plus 1/2 plus m. And so now that's plus 1. And for this guy similarly, h bar omega n plus n and now plus 1/2 again plus 3/2. This is a basis of solutions of the energy eigenfunction equations. These are the solutions of the energy eigenfunctions for the 3D harmonic oscillator. And now here's the question. The question that was asked is, look, there are no degeneracies in bound states in 1D. Here we have manifestly a 3D bound state system. Are there degeneracies? AUDIENCE: Yes. PROFESSOR: Yeah, obviously, right? So for example, if I call l1 this 0 and this 0. Or if I call this 010 or 001, those all have the same energy. They have the energy h bar omega 0 times 1 plus 3/2 or 5/2. So let's look at this in a little more detail. Let's write a list of the degeneracies as a function of the energy. So at energy what's the ground state energy for the 3D harmonic oscillator? 3 halves h bar omega. It's three times the ground state energy for the single 1D harmonic oscillator. So 3/2 h bar omega. Yeah? AUDIENCE: [INAUDIBLE]. PROFESSOR: Good, the way we arrived that this was we found that the energy-- so the energy operator acting on the 3D wave function is what I get by taking the energy operator in 1D and acting on the wave function, and the energy operator in y acting on the wave function, the energy operator in z acting on the wave function, where the energy operator for each of those is the 1D harmonic oscillator with the same frequency. OK? And then I separated. I said look, let the wave function, the 3D wave function, since I know this is separable and each separated part of the wave function satisfies the 1D harmonic oscillator equation, I know what the eigenfunctions of the 1D harmonic oscillator energy eigenvalue problem are. They are the phi n's. And so I can just take my 3D wave function. I can say the 3D wave function is going to be the separated form, the product of 1D wave function in x, a 1D wave function in y, and 1D wave function in z. Cool? So now let's look back at what let's think about what happens when I take the energy operator and I act on that [INAUDIBLE]. When I take the 3D energy operator, which is the sum of the three 1D energy operators, harmonic oscillator energy operators. When thinking I'm going to act on this guy, the first one, which only knows about the x direction, sees the y and z parts as constants. And it's a phi x, and what does it give us back? E on phi x is just h bar omega n plus one half. Ditto for this guy. And then the energy operator in 3D is the sum of the three 1D energy operators. So that tells us the energy is the sum of the three energies. Is that cool? OK good. Other questions? Yeah? AUDIENCE: Is the number of degeneracies essentially [INAUDIBLE] of number theory. PROFESSOR: Ask me that after class. So let's look at the degeneracies as a function of the energy. So at the lowest possible energy, 3/2 omega, what states can I possibly have? I'm going to label the states by the three numbers, l, m, and n. So this is just the ground state 0 0 0. So is that degenerate? No, because there's just the one state. What about at the next level? What's the next allowed energy? AUDIENCE: 5/2. PROFESSOR: 5/2. OK, 5/2. So at 5/2 what states do we have? Well, we have 1 0 0. But we also have 0, 1, 0. And we also 0 0 1. Aha, this is looking good. What does this correspond to physically? This says you have excitation, so you've got a node in the x direction. But your Gaussian in the y and z directions. This one says your Gaussian in the x direction. You have a node in the y direction as a function of y, because it's phi 1 of y. And you're Gaussian in the z directions. And this one says you have 1 excitation in the z direction. So they sound sort of rotated from each other. That sounds promising. But in particular, what we just discovered sort of by construction is that there can be degeneracies among bound states in 3D. This was not possible in 1D, but it is possible in 3D, which is cool. But we've actually learned more. What's the form of the degeneracies? So here it looks like they're just rotations of each other. You call this x, I call it y, someone else calls it z. These can't possibly look different functions because they're just rotations of each other. However, things get a little more messy when you write, well, what's the next level? What's the next energy after 5/2 h bar omega? 7/2 h bar omega, exactly. 7/2. And that's not so bad. So for that one, we get 2 0 0, 0 2 0, 0 0 2. But is that it? AUDIENCE: No. PROFESSOR: What else do we get? 1 1 0, 0 1 1, 1 0 1. So first off, let's look at the number. The degeneracy number here-- I'll call this d sub 0-- the degeneracy of the ground state is 1, OK? The degeneracy-- and in fact, I'm going to write this as a table-- the degeneracy as level n. So for d0 is equal to 1. d1 is equal to 3. And d2 is equal to 6. Now it's less clear here what's going on, because is this just this guy relabelled? No. So this is weird, because we already said that the reason we expect that there might be degeneracy, is because of rotational symmetry. The system is rotationally invariant. The potential, which is the harmonic oscillator potential, doesn't care in what direction the radial displacement vector is pointing. It's rotationally symmetrical. When we have symmetry-- on general grounds, when we have a symmetry, we expect to have degeneracies. But this are kind of weird, because these don't seem to be simple rotations of each other, and yet they're degenerate. So what's up with that? Question? Yeah? AUDIENCE: [INAUDIBLE] Gaussian in certain directions? PROFESSOR: Yeah, sure OK. So let me just explain what this notation means again. So by 1 0 0, what I mean is that the number l is equal to 1, the number m is equal to 0, n is equal to 0. That means that the wave function phi 3D is equal to phi 1 of x, phi 0 of y, phi 0 of z. But what's phi 0 of z? What's a Gaussian in the z direction? Phi 0 of y. That's the ground state in the y direction of the harmonic oscillator. It's Gaussian in the y direction. If I wanted x, that's not the ground state. That's the excited state. And in particular, sort of being a Gaussian it goes through 0. It has a node. So this wave function is not rotationally invariant. It as a node in the x direction, but no nodes and y and z direction. And similarly for these guys. Did that answer your questions? Great. OK, so we have these degeneracies, and they beg an explanation. And if you look at the next level, it turns out that d3-- and you can do this quickly on a scrap of paper-- d3 is 10, OK? And they go on. And if you keep writing this list out, I guess it goes up-- what's the next one? 15. 21, yeah. So this has a simple mathematical structure, and you can very quickly convince yourself of the form of this degeneracy. dn is n n plus 1 over 2. So let's just make sure that works. 1 1 plus 1 over 2. Sorry I should really call this 1. n plus 1, n plus 2 if I count from 0. So for 0, this is going to give us 1 times 2 over 2. That's 1. That works. So for 1 that gives 2 times 3 over 2, which is 3, and so on and so forth. So where did this come from? This is something we're going to have to answer. Why that degeneracy? That seems important. Why is it that number? Why do we have that much degeneracy? But the thing I really want to emphasize at this point is that there's an absolutely essential deep connection between symmetries and degeneracies. If we didn't have symmetry, we wouldn't have degeneracy, and we can see that very easily here. Imagine that this potential was not exactly symmetric. Imagine we made it slightly different by adding a little bit of extra frequency to z direction. Make the z frequency slightly different. Plus m omega tilde squared upon 2 z squared, where omega tilde is not equal to omega 0. OK? The system is still separable, but this guy has frequency omega 0. The x part has omega 0. This has omega 0. But this has omega tilde. OK? And so exactly the same argument is going to go through, but the energy now is going to have a different form. The energy is going to have h bar omega-- h bar omega 0 times l plus m plus 1. But from the z part it's going to have plus h bar omega tilde times n plus 1/2. And now these degeneracies are going to be broken, because this state will not have the same energy as these two. Everyone see that? When you have symmetry, you get to degeneracy. When you don't have a symmetry, you do not get degeneracy. This connection is extremely important, because it allows you to do two things. It allows you to first not solve things you don't need to solve for. If you know there's a symmetry, solve it once and then compute the degeneracy and you're done. On the other hand, if you have a system and you see just manifestly you measure the energies, and you measure that the energies are degenerate, you know there's a symmetry protecting those degeneracies. You actually can't be 100% confident, because I didn't prove that these are related to each other, but you should be highly suspicious. And in fact, this is an incredibly powerful tool in building models of physical systems. If you see a degeneracy or an approximate degeneracy, you can exploit that to learn things about the underlying system. Yeah? AUDIENCE: So we just add the different omega to each omega [INAUDIBLE] number there is still a possibility to get a degeneracy. PROFESSOR: Exactly. So it's possible for these omegas to be specially tuned so that rational combinations of them give you a degeneracy. But it's extraordinarily unlikely for that to happen accidentally, because they have to be rationally related to each other, and the rationals are a set of measures 0 in the reels. So if you just randomly pick some frequencies, they'll be totally incommensurate, and you'll never get a degeneracy. So it is possible to have an accidental degeneracy. Whoops, just pure coincidence. But it's extraordinarily unlikely. And as you'll see when you get to perturbation theory, it's more than unlikely. It's almost impossible. So it's very rare that you get accidental degeneracy. It happens, but it's rare. Other questions? OK, nothing? OK so here we're now going to launch into-- so this leads us into a very simple question. At the end of the day, the degeneracies that we see for the 3D free particle, which is a whole sphere's worth of degeneracy, and the degeneracy we see for the 3D harmonic oscillator, the bound states, which is discrete, but with more and more degeneracy the higher and higher energy you go. Those we're blaming, at the moment, on a symmetry, on rotational symmetry, rotational invariance. So it seems wise to study rotations, to study rotational invariance and rotational transformations in the first place. In the first part of the course, in 1D quantum mechanics, we got an awful lot of juice out of studying translations. And the generator of translations was momentum. So we're going to do the same thing now. We're going to study rotations and the generators of rotations, which are the angular momentum operators, and that's going to occupy us for the rest of today and Thursday. Yeah? AUDIENCE: So your rotational symmetry will explain a factor of three in your degeneracy, right? But what's the symmetry that explains the way this grows. Because this very clearly appears there's 1 up to a factor of three. And then there's 2 up to a factor of three. And there's even more that's not even a multiple of three. PROFESSOR: Right, actually so here's a very tempting bit of intuition. Very tempting bit of intuition is going to say the following. Look, rotational invariance, there's x, there's y, and there's z. It's going to explain rotations amongst those three. So that could only possibly give you a factor of three. But it's important to keep in mind that that's not correct intuition. It's tempting intuition, but it's not correct. And an easy way to see that its not correct is that for the free particle, there is a continuum, a whole sphere's worth of degenerate states in any energy. And all of those are related to each other by simple rotation of the k vector, of the wave vector, right? So the rotational symmetry is giving us a lot more than a factor of three. And in fact, as we'll see, it's going to explain exactly the n plus 1 n plus 2 over 2. OK, so with that motivation let's start talking about angular momentum. So I found this topic to be not obviously the most powerful or interesting thing in the world when I first studied it. And my professor was like no, no, no. This is the deepest thing. And recently I had a fun conversation with one of my colleagues, Frank Wilczeck, who said yeah, in intro to quantum mechanics the single most interesting thing is the angular momentum and the addition of angular momentum. And something has happened to me in the intervening 20 years that I totally agree with him. So I will attempt to convey to you the awesomeness of this. But you have to buy in a little. So work with me in the math at the beginning of this, and it has a great payoff. OK so the question is, what is the operator. So we're going to talk about angular momentum now. And I want to start with the following question. In the same sense as we started out by asking what represents position and momentum, linear momentum, in quantum mechanics, what represents what operator by our first, second, or third postulate-- I don't even remember the order now. What operator represents angular momentum in quantum mechanics? And let's start by remembering what angular momentum is in classical mechanics. So L in classical mechanics is r cross p. In classical mechanics. So let's just try this. Let's construct that operator. This is not the world's most beautiful way of deriving this, but let's just write down natural guess. For in quantum mechanics what's the operator we want? Well, we want a vectors worth of operators, because angular momentum is a vector. It's a vector of operators, three operators. And I'm going to write these as r vector the operators x, y, z cross p the vector of momentum operators. So at this point, you should really worry, because do r and p commute? Not so much, right? However, the situation is better than it first appears. Let's write this out in terms of components. So this is in components. And I'm going to work, for the moment, in Cartesian coordinates. So Lx is equal to? Lx is equal to? You all took mechanics. Lx is equal to? AUDIENCE: [INAUDIBLE]. PROFESSOR: Thank you. YPz minus ZPy. And that's the curl, the x component of the curl. And similarly, the x component-- so the way to remember this is that its cyclic. x, y, z. y, z, x. So z, p, y. PX minus XPz. And then we have z XPy minus YPx. So we were worried here about maybe an ordering problem. Is there an ordering problem here? Does it matter if I write YPz or PZy? AUDIENCE: No. PROFESSOR: No, because they commute with each other. PZ is momentum for the z-coordinate, not the y-coordinate, and they commute with each other. So there's no ambiguity. It's perfectly well defined. So we're just going to take this to be the definition of the components of the angular momentum operator Lx, Ly, and Lz. And just for fun, I want to write this out. So because we know that Px, Py, and Pz can be expressed in terms of derivatives or differential operators, we can write the same operator in Cartesian coordinates in the following way. So clearly we could write this as Y d dx i upon h bar-- or sorry, h bar upon i. And z h bar upon i d dy. So we could write that in Cartesian coordinates as a differential operator. But we can also write this in spherical coordinates. I'm just going to take a quick side note just to write down what it is. If we did this spherical coordinates, it's particularly convenient to write-- let me just write Lz for the moment. This is equal to minus i h bar derivative with respect to phi, where the coordinates, [INAUDIBLE] coordinates and spherical coordinates in this class is theta is going to be the angle down from the vertical axis, from the z-axis. And phi is going to be the angle of a period 2pi that goes around the equator, OK? Just to give you a name. This is typically what physicists call them. This is typically not what mathematicians call them. This leads to enormous confusion. I apologize for my field. So here it is. I can also construct the operator associated with the square of the momentum. And why would we care about the square of the momentum? Well, that's what shows up in the Hamiltonian, that's what shows up in the energy. So I can construct the operator for the square of the momentum and write it, and it takes a surprisingly simple form. When I see surprisingly simple, you might disagree with me, but if you actually do the derivation of this, it's much worse in between. 1 over sine theta d theta sine theta d theta plus 1 over sine squared theta d phi. Couple quick things. What are the dimensions of angular momentum? Length and momentum. What else has units of angular momentum? AUDIENCE: h bar. PROFESSOR: Solid. h bar, dimensionless. Angular momentum squared, angular momentum squared. OK, great. So that's going to be very convenient. h bars are just going to float around willy nilly. OK, so suppose I ask you the following-- bless you. Suppose I ask you the following questions. I say look, here are the operators of angular momentum. This is Lz. We could have written down the same expression for Lx, and a Ly, and L squared. What are the eigenfunctions of these operators? Suppose I ask you this question. You all know how to answer this question. You take these operators-- so for example, if I ask you what are the eigenfunctions of Lz? Well, that's not so bad, right? The eigenfunction of Lz is something where Lz on phi-- I'll call little m-- is equal to minus I h bar d d theta phi sub m. But I want the eigenvalue, so I'll call this h bar times some number. Let's call it m, because Lz is an angular momentum. It carries units of h bar, and its h bar times some number which is dimensional, so we'll call it m. And we all know the solution of this equation. The derivative is equal to a constant times-- we can lose the h bar. We get a minus i, so we pick up an i. So therefore phi sub m is equal to some constant times e to the im phi. What can we say about m? Well, heres an important thing-- oh shoot. I'm using phi in so many different ways here. Let's call this not phi, because it's going to confuse the heck out of us. Let's call it Y. So we'll call it Y sub m. Why not? It's not my joke. This goes back to a bunch of-- yeah well, it goes way back. So phi is the variable. Y is the deciding eigenfunction of Lz, and that's great. But what can you say about m? Well phi is the variable around the equator is periodic with period 2pi. And our wave function had better be single valued. So what does that tell you about m? Well, under phi goes to phi plus 2pi. This shifts by im 2pi. And that's only one to make a single valued if m is an integer. So m has to be an integer. m is an integer. Now we did that for Lz. We found the eigenfunctions of Lz. What about finding the eigenfunctions L squared? Exactly the same thing. We're going to solve the eigenvalue equation, but it's going to be horrible, horrible to find these functions, right? Because look at this. 1 over sine squared d d phi. And then 1 over sine d theta sine d theta. This is not going to be a fun thing to do. So we could just brute force this, but let's not. Let's all agree that that's probably a bad idea. Let's find a better way to construct the eigenfunctions of the angular momentum operators. So let's do it. So we ran into a situation like this before when we dealt with the harmonic oscillator. There was a differential equation that we wanted to solve. And OK, this one isn't nearly as bad, not nearly as bad as that one would have been. But still it was more useful to work with operator methods. So let's take a hint from that and work with operator methods. So now we need to study the operators of angular momentum. So let's study them in a little more detail. So something you're going to show in your problem set is the following. The commutator of Lx with Ly takes a really simple form. This is equal to i h bar-- let's just do this out. Let's do this commutator. We're OK. So Lx with Ly, this is equal to the commutator of YPz minus ZPy with Ly ZPx minus XPz. So let's look at these term by term. So the first one is YPz ZPx. YPz ZPx. That's a Px. Sorry, ZPy. ZPx, this term. X, good. That's the first commutator. The second commutator, it can be YPz an XPz minus YPz XPz. And then these commutators, the next two, are going to be ZPy with ZPx. And finally ZPy. And that's minus and this is a plus. ZPy and XPz. OK, so let's look at these. These look kind of scary at first. But in this one notice the following. This is XPz ZPx minus ZPx YPz. But what you say about Y with all these other operators? Y commutes with all of them. So in each term I could just pull Y all the way out to one side. Yeah? So I could just pull out this Y. So for the first term, I'm going to write this as Y commutator PZ with ZPx. And let me just do that explicitly. There's no reason to. So this is YPz ZPx minus ZPx YPz. And I can pull the Y out front, because this commutes with Px and with Z. So I can make this Y times PZ ZPx minus ZPx Pz. But now note that I can do exactly the same thing with the Px. Px commutes with z. Px commutes with Pz. And it commutes with y. So I can pull the px from each term out. Px Y. And now I lose the Px. I lose the Px. But now this is looking good. This is Px times Y. And Pz minus ZPz PZz minus ZPz. This is also known as PXy times commutator of PZ with Z. And what is this equal to? Let's get our signs right. This is Px with Y time-- OK, we all agree that this is going to have an h bar in it. Let's write units. There's going to be an i. And is it a plus or minus? Minus. OK, good. So we get minus h bar XPy. So this term is going to give us minus i h bar Py XPx times Y. And let's look at this term. OK, this is Y, and Y commutes with PZx and PZ, right? So I could just pull out the y. And x commutes with everything, so I can pull out the x. And I'm left with the commutator of Pz with Pz. What's the commutator of Pz with Pz? AUDIENCE: 0. PROFESSOR: 0. This term gives me a 0. Similarly here, Py commutes with everything. Z ZPx. Px commutes with everything. Z ZPy. So I can pull out the Px Py, and I get Z commutator Z, and what's that? AUDIENCE: 0. PROFESSOR: 0. So this gives me 0. And now this term, ZPy XPz, the only two things that don't commute with each other are the Z and the Pz. The Py and the X I pull out, so I get it a term that's XPy and the commutator of Z with Pz. And what is that going to give me? PyX i h bar. Aha, look at what we got. This is equal to i h bar times-- did I screw up the signs? XPy minus YPz. Oh sorry, Px. And what is this equal to? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah, i h bar Lz. And more generally, as you'll show on the problem set, you get the following commutators. Once you've done this once, you can do the rest very easily. Lx Ly is-- so Lx with Ly is i h bar Lz. And then the rest can be got from cyclic rotations Ly with Lz is i h bar Lx and Lz with Lx is i h bar Ly. And now here's a fancier one. This is less obvious, but exactly the same machinations will give you this result, and you'll do this again on the problems set. If I take L squared, k and L squared here is going to be L squared, I just mean Lx squared plus Ly squared plus Lz squared. This is the norm squared of the vector, the operator form. L squared with Lz-- or sorry, with Lx. Yeah, fine. Lx is equal to 0. So Lx commutes with the magnitude L squared. Similarly L squared, now just by rotational invariance, if Lx commutes with it, Ly and Lz had better also commute with it, because of who you to see what's Lx. And L squared with Lz must be equal to 0. Yeah? Everyone cool with that? And the thing is we've just-- and I'm going to put big flags around this. We've just learned a tremendous amount about the eigenfunctions of the angular momentum operators. Why? Let's leave that up. So what have we just learned about the eigenfunctions of the angular momentum operators Lx, Ly, Lz, and L squared? Anyone? We just learned something totally awesome about them. Anyone? So can you find simultaneous eigenfunctions of Lx and Ly? AUDIENCE: No. PROFESSOR: Not so much. They don't commute to 0. What about Ly and Lz? Nope. Lz Lx, nope. So you cannot find simultaneous eigenfunctions of Lx and Ly. What about Lx and L squared? Yes. So we can find simultaneously eigenfunctions of L squared and Lx. OK, what about can we find simultaneous eigenfunctions of L squared and Ly? Yep. Exactly. What about L squared, Ly, and Lz? Nope. No such luck. And this leads us to the following idea. The idea is a complete set of commuting observables. And here's what this idea is meant to contain. You can always write down a lot of operators. So let me step back and ask a classical question. Classically, suppose I have a particle in three dimensions, a particle moving around in this room, non relativistic, familiar 801. I have a particle moving around in this room. How much data must I specify to specify the configuration of this system? Well, I have to tell you where the particle is, and I have to tell you what its momentum is, right? So I have to tell you the three coordinates and the three momenta. If I give you five numbers, that's not enough, right? I need to give you six bits of data. On the other hand, if I give you seven numbers, like I give x, y, z, Px, Py, Pz, and e, that's over complete. Right? That was unnecessary. So in classical mechanics, you can ask what data must you specify to completely specify the state of the system. And that's usually pretty easy. You specify the number coordinates and the number of momenta. In quantum mechanics, we ask a slightly different question. We ask, in order to specify the state of a system, we say we want to specify which state it is. You can specify that by saying which superposition in a particular basis. So you pick a basis, and you specify which particular superposition, what are the eigenvalues of the operators you've diagonalized in that basis? So another way to phrase this is for a 1D problem, say we have just a simple 1D problem. We have X and we have P. What is a complete set of commuting operators? Well, you have to have enough operators so that the eigenvalue specifies a state. So X would be-- is that enough? So if I take the operator X and I say look, my system is in the state with an eigenvalue X not of X, does that specify the state of the system? It tells you the particles are in a delta function state right here. Does that specify the state? AUDIENCE: Yes. PROFESSOR: Fantastic, it does. Now what if I tell you instead, oh it's in a state of definite P. Does that specify the state? Absolutely. Can I say it's in a state with definite X and definite P? AUDIENCE: No. PROFESSOR: No. These don't commute. So a complete set of commuting observables in this case would be either X or P, but not both. Yeah? Now if we're in three dimensions, is x a complete set of commuting observables? AUDIENCE: No. PROFESSOR: No, because it's not enough. You tell me that it's X, that doesn't tell me what state it is because it could have y dependence or z dependence. So in 3D, we take, for example, x, and y, and z. Or Px, and Py, and Pz. We could also pick z, and Px, and y. Are these complete? If I tell you I have definite position in z, definite position in y, and definite momentum in x? Does that completely specify my state? AUDIENCE: Yes. PROFESSOR: Yeah, totally unambiguously. e to the ikx delta of y delta of z totally fixes my function, completely specifies it. If I'd only picked two of these, it would not have been complete. It would have told me, for example, e to the ikx delta of y, but it doesn't tell me how it depends on z, how the wave functions on z. And if I added another operator, for example, Py, this is no longer commuting. These two operators don't commute. So a complete set of commuting observables can be thought of as the most operators you can write down that all commute with each other and the minimum number whose eigenvalues completely specify the state of the system. Cool? OK. So with all that said, what is a complete set of commuting observables for the angular momentum system? Well, it can't be any two of Lx, Ly, and Lz. So let's just pick one. I'll call it Lz. I could've called it Lx. It doesn't matter. It's up to you what axis is what. I'll just call it Lz conventionally. And then L squared also commutes, because L squared commutes with Lx. It commutes Ly and with Lz. So this actually forms a complete set of commuting observables for the angular momentum system. Complete set of commuting observables for angular momentum. So this idea will come up more in the future. And here is going to be the key [INAUDIBLE]. So we'd like to use the following fact. We want to construct the eigenfunctions of our complete set of-- yeah, question? AUDIENCE: Really quick can you explain how you got that L squared and Lx commute? PROFESSOR: Yeah, I got it by knowing what you're going to write on your solution set. So this is on your problem set. So the way it goes-- so there are fancy ways of doing it, but the just direct way of doing, how do you construct these commutators? Is you know what the operators are. You know what L squared is. And you know that L squared is Lx squared plus Ly squared plus Lz squared. It's built in that fashion out of x and Py. And then I literally just put in the definitions of Lx, Ly, and Lz into that expression for L squared and compute the commutator with Lx, again, using the definition in terms of Py and z. And then you just chug through the commutators. Yeah, it just works out. So it's not obvious from the way I just phrased it that it works out like that. Later on you'll probably develop some intuition that it should be obvious. But for the moment, I'm just going to call it a brute force computation. And that's how you're going to do it on your problem set. So let me tell you where we're going next. So the question I really want to deal with is what are the eigenfunctions? So this is where we're leaving off. What are the eigenfunctions of our complete set of commuting variable L squared and Lz. What are these guys? And we know that we could solve them by solving the differential equations using those operators, but that would be horrible. We'd like to do something a little smarter. We'd like to use the commutation relations and the algebra. And here a really beautiful thing is going to happen. When you look at these commutation relations, one thing they're telling us is that we can't simultaneously have eigenfunctions of Lx and Ly. However, the way that Lx and Ly commute together is to form Lz. That gives us some information. That gives us some magic, some power. And in particular, much like that moment in the harmonic oscillator I said, well look, we could write down these operators as a. Well look, we can write down these operators, which I'm going to call L plus and L minus. L plus is going to be equal to Lx plus i Ly. And L minus is going to be equal to Lx minus I Ly. Now Lx, Ly, those are observables? What can you say about them as operators? What kind of operators are they since they're observables? Hermitian, exactly. So what's the Hermitian adjoint of Hermitian plus i Hermitian? Hermitian minus i Hermitian. OK, good. So L minus is the adjoint of L plus. So this is just going to be a definition. Let's take these to be the definitions of these guys. If we take their commutator, something totally lovely happens. I'm not going to write all the commutators. I'm just going to write a couple. The first is if I take L squared and I commute with L plus, well, L plus is Lx plus Ly, and we already know that L squared commutes with Lx and it commutes with Ly. So L squared commutes with L plus. And similarly for L minus, it commutes with each term. Question? Oh, I'm sorry. Lx. Shoot, that was an x. It just didn't look like it. Lx minus i Ly. So it commutes both L plus and L minus. But here's the real beauty of it. Lz with L plus is equal to-- well, let's just do dimensional analysis. L plus is Lx plus i Ly. We know that Lz with Lx is something like Ly. And Lz with Ly is something like Lx. With factors of i's an h bars. And when you work out the commutator, which should only take you second, you get i h bar L plus. And similarly, when we construct Lz and L minus, we get minus h bar L minus. Are L plus an L minus Hermitian? No, they're each other's adjoints. Lz is Hermitian. And look at this commutation relation. What does that tell you? From the first observation, if we have an energy or if we have an operator e and an operator a that commute in this fashion, then this tells you that the eigenfunctions of this operator are staggered in a the ladder spaced by h bar. The eigenvalues of Lz come in a ladder spaced by h bar. We can raise with L plus, and we can lower with L minus just like in the harmonic oscillator problem. And we'll exploit the rest of the-- we'll deduce the rest of the structure of the angular momentum operator eigenfunctions next time using this computation relation. See you next time.
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_22_Metals_Insulators_and_Semiconductors.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Hi everyone. Spring has regressed. So, we have-- we're going to have a guest at the end of lecture today, which should kind of entertaining. Just as a warning, if you see someone come in. So questions, before we get started? No questions about anything? At all? Math? Nothing? Yeah? AUDIENCE: Can you explain the physical significance of the crystal momentum? PROFESSOR: Yeah. OK. Let me go over that. That's a good question. So the question is what again is the significance of the crystal momentum? So let me answer that in a slightly backward way. So this is a form of the explanation I haven't given you. It's going to be a slightly different one. Let's step back and think about the momentum, and ask what the momentum is. Now you guys showed on a problem set, the following fact. That if you have a wave function, sine of x, such that, the expectation value in the state SI of x in the state SI is equal to x naught, and the expectation value of p in the state SI is p naught. Hat, hat. Then if you want to change the momentum, increase momentum by h bar k, the way to do that is to take SI and build a new wave function, SI tilda, is equal to e to the i, k x, SI of x. And then the expectation value of x is the same, SI tilda, still equal to x naught, because this phase goes away from the two complex, from the wave function is complex content we give the inner product. But the expectation value, the momentum is shifted in state SI tilda, is shifted by each h bar k, p naught plus h bar k. So all the intuition you have about momentum, you can translate into intuition about the spatial variation of the phase of the wave function. Yeah? AUDIENCE: [INAUDIBLE] PROFESSOR: OK, good. OK, good, So we have a sneaky [INAUDIBLE]. So, the information about the momentum can be encoded in these spatial variation of the phase of the wave function. So another way to answer the question of what is momentum, apart from it's the thing that-- so what are ways to answer the question, what is momentum, you could ask well what is momentum? It's the thing that commutes with p or with x by i h bar. That's one way to answer it. Another way to answer is to say that translations by l can be expressed in terms of momentum as e to the minus i upon h bar p l. So these are both ways of describing what the momentum is. But another way of talking about the momentum is the momentum p governance the spatial variation, the x dependence of the phase of the wave function. So these are always talking about what the momentum is. So now let's turn this around, and let's ask about the crystal momentum. Oh, and one last thing, a last defining property of the momentum, a central property from the Schrodinger equation is at the time variation d dt of p is equal to the expectation value of minus d the potential of x d x. Also known as the force. So this is the Ehrenfest Theorem Statement that the classical equation of motion, p dot, is equal to the minus d v d x is equal to the force, Ehrenfest's Theorem tells us that the classical equations of motion are realized as expectation values. And equivantly, if there's no potential, the potential is constant, this tells us that the momentum expectation value is time independent. Right? A familiar fact. So these are all true lovely and things about the momentum. So let's turn all these facts around into the crystal momentum. So let's talk about crystal momentum. Which was the question, what is the crystal momentum? So the crystal momentum is defined from beginning, from the following property. If we have a potential v of x, which is invariant under shifting, by one lattice spacing, by some l, v of x, then this tells us that the energy operator is invariant if we shift by l. If we translate by l equals zero. And from this fact, we deduced via block or a la block, that the wave functions are really the energy eigenfunctions, can be written in the form e cubed is equal to e to the i q x, u of x, where u, we're going to take to be a periodic function. So what is this parameter q doing? Q is governing the spatial variation of the phase of the wave function. Cool? So in precisely this sense, the momentum difference is space of the wave function. Here, in the case of a periodic potential, the crystal momentum q is governing the spatial variation of the phase of the wave function. So q is the thing the governs the phase as a function of x. Well what about-- another fact about the crystal momentum which you show in your problems set, is that if you impose an external force d q d t, and really d h bar q. d t is equal to-- d dt of the expectation value of h bar q, is equal to the expectation value of the force. I'll just write-- OK? So again, this is a quantity, and this was assuming that we had a sharply peaked wave packet. So this is for a wave packet sharply peaked at q naught. And so let me just write this as h bar q naught. So the central value of your wave packet-- so this is what you've shown on the problem set that the central value of your wave packet, the peak of your wave packet varies in time according to the external force. And so in particular, if the force is zero, we turn no external driving force, your wave packet maintains its crystal momentum. It's time independent. So the crystal momentum is something that time independent, unless an external force is applied, just like the momentum. And it's something that governs the phase of the wave function just like the momentum. However, it's different in a crucial way. It is not the eigenvalue p on five sub e q is not equal to a constant p naught times 5 sub e q. Because when we take-- when we active p or we active the derivative, you pick up a term from here, which gives us a constant, but we also have this overall periodic piece. And its spatial variation is generically non-zero. And if the potential is nontrivial, it's always non constant. So when the momentum operator hits this guy, it will generically not give us zero. It'll get two terms and we will not get an eigenvalue equation. So q is not the eigenvalue h bar q is not the eigenvalue of p. And what's the last important property of q that's different from the momentum? It comes from the commutator, which tells us that the thing that's conserved is the expectation value of p l is really the precise statement. And in particular, what this tells us is that the eigenfunction, or the eigenvalue of our wave function, under translations by l, is a quantity that can be determined simultaneously with knowing the energy. However, the eigenvalue of t sub l, on this state, is equal to e to the i q l. Which means that q is only defined for determining the eigenvalue up to 2 pi over l. If you have q, which is 0, and you increase it to pi over l, that value, pi over l, is effectively the same as the value minus pi over l. Because at least they're the same eigenvalue. But that's really strange because that means that q itself, it's not strictly conserved. It's conserved mod 2 pi over l. When you have momentum conservation, momentum is strictly conserved if there's no force. And even if there is a force, it's increasing control by the force as you turn on the force, it just constantly increases. For the crystal momentum, that's not the case. You turn on a force, it increases according to the conservation law. But it's not increasing constantly. It's periodic. It's periodically defined. So it increases then it ends up at a smaller value. It increases and ends up at a smaller value. OK? So it carries many of the same properties. It governs the phase. It's time independent unless there's an external force applied. It's the eigenvalue. Controls the eigenvalue of an operator that commutes with the energy when you have a periodic potential, in the same way that the momentum commutes with the energy when you have no external force, when you have a constant potential. Does that help? Good. OK. So developing an intuition for the crystal momentum, I think, is best done by just playing with examples. And you'll do that more in the course on solids, which I encourage you all to take. Because it's really beautiful stuff. But for our purposes, this is going to be the full set of ideas we'll need for 8.04. Yeah? AUDIENCE: [INAUDIBLE] PROFESSOR: Ah. So good. So thank you. So this involves a slight subtlety, which I've been glossing over in the entire story here. Which of the following. So, is u of x a real function? Well, so when we started out asking what are the eigenfunctions of the transit by l operator, all we showed was that, and I'm going to do this on a separate board just to make it clearer. Tell me if this turns off, because it kept bumping. OK. So when we started with translate by l, and we constructed it's eigenfunctions, we said that translate by l q Phi sub cubed is equal to some phase, and this is unitary, so we're talking about they must be an actual phase in the i alpha of Phi sub q of x. And let's just suppose that this is true. Then this tells us that Phi sub q times e to the minus i q l equals u. So, I'm just going to use this to define a new function, u sub q. Or just u. I'll use sub q. Fine. of x. So this defines a new function, sub q. I take an eigenfunction, I multiply it by some phase. Sorry, minus i q x. If we choose q l to be equal to alpha, then acting on u sub q, by translate by l, on u sub q, of x, is equal to-- well, if we act on Phi sub q with translate by l, what happens to Phi sub q we pick up a phase e d i alpha. What happens to e to the minus i q x? x goes to x plus l. We pick up a phase e to the minus i q l. So if q l is equal to alpha, those two phases cancel, and we just get u back. u sub q of x. But translate by l, if u sub q, by definition, is equal to u sub q of x plus l. So we've determined is that if we take q l is equal to alpha, then Phi sub q if eigenvalue label by its eigenvalue, q, can be written in the form e to the i q x u sub q of x, where this is periodic. Everybody agree with that? OK. So that's step one. Step two is to say well look, since the eigenvalue of this guy, under t sub l, e d i alpha is equal to e to the i q l. Since this is periodic under shifts of q, by 2 pi upon l, I can just choose to define q up to 2 pi over l. So 2 q, I will take to be equivalent to q plus 2 pi over l. And the reason I'm going to do that is because it gives the same eigenvalue, and if I want to label things by eigenvalues, it's sort of redundant to give multiple values to the same eigenvalue. Now there's a subtlety, here though. And this little thing here is this. Suppose we have a free particle. Does a free particle respect translation by l? So if we have a free particle, the potential is zero. That constant function is also periodic under shifts by l. Right? Because it's just zero. So it's stupidly periodic, but it's periodic nonetheless. So now I'm going to ask the following question. What are the common eigenfunctions of the energy and translate by l for the free particle? We did this last time. So the common eigenfunctions of translate by l and the energy are the wave functions Phi sub q, comma e, are equal to e to the i q x times some function u of x, on general grounds. But we know what these eigenfunctions are. They're just e to the i k x. Where k squared upon 2 m is e. [INAUDIBLE] So we know that these are the correct eigenfunctions, but we're writing them in the form e v i q x u. Now you say that's fine. There's nothing wrong with this. We just say u is constant and q is equal to k. These functions are of this form, but they're of this form with e v i q x being e d i k x and u of x being constant. Right? There's nothing wrong with that. Everyone agree? Perfectly consistent. However, I thought we said that q is periodic by 2 pi? If q is periodic by 2 pi, then that would seem to imply that k is periodic by 2 pi, and we know that's not true because any k is allowed for a free particle. So if we want to think about q is periodic by 2 pi upon l, then we cannot require that u is real. Because it must be the phase that makes this up. It must be, so I can always write this as e to the i q x where q is less than 2 pi upon l. I'm sorry, where q is between 2 pi or pi upon l and minus pi upon l. So that it's defined only after this periodically thing. But times some additional phase, e to the i k minus q x This is trivially equal to e to the i k x. But now u is not a real function. On the other hand, if we hadn't imposed the requirement that q is periodic, we wouldn't have needed to make u real. We could just taken q to be equal to k, for any value k, and then u would be constant. u would be real. So this is important for answering the excellent question that our fearless restation instructor provoked me to answer. Which is that so what-- we'll come back to the question in just a second. But what I want to emphasize this, that if we're going to take q to be not periodic, Sorry. If we're going to take q to be defined only up to shifts by 2 pi over l, it's important that we allow u to be not real. It must be able to be an overall phase. But if we want u to be always real, we can do that. We just can't impose this periodicity. Different values of q mean different wave functions. And this is really what's going on when you see those plots, sometimes you see the plots as parabolas. The bands are represented by parabolas with wiggles, and sometimes they're folded up. And that's the difference. The difference is that when you fold them up, you're imposing this periodicity and you're labeling the eigenfunctions by q, and the overall amount of the number effectively of k phases that you're subtracting off. Yeah? AUDIENCE: So is this an arbitrary choice? [INAUDIBLE] PROFESSOR: Yeah. I mean, how to say? It's exactly akin to a choice of variables. In describing the position of this particle, should we use Cartesian coordinates, or should we use Spherical coordinates? Well it can't possibly matter. And so you'd better make sure in any description of your system, that changing your coordinates doesn't change your results. And here, that's exactly what's going on. Do we want to define our variable to be periodic by 2 pi upon l? Well, OK then. But u can't be real. Or we could take q to be not periodic by 2 pi l and impose that u is real. It's just a choice of variables. But it can't possibly give different answers. The point is, this is a subtle little distinction it we gloss over, and is glossed over into my knowledge every book on intro to quantum mechanics that even covers periodic potentials. It can be very confusing. Anyway, the reason that I had to go through all this, is that in order to answer the very, very good question professor Evans posed, I'm going to need to deal with this fact. So for the moment, let me deal with-- let's work with u real. And q, q an unconstrained, real number. OK. So not periodic. Are we cool with that for the moment? So if we do that, then notice the falling nice property of our wave function. Our wave function, Phi sub q, is equal e to the i q x times u of q, or u of x. Which is real. So when we can construct the current-- remember that j boils down to the imaginary part, h bar over 2 m i. Well, h bar over m times the imaginary part of SI complex conjugate derivative, with respect to x, which is the current, in the x direction of SI. And we need this to be imaginary, or we will get no current. You show this in a problem set, if you have a pure, real wave function, for example. A single real exponential, that's decaying, as on the wrong side of a barrier. Then you get no current. Nothing flows. And that make sense. It's exponentially decaying. Nothing gets across. So we need the wave function to be real. So if q were zero we would get zero. And what you can immediately do from this, compute from this, is that while the derivative, if the derivative doesn't hit e to the i q x, if it hits u, than the phase e to the i q x cancels. And so the contribution from that term vanishes. So the only term that's going to contribute in here is when the derivative hits the e to the i q x. But then this is going to be equal to h bar q. And we want the imaginary parts, that's going to be e to the I over m. And then we're left with u squared of x. So this is the current, but we have to do it-- we had take advantage in order for this to be sort of clean, we had to take advantage of u being real. Everybody cool with that? Now there's one last twist on this, which is that if I have k-- if I have q. So this is a side note. Going back up here, to this logic. If I have q, and I want, I can always write it as some q naught plus n pi over l. And so now what I want to do is I want to take sort of a hybrid of these two pictures. And I want to say Phi sub q is going to be equal to e to the i q naught x. Where this is the value that's periodic by 2 pi. e to the I n pi over l x u. And so now really what's going to happen, what I'm doing here is I'm labeling q, not by a single number. I'm labeling my wave function not by single number q, but by q naught and an integer n. Comma n. So q naught and n. So now q naught is periodic. It's defined up to shifts by 2 pi. n is an additional integer, and what it's telling you is how many times did you have to shift back to get into that fundamental zone between pi and minus pi. And this fits nicely into this story, because now all we're going to get here is q, which is q naught plus n pi. So the current depends on both the part defined mod 2 pi over l, and the integer, which tells you how many factors of 2 pi over l did you have to subtract off to get into that fundamental domain. So let's think back to our band structure. So what is this n quantity? Let's think back to our band structure. In our band structure, we had something that looks like this. And here's the value of q. But am I plotting q? No. I'm plotting here q naught. I'm plotting the part that's periodically defined up to 2 pi over l. So this is pi over l. This is minus 2 pi over l. Or minus pi over l. OK. And what we see is that there isn't a single energy. Because this is the energy the vertical direction for the band pictures. There isn't a single energy for a given value of q. In fact, the set of energy eigenvalue-- or the set of allowed states or energy eigenvalues for an allowed value of q would say this particular value of q naught, how many of them are there. Well, there are as many as there are integers. One, two, three, four, count. So to specify a state, I don't just have to specify q NAUGHT, I also have to specify N. Which one of these guys I'm hitting. And when you unfold this into the parabola picture, remember where these came from. These came from these curves. Came from shifting over. And the higher up you go, the more you had to shift over. And that's exactly the integer piece in n pi over l. And so we can write the current now, in terms of h bar q naught upon m, u squared-- I'm sorry. h bar q naught upon m plus n pi h bar upon m u squared of x. So we get a contribution from the crystal momentum and from which we're in. OK? So sort of an elaborate story to answer the phase question. Yeah? AUDIENCE: [INAUDIBLE] PROFESSOR: Good. So here we had SI-- so SI-- I'm sorry. I should have done this for Phi. But I meant this wave function, right. This is Phi, this is Phi q. So from here we're going to get the imaginary part. So we get the imaginary part of this wave function which is u to the minus i q x u of x derivative of e to the i q x u of x. Now the term that contributes is when the derivative hits the e to the i q. x pulls down a factor of i q, and the two phases cancel from these guys, leaving us with a u of x here, and a u of x here. AUDIENCE: [INAUDIBLE] PROFESSOR: Oh sorry. This is a potential. Good. That's the point. So this is the potential. So in this statement that what we have this translation by x. So this is just some function. It has nothing to the potential. It's defined in terms of the wave function. The eigenfunction of translate by l. So the logic here goes, if we know we have a function of translate by l, then I construct a new function u. Nothing to do with the potential, just a new function. Which is e to the minus i q x times it. You can't stop me. You hand me a function, I will hand you a different function. And then we pick q felicitously, to show that u is periodic. So u is just some periodic function which is contained which is defined from the wave function. From the energy. From eigenfunction of t l. Did that answer your question? OK. So here, it just came from the fact that u is Phi is the e to the i q x u, x and then a factor of u for each of these. Other questions. Yeah. AUDIENCE: [INAUDIBLE] PROFESSOR: So this picture, when it's unfolded, first off, you know what it is for a free particle. So we want the energy as a function of q. So what is it for a free particle? Parabola. Yeah, exactly. And now let's add in-- let's make this a function of q, not q naught, but so here's pi over l. Here's 2 pi over l. Here's 3 pi over l. And I need to do this carefully, because it's incredibly difficult to get the straight. OK. My artistic skills are not exactly the thing of legend. OK. So here's the parabola that would have been, if we had not turned on a periodic potential. As we turn on the periodic potential, we know that the energies change. And so in the first band it's easy to see, because for minus pi over l, it's pi over l. We don't have to do anything. So it look exactly the same as the lowest band over here. So in particular-- OK? So what about this second band? Well what I want to know what's the allowed, the other allowed energy that's say, plus pi over l. Plus pi over l, it's going to be something greater than this value. But plus pi over l, we already know the answer from that diagram, because plus pi over l is the same as minus pi over l, so what's the value over here? Well, the value over there for the second band is slightly above, and then it increases and decreases. So slightly above, and then it increases. Shift by pi over l. Whoops. Did I shift by pi over l for this guy? That's one, two. Yes. I did. Good. And it goes the other way. So just noting that it goes away from the top. I have a hard time drawing these things. So for every value of q, there's an allowed energy. But it's different than it would have been for the free particle. And then we do the same thing for the next state. And it looks like this. So now imagine what happens when we take this, and we it over one two. What we get is a band the looks-- that should look like this. That's what the second band should look like. And indeed, when we put it in the fundamental domain, this is what we get. This is what the first band and the second band together look like. And then the third band, we'll move this over once, and then twice, it's going to look like this. And this guy, move it over once, twice, looks like whoops. Yeah? AUDIENCE: If we wanted to plot u with respect to k instead, would that just be a parabola dotted line? If so, why do we not have really-- PROFESSOR: If we just wanted-- sorry. Say it again? AUDIENCE: E as a function of k instead of q. PROFESSOR: Oh. Yeah. E as a function of k is always going to look like that. But k is not a well-- so what is k? K is just defined as h bar squared, k squared upon 2 m is equal to e. So this doesn't tell you anything. Right. Because any allowed k. Sure any allowed k is some valid value of e. But this didn't tell you which values of e are allowed. Only some values of e are allowed, right? There are no values of e-- there are no energy eigenstates with energy in between here and here, right? And so that tells you they're no allowed k's because k is just defined, it's just completely defined by e. So this doesn't tell you anything about which states you're at. It just that given an e, there's some quantity that could define k. This is a definition of k, in terms of e. What this diagram is telling you is which e's are allowed. AUDIENCE: [INAUDIBLE] PROFESSOR: Yes. Yes. There should be. Let's see. What's AUDIENCE: [INAUDIBLE] PROFESSOR: Oh, here. Yes. Yes, you're absolutely right. Over out. Thank you. Excellent. That's exactly right. Yeah. Oh man. I made a dimensional mistake. Thank you. Jesus. OK. Good. Yeah. AUDIENCE: Could you like re-explain how imperfections and a lattice leads to actual conduction? PROFESSOR: Yeah. I'm going to do that. So that's an excellent question. The question is could you explain again how imperfections and a lattice leads to actual conduction. As we talked about last time, when you have a perfect lattice, there is actually no current flowing in response to an applied electromagnetic field. If you put on a capacitor, played across your perfect lattice, you don't get any current. So the particle, the charged particle in your lattice, just oscillates back and forth in a block oscillation, running up the band, and down the band, and up the band, and down the band. So, let me slightly change your question, and turn it into two other questions. The first question is given that that's obviously not what happens in real materials, why don't we just give up on quantum mechanics and say it totally failed? And so this is a totally reasonable question, and I want to emphasize something important to you. Which is the following. That model led to a prediction, which is that if you put a capacitor plate across a perfect crystal, then you would get no current flowing across, you would just see that the electron wave packets oscillate. Or block oscillations as we discussed last time. And that is manifestly what happens with copper. But the experimentalist comes back to you and says look dude. That is a ridiculous model because the copper isn't in fact perfect, it's messy. So how do you test the model? Well there are two ways to test-- to deal with the situation. One is you improve the model to incorporate properties that copper actually has. And see if you can actually get the same conductivity that you see. But the other is you could improve the material, instead of improving the theory. So let's make up what-- can we actually build a perfect crystal? This is actually something that I'm doing research on right now. Not on the building side, but on the theory side, because I'm a theorist and you should not let me in a lab. But I collaborate with experimentalists, so they're nice people. They're very good physicists. So here's something you can do. You can build a system that has exactly a periodic potential. It turns out it's very difficult to do this with quantum systems. But what you can do is you can do it with lattices not of atoms, but lattices of dielectric. So the equation. Here's a cool fact, the equation for light going through a dielectric, where the dielectric has different constants, like wave guides. You've got glass, you got air. You've got glass, you got air. That equation can be put in exactly the same form as the Schrodinger equation for the time evolution of a wave function. They're both waves. And so it's not so surprising these two wave equations are related to each other a nice way. Meanwhile, the index of the dielectric turns into the potential for the quantum mechanical problem. So if you have a periodic potential, what do you want? You want a periodic dielectric constant. Yeah. And so you can build a system which incredibly, cleanly, has a periodic dielectric constant and no disorder. And then you can put light into the system, and you can ask what happens to this system. So here's the idea, I take a system which is a periodic-- I'm going to draw the potential here. So I'm going to draw the dielectric constant. So small, large, small, large, small, large, small, large, et cetera. But instead of having it be a one dimensional lattice, I'm going to make it a two dimensional lattice. So now, basically, I've got a set of wave guides. Let me draw this differently. So does everyone get the picture here? So literally what you have, is you have glass, glass with a different index, glass, glass with a different-- if you can think of those as a line of glass fibers. Optical fibers. And you shine your light that's reasonably well localized, in both position, and in phase variation, or crystal momentum. Because you can control the phase of the light. So you send this wave packet in and you ask what happens. Well not a whole lot happens. It's a wave packet. It's going through a wave guide, but we haven't implemented an electric field. To handle an electric field, you need the potential to be constantly varying. Uh huh. So it's at a linear ramp into the potential. Instead of making it just perfectly periodic, let's make the index ramp just a little bit. And this experiment has been done. In this experiment, so as the wave packet moves along, what's discovered is that the position-- if I draw the x as a function of t, so now the role of t is being played by the distance it's moved along the wave guide, what you find is that it does this. It exhibits beautiful block oscillations. And this has been proved in a very small number of real honest quantum mechanical systems. The most elegant experiment that I know of was done by Wolfgang Ketterle, who's here at MIT. And he got three data points because it was preposterously difficult and declared victory. So I talked to him about this in the hallway one day. And he said yes, this was ridiculous, but we got three data points. We got small, we got large. Victory. We declared victory. But it really needs to be done well. So one of the interesting questions in this part of the field right now is we know that it's true. But we want to see it. We want to feel it, so various people around the world are working on making a truly beautiful demonstration of this bit of physics. Yeah. AUDIENCE: [INAUDIBLE] PROFESSOR: It's totally impractical, because any interference is just going to kill you. Unfortunately. So, you have to work ridiculously hard to make systems clean. So the question is really a question about quantum computation, which we'll come to next week. But, the basic question is how robust is this. And the answer is it's not robust at all. But which you can tell because everything in the real world has enough impurity that it conducts. Or as an insulator. Yeah. AUDIENCE: What place sort of like the larger role in sort of like the perfection of a lattice like temperature or impurities. PROFESSOR: That's a very good question. So the question is what's the most important property? What's most important disordering property that leads to conduction? And there's temperature fluctuations, there are impurities in the lattice. There are decohereing effects which is a more complicated story. And that's actually, it depends on the situation, it depends on the system. And exactly how it depends is something that is an active area of research. Now there are many, many ways to probe this physics. So we know that these block oscillations are true. We see them in all sorts of different systems that are analogous. So there's lots of [INAUDIBLE], it's not like this is an ambiguous bit of physics. But it's one that turns out to be surprisingly difficult to tease apart. The reason I bring all this up is to emphasize the following, our model made a prediction that disagreed explicitly with the connectivity property of copper and other materials. So don't throw away the model. Observe that you've modeled the wrong system. If you find a system that fits your-- that is-- that shares the assumptions of your model, that's when you ask did it work. And it worked like a champ. OK. So now let's talk about real materials. This is going to close up our discussion bands and solids. And this is actually what I wanted to get to at the beginning of the lecture. But that's OK. There are lots of questions and they were good questions. So this is an extremely brief. But I want to ask you the following question. What happens in the following three systems? So first, imagine we take why don't we take a system with built out of single wells, which have some set of energy eigenstates, and then we build the periodic array out of them. What do we expect? And let me draw this bigger. What do we expect to see when we build a lattice? We expect that this is going to-- that these states are going to spread out into bands a funny way Yeah and let's just talk about the 1 d potential. So what we'll find is that this band turns into-- I'm sorry. This state, this single state turns into a band of allowed energy eigenstates. There's now a plot of the energy. And similarly, this state is going to lead to another band with some width. And this state is going to lead to another band, which is even wider. Everyone cool with that? Quick question? In 1 d, do these bands ever overlap? No. By the node theorem. Right? OK. Now let's take a single electron, and let's put in-- let's take a single electron, and let's put it in the system. What will happen? Well if we put it in the system, what state will this single electron fall into? Yeah one event. But which state? AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah, if you kick the system around, you let it relax a little bit. It's going to fall down to the ground state. You have to couple to something else like hydrogen has to be coupled with an electromagnetic field to decay. But couple it, kick it, and let it decay. It'll settle down to its ground state. So you get an electron down here in the ground state, and looking back at that band, we know that the band for that ground state looks like this. So, here it is. There's our electron. It's sitting in the lowest energy eigenstate. Is it moving? Well, it's in a stationary state. Is the expectation value of the position changing in time? No. The expectation values don't change in time, in the stationary state. That's part of what it is to be a stationery state, to be an energy eigenstate. OK. Great. it's not moving. Now, in order to make it move, what do you have to do? What kind of state corresponds to the position changing in time? Yes. Superposition. Right? From the superpositions we'll get interference terms. So if we put in a superposition of say, this state, and this state, which corresponds to different energies. If we put it in a superposition of these guys, then it's meaningfully moving. It has some meaningful, well defined time variation of its position expectation value. So in order to induce a current, in order to induce a current of this system where the electron wave packet carries a little bit of momentum is changing in time it's position, what do I have to do to the electron in the ground state? I have to excite it, so that it's in a superposition of the grounds state and some excited state. Or more generally, into a superposition of other states. Yes? In order to induce the current, I must put the electron into a higher energy state and in a particular superposition of higher energy states. Everyone down with that? Here's why this is so important. Imagine each one of these wells is actually not some square well, but it's an atom. And let's say the atom is hydrogen, just for-- this doesn't actually happen, but just imagine-- in particular what it means is it has the ion, the nucleus is charge plus 1. And so in order for the system to be neutral, I must have one electron for every well. So if I have n wells, I must have n electrons in the system. Everybody agree with that? In order to be neutral. Otherwise, the thing's charged and all sorts of terrible things-- electrons will get ripped off from nearby cad. So we must have an electron per well. How many states are in this band? For n wells? n. Right? OK. So if I put in the n electrons I need to neutralize a system, where do those n electrons go? Yeah, they fill up the first band. And if we let the system relax with lowest energy configuration, every state in this lowest band will be filled, and none of these states will be filled. Everyone down with that? So here's my question. When I've got that ground state configuration of this lattice of atoms with one electron per well, in these distributed wave functions, filling out these bands, is anything moving? Wow, you guys are so quiet today. Is anything moving? This system is in an energy eigenstate. In particular, it's in a completely antisymmetrized configuration, because they're identical fermions. So, nothing is moving. If we want to induce a current, what do we have to do? Yeah. We have put them in a superposition. But where's the next allowed energy eigenstate? Next band. So it's in the next band. The next allowed energy eigenstate. So the configuration we have now is that these guys are all filled, these guys are all empty, but in order to take an electron from here and put it into this excited state, we have to put in a minimum amount of energy, which is the gap between those two bands. Right? So now think about it this way. Suppose I take light and I send my light at this crystal. In order for the light to scatter off the crystal, you must have electrons in superposition states so that they can have a dipole and absorb and radiate that energy. Yeah. But in order for that to happen, the light has to excite an electron across the gap. It has to give it this macroscopic amount of energy. Well, it's not macroscopic. it's large. It's not infinitesimally small. That means that there's a minimum amount of energy that that incident light must have in order to excite the electron in the first place. So very long wavelength light will never do that. Light along wavelength will not have enough energy to excite an electron across this gap into the next band to allow there to be a current, which could oppose the electric field. So the only for light to scatter off of this crystal, is if the energy, h bar omega, of the light is greater than or equal to, let's say greater than approximately, the band gap delta e. That cool? We've just discovered something. Crystals are transparent unless you look at sufficiently high frequencies. That's cool. Right? A crystal is transparent unless you look at sufficiently high frequencies. If you look at low frequencies, your crystal should be transparent. Well that's really interesting. In particular, we immediately learn something cool about two different materials. Consider diamond and copper. These are both crystals. They're solids made out of a regular array, perhaps not perfect, but extraordinarily good, regular array of atoms of the same time. Array in a particular structure. Diamond, anything and I think face inner cubic. I don't remember. I really should know that. Anyway, copper. It's a lattice. That's embarrassing. I really should know that. So we have these two materials which one has the larger band gap? Diamond, because it's transparent. At in the visible. So the band gap, delta e of diamond is much larger than the band gap for copper. But in fact, this is a little more subtle, because copper in fact, doesn't even have a band gap. We made an important assumption here. So I want to think about-- we're going to come back to copper in a second, but I want to point out the nice thing here. Which is that diamond has to have a band gap. It's transparent. It must have band gap. It must be such that when you fill up all the electrons you need for it to be neutral, there is a gap to the next energy states. And that gap must be larger than visible wavelengths of light. Yeah. That's cool. And that must be true of all the transparent crystals that you see. Otherwise, they wouldn't be transparent. They would respond by having free electrons that could respond like a metal. Yeah. AUDIENCE: So, [INAUDIBLE] PROFESSOR: Yeah. Are diamonds good conductors? No. They're terrible conductors. In fact, there preposterously-- if you compare the number of-- I'll get into this later. But yes, they're terrible. AUDIENCE: [INAUDIBLE] PROFESSOR: Uh, that's a slightly more complicated story, which let me come back to. Hold on to that, and if I don't answer today, ask me after in office hours because it's a little more-- what? Really? Wow. Well, MIT. It's all about the intellect. And everything else has to-- OK. So, this is pretty good, but here's the thing. In one dimensional crystals, the only thing that can happen is, look if you have each band come from allowed energy state and each energy state, each well comes with one electron, or two electrons, or three electrons, you will always have filled bands, and then a gap and filled bands and then a gap. Does everybody agree with that? You can't have a partially filled band if each band comes from a bouncy, in a single well, and each well comes with an integer number of electrons. You just-- you're stuck. Yeah. AUDIENCE: [INAUDIBLE] PROFESSOR: Oh. I'm lying about spin. But spin in one dimension is little-- I'm lying about spin. But do you really want me to get in spin? Man. OK. So if we include spin, and we have splitting, then it becomes a more subtle story. If we include spin, then there are two states for every allowed energy eigenstate of the potential. However, there are generically going to be interactions between the-- there are generically going to be magnetic interactions which split the energy of those two spin states. Electrons spin up, and electrons spin down, will generically have different energies. Now in 3D, this isn't such a big deal, because those splittings are tiny, and so the states can sort of overlap. But in 1D they can't. So I mean, that's also not exactly true, but it depends on exactly the details. It depends on the details of the system, is what I wanted to get to. Curse you. So let me talk about the same phenomena in an easier context, where we don't have to worry about spin, which we haven't discussed in detail, in the class. Which is in three dimensions. Where the story changes in a dramatic way. Yeah. AUDIENCE: [INAUDIBLE] PROFESSOR: Oh. It's not. Generically, no, it's not. It will depend. AUDIENCE: [INAUDIBLE] PROFESSOR: You say it happens to be the salient one. Yeah, exactly. That is exactly right. There the gaps are not the same. That they do not remain constant. OK. So let's talk more about this system, but let's talk about it in three dimensions. So in three dimensions, you guys did an interesting thing, when you studied, you didn't know this was about the structure of solids, but it really was. When you studied the rigid rotor. And when you studied the rigid rotor, you found that you had energy eigenstates and they were degenerate with degeneracy 2 l plus 1. The various different l z eigenstates. Yeah. And then we turned on an interaction which was the energy costs, the energy penalty for having angle momentum in z direction. Which added an l z term to the energy. And what you found is that as a function of the coefficient, which I think we called epsilon, of that perturbation of the energies of the energy was equal to l squared over 2 i plus epsilon l z. What you found is that these guys split. So this remained constant. And this split into, so this is the [INAUDIBLE] l equals 1. So l equals zero. So this is one, this is three, this is five. So the l equals zero state, nothing happens. l equals 1. There's one that changes, one that doesn't. And then this guy has five. One, two, three, four, five. OK. And what we found here is that these guys could cross. States from different multiplates, with different values of l, had energies that could cross as a function of the strength of the deformation of your system. Right? The deformation is where you have a sphere and you stick out your arm. So it's no longer symmetric top. So here we can have states crossing. There's no nodes here in three dimensions. So as a consequence, when you have a three dimensional material built out of atoms. So here's my sort of pictorial description of three dimensional system built out of atoms. You have a potential well, potential, potential well. Now, if the energy in one particular potential well, is like this, and like this, and like this, then when we add in a lattice we get bands again. The structure's a little more intricate because it depends on the momentum. But these bands now can overlap. OK. Everybody see that? Because there's nothing preventing states from different-- in different multiplates from having the same energy in three dimensions. There's no nodes here that tells you have to keep the ordering constant as you turn on the potential. Now we turn on the multiple particle potential, and they can interact, they can overlap. As a consequence, when we fill up, let's say we two electrons per potential, or per well, when we filled those first two bands, well, there is the first-- so the first band is now filled. The second band and part of the first-- part of the third band and most of the second band are going to be filled. But part of the second band is now available, and much of the third band is now available. We filled in 2n electrons, but we haven't filled up this band, because it's really two bands jammed together. Or really bands from two different orbitals jammed together. They happened to overlap. So as a consequence here, if this is the length of the energy of the last electron that you put in, how much energy do you have to give the system, do you have to add the system, to excite the energy-- or to excite the electrons into excited states, in particular into superpositions so that the electrons can move? AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah. Preposterously small amount. An amount that goes like one over the number of particles. So in the continuum limit, it's zero. There's an arbitrarily nearby energy. So how much energy does it take to excite an electron and cause a current that opposes the induced electric field? Nothing. Any electric field that you send in will be opposed by an induced current. So this behaves like a classical conductor. You turn on an electric field, and the charges will flow to oppose that externally imposed electric field. You get charges then building up on the walls of your capacitor plates. So, this is where we have a conductor. Because there's an unfilled band. And back here , we had an insulator because we had filled bands separated by gap. The gap between the filled band and the next available band. This is actually called a band insulator. Because there are other ways of being an insulator. So from this so far, just from the basic quantum mechanics of a particle and a periodic potential, we now understand why some crystals are transparent. Why some materials conduct. Why the materials that are transparency are also insulators. And the things that conduct are not transparent, generally. Yeah. AUDIENCE: [INAUDIBLE] PROFESSOR: Excellent. Excellent question. So what's so special about diamond and differ from copper? And so the answer goes like this. So what determined the exact band structure in for a 1D periodic potential? Two properties. One was l, the periodicity. And that came in the q l and k l. And the second is the detailed shape of the potential. Now in three dimensions, the story's going be a little more complicated. In three dimensions, the things that are going to determine the potential are not just the distance between atoms, but you have a three dimensional lattice. And the three dimensional lattice could have different shapes. It could be cubic, it could be hexagonal, could be complicating in all sorts of different ways. Right? It could be bent, it could be rhomboidal, and it could have all sorts of different crystallographic structures. So that's going to go into it, in the same way that l went into it, which is the only parameter in one dimension. In the same way that l goes into it. So the crystal structure, the shape of the lattice, is going to determine it. Secondly, the structure of the orbitals is different. Different atoms are different wells, so they'll give you different band structure. So different materials for example, diamond versus copper, are going to give you different bands allowed energies, because the potential is different. It has different shape. And so when you solve the problem for the energy eigenvalues is a function of now the three different components of the crystal momentum, you'll just get a different set of equations. And working those out is not terribly hard. But it's a computation that must be done, and it is not trivial. And so one of the sort of, I don't know if I'd say exciting, but one of the things that one does when one takes a course in solids, is you go through a bunch of materials. And you understand the relationship between the potential, at the atomic orbital structure of the individual atom, the crystal structure, and the resulting band structure. And there's some sort of nice mnemonics, and there are calculations you do to get the answer. AUDIENCE: [INAUDIBLE] PROFESSOR: You will almost always find overlapping bands in three dimensions in sufficiently high energy. I can't off the top of my head give you a theorem about that, but yeah, it's generic. Yeah. AUDIENCE: --analog to conductor in one dimension? You have these like, non-zero band depths? PROFESSOR: Yeah. And this is why Matt was barfing at me. So the answer to that is yeah. There aren't [INAUDIBLE]. But what would we need? What we need is one of two things. We need either the band gap coincidentally is ridiculously small. What's a good example of that? A free particle. In the case of a free particle, these band gaps go to 0. Right? And so that's a conductor. Just an electron. Right. It conducts, right? OK. So that can certainly happen. But that's sort of stupid. I mean, it's not totally stupid. But it's sort of stupid. But a better answer would be, well, can you have a system where there are bands but you didn't have one electron per potential well? And yeah. You could orchestrate that in lots of ways. Now it involves orchestration. So it's not the generic system that we were talking about here. But you can't orchestrate it. So spin is a useful thing that gives you an extra handle. If you have twice as many states per well then you can have half a band filled. So that's one way to do it. Then it becomes dependent on details of the system, which is what I didn't want to get into. But yeah, you can orchestrate it. It's just not a generic thing from what we've done. And it's really not for spin-less systems. On the other hand, accidental small gaps. Easy. That happens. That certainly happens. So that brings me to the last thing I wanted to talk about before getting to entanglement, which is accidental small gaps. So what happens to a system which is-- so there are some systems that are neither conductors nor insulators. They are reasonably good conductors and reasonably bad insulators. But they're not perfect. And these materials are called semiconductors. I want to talk about why they're called semiconductors and what that means. So this is going to be very brief. Then I'm going to give you-- we're going to get into entanglement. So consider a system exactly using the same logic we've used so far which has the following property. We have two bands. And the bottom band is filled because we've got just the right number of charged particles. Bottom band is filled. And this guy is empty, but the gap is tiny. OK. Delta e is very small. Now delta e has dimensions. It has units, right? So when I say small, that doesn't mean anything. I need to tell you small compared to what. So what's a salient thing that controls an energy scale for a real material? Well the temperature. If you have a hot piece of copper, then the lattice is wiggling around. And every once in a while, an ion can hit one of the electrons and excite it, give it some momentum. And so there's an available reservoir of energy for exciting individual electrons. You have it really hot, what happens is every once in a while an electron will get nailed by a little thermal fluctuation in the system and get excited above the gap. And now it's in a super-- and generically, it's going to be in a superposition state of one of these excited states. So it's in general going to be moving. It can radiate. It will eventually fall back down. But you're constantly being buffeted. The sea of electrons is constantly being buffeted by this thermal fluctuation. And as a result, you constantly have electrons being excited up, cruising around, falling back down. So you end up with some population of electrons. And they can ask-- and both when asked, although not quite in this language, how likely are you to get an electron up here? How likely is an electron to be excited up thermally? And those of you taking 8.04 will know the answer to this. The probability goes as e to the minus delta e over kt. So let's think of this where this is the Boltzmann constant. So what does this mean? At very low temperatures, if the gap isn't 0, then this is 0. It doesn't happen. But at large temperatures, the denominator here is large. If the temperature is large compared to the width of the gap, then this is a small number. And e to the minus of a small number is close to 1. So at high temperature, you're very likely to excite electrons up here. And now if you have electrons up here, you have a bunch of available states down here-- also known as holes-- and you have a bunch of available electrons up here with lots of available states. So at a high temperature, a material with a small gap-- or at least at temperatures high compared to the size of the gap-- it's basically a conductor. And at low temperatures, it's basically an insulator. This is called a semiconductor. And there are notes on the Stellar web page that discuss in a little more detail what I just went through and show you how you build a transistor out of a semiconductor. And the important bit of physics is just this. OK. So that finishes us up for the band gap systems for periodic potentials. We've done something kind of cool. We've explained why diamonds are transparent. We've explained why they don't conduct. We've explained why copper does and it's opaque. And that's pretty good for 15 minutes of work. It's not bad. But along the way, we also talked about the analogous system of what are called photonic crystals. Systems of periodic arrays of dielectrics. Like wave guides. And those have the same structure. They have bands of allowed energy and gaps of disallowed energies where no waves propagate through. So you might think that's a little bit of a ridiculous example. So just to close this off, you've all seen a good example of a photonic crystal flying past you. You know that highly reflective at very specific frequency structure on the surface of a butterfly wing that makes it shiny and blue? It looks metallic. It looks like it's a crystal reflecting in a specific frequency. At some sharp blue. And the reason is, it's a photonic crystal. It is exactly this form. If you look at it under a microscope, you see little rays of protein which have different dielectric than air. And they form exact crystals-- or not exact, but very good crystals-- that reflect at very specific wavelengths. And as a consequence, they have a metallic sheen. So why would a butterfly put a photonic crystal on its surface? Well it's extremely light. It's fairly rigid. It looks shiny and metallic without actually being shiny and metallic. And it's not a pigment, so it doesn't absorb light and decay over time. It's like the best thing you could ever do if you wanted to be a shiny, fluttery, flying thing. Anyway. So there's an incredible amount of physics in this story of the band gaps. And consider this an introduction to the topic. OK. So that's it for band gaps. And I want to move on to the remainder, the last topic of our course. Which is going to be entanglement and quantum computation. And here I need to give you one quick observation and then move on to the punchline of today. The one quick observation is this. We've talked about identical particles before. And we've talked about identical particles in funny states. So for example, imagine I have two particles described by a wave function where the first particle could be in the state a and the second particles in the state b. And I can build a wave function for the first particle being in state a and the second particle in state b in the following way. So let's say position a and position b. I could take a single particle wave function, chi of a, and a single particle wave functions phi of b. And we've talked about what this tells us. And you've studied this on your problem set. What this tells you is that the probability of finding the particle at point A is given by chi a squared. And this is normalized, so when we integrate against it, we get 1. And similarly, the probability that we find the second particle at b is this thing norm squared. And it's independent of what a is. But we also studied-- and so this was called the distinguishable. We also studied the symmetric configuration, which was equal to 1 over root phi, root 2. Chi of a. Phi of b. Symmetric, plus chi of b phi of a. And this tells us something totally awesome. What's the probability that I find the first particle at a? It's the norm squared of chi of a phi of b, right? If we integrate over all phi b, this is the norm squared integrates to 1. So it's fine. So there's a factor of one half. We either find it at chi of a or chi of b. However if I tell you that I've measured the first particle and I find it in the state chi, what can you say about the second particle? It's in the state phi. If you know the first particle's in the state chi, the second part is in the state phi. Because we measured it and it's not in the state-- the first particle's not in the state phi. So measuring one particle tells you something about the second particle. And this is deeply disconcerting, because I could've taken these particles, put them in this entangled state, and sent one particle off to a distant planet and the second particle to my sister in DC. And my sister measures this second particle and determines what state it's in and is immediately determined what state the first particle is in over in this distant planet Zorg, right? So that's deeply disconcerting. And to those of us who have studied quantum mechanics up to this point-- which we all in this room have-- to those of us who have studied quantum mechanics to this level of development and understand that it is a correct description of many experiments, this should be yet another moment of serious discomfort. We've run into a bunch of these over the semester. But this one should be troubling to you. Because look. How can something here dramatically change the state, the configuration, the initial configuration, of a particle arbitrarily far away? Isn't that deeply concerning? And if you think about relativity, this should be all the more deeply disconcerting. Because how does relativistic causality fit into this? So there was a person that roughly this time, a little earlier, who was troubled by this problem. And his name was Einstein. And so one of the things that's kind of amazing is that he created a thought experiment which we're going to study in detail next week called the EPR experiment. And there's a beautiful historical story about the setting and the meaning and the particular person. And unfortunately, I'm not a historian so I can't tell you that story. It sure would be nice if we had someone who wrote a biography of Einstein to tell you a little bit about that story. Oh look, it's Tom Levenson who wrote a biography about Einstein. So Tom is-- TOM LEVENSON: Oh, I need a microphone. Those of who have taken courses in [INAUDIBLE]-- and I'm sure that's all of you because of the GIRs-- know this is larger than the usual [INAUDIBLE] class. So I'm very used to microphones, but not in this context. OK. Is this-- yeah, it's on. Can you hear me? All right. So there are lots of ways to slice the story of Einstein by the time he reaches the EPR experiment, which is Einstein, Podolsky, and Rosen for the three people who actually wrote the paper. Just to dot the I's and cross the T's on the paper itself, Rosen is apparently for person who first came to Einstein. Podolsky and Rosen were two young physicists in Princeton after Einstein moved to Princeton. Einstein moved to Princeton in 1933. About three weeks before-- I'm sorry, he moved to Princeton '33. He left Germany in 1932 December, about three weeks before Hitler took power. And he did so with decisiveness and dispatch and a head of almost all of his-- in fact, I think all of his German-Jewish physicist colleagues and those German physicists for whom the Hitler regime was unacceptable. Which shows that Einstein really was smarter than most of his peers. That's one of many different ways you can ascertain that. And so he came to Princeton in '33. He actually went to Caltech before we went to Princeton. As part of an ongoing visitor-ship he had there. Came back to Europe, hung with the queen of Belgium who was a friend of his. Went to England. And then headed across the Atlantic and took up residency in Princeton at the Institute for Advanced Studies where he stayed for the rest of his life. And over the course of the-- that was '33, he died in '55, I think. I should know that, but I think that's right. 22 years. He worked with a lot of different, mostly younger physicists. And Podolsky and Rosen were early members of that chain. So Rosen was talking with him some day and starts to frame this experiment. Einstein develops it. The three of them talk about it. They write the paper and they put it out. And I want to share with you, actually, a really lovely description of the way the problem was represented in a way by-- this is from a book that I recommend to all of you. It's actually really hard to find. It's really sweet. Jeremy Bernstein, who is a physicist. He's sort of been around. A physicist and writer. He's in his eighties now. He lives in Aspen. He worked with CERN for a number of years. He's always been independent. He wrote for the New Yorker. Anyway. So you've all heard of the physicist Bell, I assume? Bell's inequality? OK. So Bell had a lovely way to describe-- I'm trying to find. I had this marked and then I lost my piece of paper. I have already lost it. That's terrible. So Bell has this wonderful way of describing this problem of entanglement. And it's based on his description of an actual person. I was going to read you his actual quote. Now I'm just going to paraphrase it for you. He had a friend named, I think, Bartelstein. Or at least someone known to him. Who had two quirks. An unusual color sense and a taste for mismatched socks. And so Bell used to say, if you saw Bartelstein and you could only see one leg and that sock was pink, you knew to a certainty that the other sock was not pink. He comes up I think-- I'm trying to remember who this is originally attributed to. Same thing. If you have a coin and you cut it in half down the-- so you've got two coin shape disks. You cut the disk in half, not-- and you have one side that's the head and the other side that's the tail. And they're separated. They get handed to two different gamblers. And one gambler tries to cheat the gambling establishment by tossing in his half coin. And you see the head that you know somebody-- somebody at some other casino is cheating by tossing in the half coin that only has a tail on it. So there are lots of ways to represent this. And many physicists being very witty indeed have come up with different metaphors for it. So Allan just described for you the basic claim in EPR. Its weirdness. That you have two particles that are entangled in some way and then go their separate ways. And thus you have-- if you have knowledge of what's the state of one, you have certain knowledge of the state of the other, violating relativistic ideas of locality. And just kind of making you queasy if you're sort of approaching it naively. What Einstein, Podolsky, and Rosen argued was actually something a little bit-- in fact, the paper comes to an end on that note of queasiness. But what they argue is a little bit more subtle. Because what they said is, OK. You perform this thought experiment. You send the two particles off. You measure position of one, you know absolutely the position of the other. You've conferred-- and the paper turns on a discussion of the connection between a measurement-- a physical measurement-- and a property of physical reality. And they have definition for what reality is. And that is something whose-- if you can perform a measurement, you know that quantity absolutely. I don't have the mathematics to express that properly. But that'll do for this hand waving. You can then do another experiment and measure a complementary property. And you know that piece of reality. But you can't do the-- so on the one hand, quantum mechanics says you can't know physical reality to this level of precision. And on the other hand, the fact that you can do that measurement violates the relativistic picture of reality. So you have what they claimed was a paradox. And this paper was published. And it received a range of reactions from indifference by younger physicists who said, we don't care that it's weird. We're going to keep on doing quantum mechanics and performing experiments and making measurements. And just see where this leads us. Remember, this is happening in the mid '30s. 1935. One of these three books will tell me precisely in a moment. And the quantum theory, as it turned into quantum mechanics, developed in its first period between '23 and '27. And by '35, you have enormous numbers of productive results and unexpected things and the prediction of the positron and then its observation. And I mean, the theory is enormously, dramatically, excitingly productive. So those who are really heads down doing the work are, for the most part, saying, this is fine. We'll get back to it when we're old and retired and bored. But that wasn't the uniform case. And most notably Niels Bohr found this paper really troubling. And spent about six weeks, apparently, discussing this and trying to come up with a response to it. And what he responded was essentially that-- in some ways, it was the same reaction as his younger colleagues. Get over it. But more precisely, it was he said, there's no description of reality that excludes the measuring apparatus anymore. You can't make statements about physical reality unless you include a description of the measuring apparatus. And you've said that we can measure this one quantity with precision and know the other thing. And then we can subsequently, in a separate observation, measure a complimentary quality and know the other one. You still can't know much them at the same time. It's still true that the complementarity in essence means that once you know one part of the picture, you know some other part the picture. And that's just the nature of the quantum world. Einstein had argued that the EPR paradox suggested that quantum mechanics was incomplete. And Bohr essentially responded in effect that Einstein's description of quantum mechanical explanation was inadequate. The important thing to remember-- and I want to just spend a couple minutes going back into the pre-history of all this, and then a couple minutes speculating on why Einstein reached the position he did. And what that might tell you about the practice of science as a lived experience as opposed to one reflected in your textbooks. But the thing to remember is that there's nothing logically wrong with the EPR paper. Right? You know. It does what it says it does and there's no overt error in it. And there's nothing wrong with Bohr's response. And in fact, when the experiments were-- Bell formalized the-- what Bell's inequality really does is it formalized the two arguments. It says, if Bohr is right, you will observe this in the experiment. And if Einstein is right, you would observe something different. The experiments were done, and I imagine are still being done, as sort of demonstrations. And they showed that Bohr's interpretation was correct and that yes, quantum mechanics produces results that are non-local just as Allan described to you. And that the world really is as strange as people first glimpsed in 1925, '26, and '27. And the question of whether or not that strangeness is adequately explained without the explanations that you're going to learn in this class and subsequent ones are quote "complete" or not. And completeness is a very funny, very, very tricky concept. But the question whether or not the framework of quantum mechanics is somehow unsatisfactory in any kind of formal a technical sense is one that's at least partly dependent on your scientific temperament, I think. So that's the cartoon version of what happened in '35. Einstein with his two young colleagues proposes-- really you should understand the EPR paper as a description in detail of a consequence of quantum theory as it was then expressed with the conclusion-- and I just want to read you this thing-- no reasonable definition of reality could be expected to permit such a result. In fact, it's called a paradox, but it isn't. It's a complaint. You know, it's a memo to the Flying Spaghetti Monster that the universe shouldn't be this way if, in fact, experiments turn out to show that it is and they have. The oddity here for a biographer of Einstein opposed to a physicist is given what you know about Einstein between 1879 when he's born and say, 1925 or so when he completes the last of his really great physics. How could-- I mean, I actually keep-- I've been working on Einstein off and on for years and years. I keep finding out new ways in which he's just inconceivably bright and on target and with a nose for the right problem and insightful. And yet by the 1935, 10 years later, he's still a relatively young man. He's in his '50s. Which being in my '50s I think is an extremely young man. Just 10 years after doing work that's right on the edge of modern quantum mechanics that is essential to its foundation. That's really extraordinary. 10 years after that, he's saying, no reasonable definition of reality should be permitted to behave this way. Where does that come from? Well the first thing I want to tell you-- again, this is all going to be a really cartoon version. Because there's not much time, I understand. Is that Einstein-- I mean, how much are you aware of Einstein's role in the creation of the quantum theory? A lot. I mean, a lot? None? OK. I mean, I'm going to make a claim that except for Heisenberg, Schrodinger, maybe Bohr. Maybe Born. Maybe a couple of others. There's no one more important to the quantum theory than Einstein. And you could maybe even argue that from a sort of foundational point of view that without Einstein, rigorous thinking about quantum mechanics would have taken much, much longer. I mean, he's really central to it. Planck in 1900 publishes as an ad hoc solution to the black body problem the first quantum theory. In 1905, Einstein says it's not an ad hoc thing. If you look at the photoelectric effect is the particular problem he's dealing with in explanation. But that's really-- the behavior of the photoelectric effect is really presented as the confirmation of this idea that light exists as quanta with particular kinds of behavior. And from 1905 on, he spends probably more time on quantum problems than he did on any other physics problems. Certainly more than on relativity. Though he spent enormous energy on special and general relativity. One of most amazing things about Einstein, in fact, is that despite the fact that he's seen and appears by 1935 to be this hidebound old guy who can't accommodate himself to the new world is he had an extraordinary capacity to do what the Red Queen did in Alice in Wonderland. and believe two impossible things before breakfast. Just think in 1905. April, he publishes The Quantum Theory Of Light. June he publishes Special Relativity, which treats light as a wave. And makes no mention of his revolutionary-- I mean, he called it revolutionary in a private letter in 1905. So he knew what he had in the quantum theory of light. But in special relativity, go read the special relativity paper. It's actually lovely reading. And you'll see he doesn't even nod in that direction. He doesn't say, you know this is a hero-- he says nothing. So he's capable of doing excellent-- and there's a reason that year is called the annus mirabilis, the year of miracles. And in part it's because Einstein is able to actually really focus on these things. And I realize class is almost over. So there's several more steps in Einstein's quantum journey. But you know, what you should take away is that Einstein's ability to deal with the problems of quantum pictures extends to the point-- he's the first person to suggest there might be a problem with causality in quantum mechanics. He does this in 1917, eight years before quantum mechanics is invented. When he starts looking at what the quote "classical" quantum theory tells you about the emission of radiation from an excited atom. He realized you can't predict it precisely. Radioactive decay has the same problem. He says, well-- he writes in a letter. I don't want to give up causality, but we may have to. So he's aware of these things. So I see class is over now at 12:30. OK, sorry. So juts to finish off, the question here is why does Einstein give up on this. And the answer, I think, is because in addition to his-- as he started at the beginning of his career, he says with the quantum theory of light and with special relativity, ignore your physical pictures. Try and look at the phenomena and explain those. And by 1935, that becomes very difficult for him. Because the phenomenology becomes too strange. One of the things that quantum mechanics does is it takes away the immediate ability to visualize physical systems. There. English is my first language sometimes. And that's an aesthetic failure on Einstein's part. He had the intellectual capacity and explicitly said, quantum mechanics is a logically consistent theory that incredibly powerfully describes lots of problems. He said that in print. He nominated Schrodinger and Heisenberg for Nobel Prizes twice. I mean, he wasn't stupid. He was Albert Einstein. But he was aesthetically incapable of pursuing this new physics in ways that were possible under the research possibilities of the time. And that is what I would leave you with. Physics is an aesthetic as well as an intellectual pursuit. So thank you all. [APPLAUSE]
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_3_The_Wave_Function.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So today, before we get into the meat of today's lecture, Matt has very kindly-- Professor Evans has very kindly agreed to do an experiment. Yeah, so for those of you all who are in recitations both he and Barton talked about polarization in recitation last week. And Matt will pick it up from there. MATTHEW EVANS: So back to the ancient past-- this was a week ago. We had our hyper-intelligent monkeys that were sorting things. It all seemed very theoretical. And in recitation, I said things about polarizers. And I said, look, if we use polarizers, we can do exactly the same thing as these monkeys. We just need to set up a little polarization experiment and the results are identical. You can use the one figure out the other. But I didn't have this or a nice polarizer ready then to give a demo, so here we go. What I'm going to show you is that, if we start with something polarized here with all white-- and right now I have all vertical polarization here-- and if I just put on this other box there, which is going to be another polarizer, if I put it the same way, this is all of our white electrons coming through all white. See? It doesn't really do much. And if I look at the black output over here of the second Keller sorting box, that's the same as turning my polarizer 90 degrees, so nothing comes out black. So if we remove this guy from the middle, you have just exactly what you'd expect. You sort here, you have white, and you get all white out. Great. Everyone thought that was easy. We all had that figured out. This box got thrown in the center here and it became sort of confusing, because you thought, well, they were white-- I'm going to throw my box in the middle here-- that's this guy at 45 degrees. And then if I throw this guy on the end again, the idea was, well, they were all white here, so maybe this guy identified the soft ones from the white ones. And now we have white and soft. And it should still all be white, right? So I put this guy on up here. They should all come out but they sort of don't. And if you say, well, are they black? Well, no, they're not really black, either. They're some sort of strange combination of the two. All right, so that's this experiment done in polarizers. But let me just play the polarizer trick a little bit, because it's fun. So this is if I say, vertical polarization and how many of them come out horizontal? So here I'm saying, white, and how many of them will come out black? That's the analogy. The answer is none of them. And strangely, if I take this thing, which seems to just attenuate-- this is our middle box here-- and I just stuff it in between them, I can get something to come out even though I still have crossed polarizers on the side. So you can see the middle region is now brighter and you can still see the dark corners there of the crossed polarizers. And as I turn this guy around, I can make that better or worse. The maximum is somewhere right there, and then it goes off again. So this is a way of understanding our electron-sorting, hyper-intelligent monkeys in terms of polarizations. And here it's just a vector projected on another vector projected on another vector-- something everybody knows how to do. So here's the polarization analogy of the Stern-Gerlach experiment. PROFESSOR: Awesome. So the polarization analogy for interference effects in quantum mechanics is a canonical one in the texts of quantum mechanics. So you'll find lots of books talking about this. It's a very useful analogy, and I encourage you to read more about it. We won't talk about it a whole lot more, but it's a useful one. All right, before I get going, any questions from last lecture? Last lecture was pretty much self-contained. It was experimental results. No, nothing? All right. The one thing that I want to add to the last lecture-- one last experimental observation. I glossed over something that's kind of important, which is the following. So we started off by saying, look, we know that if I have a ray of light, it's an electromagnetic wave, and it has some wavelength lambda. And yet the photoelectric effect tells us that, in addition to having the wavelength lambda, the energy-- it has a frequency, as well, a frequency in time. And the photoelectric effect suggested that the energy is proportional to the frequency. And we write this as h nu and h bar is equal to h upon 2 pi and omega is equal to 2 pi nu. So this is just the angular frequency, rather than the number-per-time frequency. And h bar is the reduced Planck constant. So I'll typically write h bar omega rather than h nu, because these two pi's will just cause us endless pain if we don't use the bar. Anyway, so to an electromagnetic wave, we have a wavelength and a frequency and the photoelectric effect led us to predict that the energy is linearly proportional to the frequency, with the linear proportionality coefficient h bar-- Planck constant-- and the momentum is equal to h upon lambda, also known as-- I'm going to write this as h bar k, which is equal to h upon lambda, where here, again, h bar is h upon 2 pi. And so k is equal to 2 pi upon lambda. So k is called the wave number and you should have seen this in 8.03. So these are our basic relations for light. We know that light, as an electromagnetic wave, has a frequency and a wavelength-- or a wave number, an inverse wavelength. And the claim of the photoelectric effect is that the energy and the momenta of that light are thus quantized, that light comes in chunks. So it has a wave-like aspect and it also has properties that are more familiar from particles. Now, early on shortly after Einstein proposed this, a young French physicist named de Broglie said, well, look, OK, this is true of light. Light has both wave-like and particle-like properties. Why is it just light? The world would be much more parsimonious if this relation were true not just of light, but also of all particles. I am thus conjecturing, with no evidence whatsoever, that, in fact, this relation holds not just for light, but for any object. Any object with momentum p has associated to it a wavelength or a wave number, which is p upon h bar. Every object that has energy E has associated with it a wave with frequency omega. To those electrons that we send through the Davisson-Germer experiment apparatus, which are sent in with definite energy, there must be a frequency associated with it, omega and a wavelength lambda associated with it. And what we saw from the Davisson-Germer experiment was experimental confirmation of that prediction-- that electrons have both particulate and wave-like features simultaneously. So these relations are called the de Broglie relations or "de BROG-lee"-- I leave it up to you to decide how to pronounce that. And those relations are going to play an important role for us in the next few lectures. I just wanted to give them a name and a little context. This is a good example of parsimony and elegance-- the theoretical elegance leading you to an idea that turns out to be true of the world. Now, that's a dangerous strategy for finding truth. Boy, wouldn't it be nice if--? Wouldn't it be nice if we didn't have to pay taxes but we also had Medicare? So it's not a terribly useful guide all the time, but sometimes it really does lead you in the right direction. And this is a great example of physical intuition, wildly divorced from experiment, pushing you in the right direction. I'm making it sound a little more shocking than-- well, it was shocking. It was just shocking. OK, so with that said, let me introduce the moves for the next few lectures. For the next several lectures, here's what we're going to do. I am not going to give you experimental motivation. I've given you experimental motivation. I'm going to give you a set of rules, a set of postulates. These are going to be the rules of quantum mechanics. And what quantum mechanics is for us is a set of rules to allow us to make predictions about the world. And these rules will be awesome if their predictions are good. And if their predictions are bad, these rules will suck. We will avoid bad rules to the degree possible. I'm going to give you what we've learned over the past 100 years-- wow-- of developing quantum mechanics. That is amazing. Wow. OK, yeah, over the past 100 years of developing quantum mechanics. And I'm going to give them to you as a series of postulates and then we're going to work through the consequences, and then we're going to spend the rest of the semester studying examples to develop an understanding for what the rules of quantum mechanics are giving you. So we're just going to scrap classical mechanics and start over from scratch. So let me do that. And to begin, let me start with the definition of a system. And to understand that definition, I want to start with classical mechanics as a guide. So in classical mechanics-- let's think about the easiest classical system you can-- just a single particle sitting somewhere. In classical mechanics of a single particle, how do you specify the configuration, or the state-- just different words for the same thing-- how do you specify the configuration or state of the system? AUDIENCE: By position and momentum. PROFESSOR: Specify the position and momentum, exactly. So in classical mechanics, if you want to completely specify the configuration of a system, all you have to do is give me x and p for my particle. And if you tell me this, I know everything. If you know these numbers, you know everything. In particular, what I mean by saying you know everything is that, if there's anything else you want to measure-- the energy, for example. The energy is just some function of the position and momentum. And if you know the position and momentum, you can unambiguously calculate the energy. Similarly, the angular momentum, which is a vector, you can calculate it if you know x and p-- which is just r cross p. So this gives you complete knowledge of the system. There's nothing more to know if you know that data. Now, there are certainly still questions that you can't answer given knowledge of x and p. For example, are there 14 invisible monkeys standing behind me? I'm here. I'm not moving. Are there 14 invisible monkeys standing behind me? You can't answer that. It's a stupid question, right? OK, let me give you another example. The electron is x and p, some position. Is it happy? Right, so there are still questions you can't answer. The point is, complete knowledge of the system to answer any physically observable question-- any question that could be meaningfully turned into an experiment, the answer is contained in knowing the state of the system. But this can't possibly be true in quantum mechanics, because, as you saw in the problem set and as we've discussed previously, there's an uncertainty relation which says that your knowledge-- or your uncertainty, rather, in the position of a particle, quantum mechanically-- I'm not even going to say quantum mechanically. I'm just going to say the real world. So in the real world, our uncertainty in the position of our point-like object and our uncertainty in the momentum is always greater than or roughly equal to something that's proportional to Planck's constant. You can't be arbitrarily confident of the position and of the momentum simultaneously. You worked through a good example of this on the problem set. We saw this in the two-slit experiment and the interference of electrons. This is something we're going to have to deal with. So as a consequence, you can't possibly specify the position and the momentum with confidence of a system. You can't do it. This was a myth. It was a good approximation-- turned out to be false. So the first thing we need is to specify the state, the configuration of a system. So what specifies the configuration of a system? And so this brings us to the first postulate. The configuration, or state, of a system-- and here again, just for simplicity, I'm going to talk about a single object-- of a quantum object is completely specified by a single function, a wave function, which I will denote generally psi of x, which is a complex function. The state of the quantum object is completely specified once you know the wave function of the system, which is a function of position. Let me emphasize that this is a first pass at the postulates. What we're going to do is go through the basic postulates of quantum mechanics, then we'll go through them again and give them a little more generality. And then we'll go through them again and give them full generality. That last pass is 8.05. So let me give you some examples. Let me just draw some characteristic wave functions. And these are going to turn out to be useful for us. So for example, consider the following function. So here is 0 and we're plotting as a function of x. And then plotting the real part of psi of x. So first consider a very narrowly supported function. It's basically 0 everywhere, except it has some particular spot at what I'll call x1. Here's another wave function-- 0. It's basically 0 except for some special spot at x2. And again, I'm plotting the real part of psi. And I'm plotting the real part of psi because A, psi is a complex function-- at every point it specifies a complex number. And B, I can't draw complex numbers. So to keep my head from exploding, I'm just plotting the real part of the wave function. But you should never forget that the wave function is complex. So for the moment, I'm going to assume that the imaginary part is 0. I'm just going to draw the real parts. So let me draw a couple more examples. What else could be a good wave function? Well, those are fine. What about-- again, we want a function of x and I'm going to draw the real part. And another one. So this is going to be a perfectly good wave function. And let me draw two more. So what else could be a reasonable wave function? Well-- this is harder than you'd think. Oh, God. OK, so that could be the wave function, I don't know. That is actually my signature. My wife calls it a little [INAUDIBLE]. OK, so here's the deal. Psi is a complex function. Psi also needs to not be a stupid function. OK so you have to ask me-- look, could it be any function? Any arbitrary function? So this is going to be a job for us. We're going to define what it means to be not-stupid function. Well, this is a completely reasonable function-- it's fine. This is a reasonable function. Another reasonable function. Reasonable. That's a little weird, but it's not horrible. That's stupid. So we're going to have to come up with a good definition of what not stupid means. So fine, these are all functions. One of them is multivalued and that looks a little worrying, but they're all functions. So here's the problem. What does it mean? So postulate 2-- The meaning of the wave function is that the probability that upon measurement the object is found at the position x is equal to the norm squared of psi of x. If you know the system is ascribed to the wave function psi, and you want to look at point x, you want to know with what probability will I find the particle there, the answer is psi squared. Notice that this is a complex number, but absolute value squared, or norm squared, of a complex number is always a real, non-negative number. And that's important because we want our probabilities to be real, non-negative numbers. Could be 0, right? Could be 0 chance of something. Can't be negative 7 chance. Incidentally, there also can't be probability 2. So that means that the total probability had better be normalized. So let me just say this in words, though, first. So P, which is the norm squared of psi, determines the probability-- and, in particular, the probability density-- that the object in state psi, in the state given by the wave function psi of x, will be found at x. So there's the second postulate. So in particular, when I say it's a probability density, what I mean is the probability that it is found between the position x and x plus dx is equal to P of x dx, which is equal to psi of x squared dx. Does that make sense? So the probability that it's found in this infinitesimal interval is equal to this density times dx or psi squared dx. Now again, it's crucial that the wave function is in fact properly normalized. Because if I say, look, something could either be here or it could be here, what's the sum of the probability that it's here plus the probability that it's here? It had better be 1, or there's some other possibility. So probabilities have to sum to 1. Total probability that you find something somewhere must be 1. So what that tells you is that total probability, which is equal to the integral over all possible values of x-- so if I sum over all possible values of P of x-- all values-- should be equal to 1. And we can write this as integral dx over all values of x. And I write "all" here rather than putting minus infinity to infinity because some systems will be defined from 1 to minus 1, some systems will be defined from minus infinity to infinity-- all just means integrate over all possible values-- hold on one sec-- of psi squared. AUDIENCE: Are you going to use different notation for probability density than probability? PROFESSOR: I'm not going to. Probability density is going to have just one argument, and total probability is going to have an interval as an argument. So they're distinct and this is just the notation I like. Other questions? Just as a side note, what are the dimensions of the wave function? So everyone think about this one for second. What are the dimensions? AUDIENCE: Is it 1 over square root length PROFESSOR: Awesome. Yes. It's 1 over root length. The dimensions of psi are 1 over root length. And the way to see that is that this should be equal to 1. It's a total probability. This is an infinitesimal length, so this has dimensions of length. This has no dimension, so this must have dimensions of 1 over length. And so psi itself of x most have dimensions of 1 over length. Now, something I want to emphasize, I'm going to emphasize, over and over in this class is dimensional analysis. You need to become comfortable with dimensional analysis. It's absolutely essential. It's essential for two reasons. First off, it's essential because I'm going to be merciless in taking off points if you do write down a dimensionally false thing. If you write down something on a problem set or an exam that's like, a length is equal to a velocity-- ooh, not good. But the second thing is, forget the fact that I'm going to take off points. Dimensional analysis is an incredibly powerful tool for you. You can check something that you've just calculated and, better yet, sometimes you can just avoid a calculation entirely by doing a dimensional analysis and seeing that there's only one possible way to build something of dimensions length in your system. So we'll do that over and over again. But this is a question I want you guys to start asking yourselves at every step along the way of a calculation-- what are the dimensions of all the objects in my system? Something smells like smoke. So with that said, if that's the meaning of the wave function, what physically can we take away from knowing these wave functions? Well, if this is the wave function, let's draw the probability distribution. What's the probability distribution? P of x. And the probability distribution here is really very simple. It's again 0 squared is still 0 so it's still just a big spike at x1 and this one is a big spike at x2. Everyone cool with that? So what do you know when I tell you that this is the wave function describing your system? You know that with great confidence, you will find the particle to be sitting at x1 if you look. So what this is telling you is you expect x is roughly x1 and our uncertainty in x is small. Everyone cool with that? Similarly, here you see that the position is likely to be x2, and your uncertainty in your measurement-- your confidence in your prediction is another way to say it-- is quite good, so your uncertainty is small. Now what about these guys? Well, now it's norm squared. I need to tell you what the wave function is. Here, the wave function that I want-- so here is 0-- is e to the i k1 x. And here the wave function is equal to e to the i k2 x. And remember, I'm drawing the real part because of practical limitations. So the real part is just a sinusoid-- or, in fact, the cosine-- and similarly, here, the real part is a cosine. And I really should put 0 in the appropriate place, but-- that worked out well. So now the question is, what's the probability distribution, P of x, associated to these wave functions? So what's the norm squared of minus e to the i k1 x? If I have a complex number of phase e to the i alpha, and I take its norm squared, what do I get? 1. Right? But remember complex numbers. If we have a complex number alpha-- or sorry, if we have a complex number beta, then beta squared is by definition beta complex conjugate times beta. So e to the i alpha, if the complex conjugate is e to the minus i alpha, e to the i alpha times e to the minus i alpha, they cancel out-- that's 1. So if this is the wave function, what's the probability distribution? Well, it's 1. It's independent of x. So from this we've learned two important things. The first is, this is not properly normalized. That's not so key. But the most important thing is, if this is our wave function, and we subsequently measure the position of the particle-- we look at it, we say ah, there's the particle-- where are we likely to find it? Yeah, it could be anywhere. So what's the value of x you expect-- typical x? I have no idea, no information whatsoever. None. But and correspondingly, what is our uncertainty in the position of x that we'll measure? It's very large, exactly. Now, in order to tell you it's actually infinite, I need to stretch this off and tell you that it's actually constant off to infinity, and my arms aren't that long, so I'll just say large. Similarly here, if our wave function is e to the i k2 x-- here k2 is larger, the wavelength is shorter-- what's the probability distribution? It's, again, constant. So-- this is 0, 0. So again, x-- we have no idea, and our uncertainty in the x is large. And in fact it's very large. Questions? What about these guys? OK, this is the real challenge. OK, so if this is our wave function, and let's just say that it's real-- hard as it is to believe that-- then what's our probably distribution? Well, something like-- I don't know, something-- you get the point. OK, so if this is our probability distribution, where are we likely to find the particle? Well, now it's a little more difficult, right? Because we're unlikely to find it here, while it's reasonably likely to find here, unlikely here, reasonably likely, unlikely, like-- you know, it's a mess. So where is this? I'm not really sure. What's our uncertainty? Well, our uncertainty is not infinite because-- OK, my name ends at some point. So this is going to go to 0. So whatever else we know, we know it's in this region. So it's not infinite, it's not small, we'll say. But it's not arbitrarily small-- it's not tiny. Or sorry, it's not gigantic is what I meant. Our uncertainty is not gigantic. But it's still pretty nontrivial, because I can say with some confidence that it's more likely to be here than here, but I really don't know which of those peaks it's going to be found. OK, now what about this guy? What's the probability distribution well now you see why this is a stupid wave function, because it's multiply valued. It has multiple different values at every value of x. So what's the probability? Well, it might be root 2, maybe it's 1 over root 3. I'm really not sure. So this tells us an important lesson-- this is stupid. And what I mean by stupid is, it is multiply valued. So the wave function-- we just learned a lesson-- should be single valued. And we will explore some more on your problem set, which will be posted immediately after lecture. There are problems that walk you through a variety of other potential pathologies of the wave function and guide you to some more intuition. For example, the wave function really needs to be continuous as well. You'll see why. All right. Questions at this point? No? OK. So these look like pretty useful wave functions, because they corresponded to the particle being at some definite spot. And I, for example, am at a reasonably definite spot. These two wave functions, though, look pretty much useless, because they give us no information whatsoever about what the position is. Everyone agree with that? Except-- remember the de Broglie relations. The de Broglie relations say that associated to a particle is also some wave. And the momentum of that particle is determined by the wavelength. It's inversely related to the wavelength. It's proportional to the wave number. Any energy is proportional to the frequency. Now, look at those wave functions. Those wave functions give us no position information whatsoever, but they have very definite wavelengths. Those are periodic functions with definite wavelengths. In particular, this guy has a wavelength of from here to here. It has a wave number k1. So that tells us that if we measure the momentum of this particle, we can be pretty confident, because it has a reasonably well-defined wavelength corresponding to some wave number k-- 2 pi upon the wavelength. It has some momentum, and if we measure it, we should be pretty confident that the momentum will be h-bar k1. Everybody agree with that? Looks like a sine wave. And de Broglie tells us that if you have a wave of wavelength lambda, that corresponds to a particle having momentum p. Now, how confident can we be in that estimation of the momentum? Well, if I tell you it's e to the i k x, that's exactly a periodic function with wavelength lambda 2 pi upon k. So how confident are we? Pretty confident. So our uncertainty in the momentum is tiny. Everyone agree? Similarly, for this wave, again we have a wavelength-- it's a periodic function, but the wavelength is much shorter. If the wavelength is much shorter, then k is much larger-- the momentum is much larger. So the momentum we expect to measure, which is roughly h-bar k2, is going to be much larger. What about our uncertainty? Again, it's a perfect periodic function so our uncertainty in the momentum is small. Everyone cool with that? And that comes, again, from the de Broglie relations. So questions at this point? You guys are real quiet today. Questions? AUDIENCE: So delta P is 0, basically? PROFESSOR: It's pretty small. Now, again, I haven't drawn this off to infinity, but if it's exactly the i k x, then yeah, it turns out to be 0. Now, an important thing, so let me rephrase your question slightly. So the question was, is delta P 0? Is it really 0? So here's a problem for us right now. We don't have a definition for delta P. So what is the definition of delta P? I haven't given you one. So here, when I said delta P is small, what I mean is, intuitively, just by eyeball, our confidence in that momentum is pretty good, using the de Broglie relations. I have not given you a definition, and that will be part of my job over the next couple of lectures. Very good question. Yeah. AUDIENCE: How do you code noise in that function? PROFESSOR: Awesome. AUDIENCE: Do you just have different wavelengths PROFESSOR: Yeah AUDIENCE: As you go along? PROFESSOR: Awesome. So for example, this-- does it have a definite wavelength? Not so much. So hold that question and wait until you see the next examples that I put on this board, and if that doesn't answer your question, ask it again, becayse it's a very important question. OK. AUDIENCE: When you talk about a photon, you always say a photon has a certain frequency. Doesn't that mean that it must be a wave because you have to fix the wave number k? PROFESSOR: Awesome question. Does every wave packet of light that hits your eye, does it always have a single, unique frequency? No, you can take multiple frequency sources and superpose them. An interesting choice of words I used there. All right, so the question is, since light has some wavelength, does every chunk of light have a definite-- this is the question, roughly. Yeah, so and the answer is, light doesn't always have a single-- You can have light coming at you that has many different wavelengths and put it in a prism and break it up into its various components. So you can have a superposition of different frequencies of light. We'll see the same effect happening for us. OK, so again, de Broglie made this conjecture that E is h-bar omega and P is h-bar k. This was verified in the Davisson-Germer experiment that we ran. But here, one of the things that's sort of latent in this is, what he means is, look, associated to every particle with energy N and momentum P is a plane wave of the form e to the i kx minus omega t. And this, properly, in three dimensions should be k dot x. But at this point, this is an important simplification. For the rest of 8.04, until otherwise specified, we are going to be doing one-dimensional quantum mechanics. So I'm going to remove arrow marks and dot products. There's going to be one spatial dimension and one time dimension. We're always going to have just one time dimension, but sometimes we'll have more spatial dimensions. But it's going to be a while until we get there. So for now, we're just going to have kx. So this is a general plane wave. And what de Broglie really was saying is that, somehow, associated to the particle with energy E and momentum P should be some wave, a plane wave, with wave number k and frequency omega. And that's the wave function associated to it. The thing is, not every wave function is a plane wave. Some wave functions are well localized. Some of them are just complicated morasses. Some of them are just a mess. So now is the most important postulate in quantum mechanics. I remember vividly, vividly, when I took the analog of this class. It was called Physics 143A at Harvard. And the professor at this point said-- I know him well now, he's a friend-- he said, this is what quantum mechanics is all about. And I was so psyched. And then he told me And it was like, that's ridiculous. Seriously? That's what quantum mechanics is all about? So I always felt like this is some weird thing, where old physicists go crazy. But it turns out I'm going to say exactly the same thing. This is the most important thing in all of quantum mechanics. It is all contained in the following proposition. Everything-- the two slit experiments, the box experiments, all the cool stuff in quantum mechanics, all the strange and counter intuitive stuff comes directly from the next postulate. So here it is. I love this. Three-- put a star on it. Given two possible wave functions or states-- I'll say configurations-- of a quantum system-- I wish there was "Ride of the Valkyries" playing in the background-- corresponding to two distinct wave functions-- f with an upper ns is going to be my notation for functions because I have to write it a lot-- psi1 and psi2-- and I'll say, of x-- the system-- is down-- can also be in a superposition. of psi1 and psi2, where alpha and beta are complex numbers. Given any two possible configurations of the system, there is also an allowed configuration of the system corresponding to being in an arbitrary superposition of them. If an electron can be hard and it can be soft, it can also be in an arbitrary superposition of being hard and soft. And what I mean by that is that hard corresponds to some particular wave function, soft will correspond to some particular wave function, and the superposition corresponds to a different wave function which is a linear combination of them. AUDIENCE: [INAUDIBLE] combination also have to be normalized? PROFESSOR: Yeah, OK, that's a very good question. So and alpha and beta are some complex numbers subject to the normalization condition. So indeed, this wave function should be properly normalized. Now, let me step back for second. There's an alternate way to phrase the probability distribution here, which goes like this, and I'm going to put it here. The alternate statement of the probability distribution is that the probability density at x is equal to psi of x norm squared divided by the integral over all x dx of psi squared. So notice that, if we properly normalize the wave function, this denominator is equal to 1-- and so it's not there, right, and then it's equivalent. But if we haven't properly normalized it, then this probability distribution is automatically properly normalized. Because this is a constant, when we integrate the top, that's equal to the bottom, it integrates to 1. So I prefer, personally, in thinking about this for the first pass to just require that we always be careful to choose some normalization. That won't always be easy, and so sometimes it's useful to forget about normalizing and just define the probability distribution that way. Is that cool? OK. This is the beating soul of quantum mechanics. Everything in quantum mechanics is in here. Everything in quantum mechanics is forced on us from these few principles and a couple of requirements of matching to reality. AUDIENCE: When you do this-- some of linear, some of two wave functions, can you get interference? PROFESSOR: Yes. Excellent. So the question is, when you have a sum of two wave functions, can you get some sort of interference effect? And the answer is, absolutely. And that's exactly we're going to do next. So in particular, let me look at a particular pair of superpositions. So let's swap these boards around so the parallelism is a little more obvious. So let's scrap these rather silly wave functions and come up with something that's a little more interesting. So instead of using those as characteristic wave functions, I want to build superpositions. So in particular, I want to start by taking an arbitrary-- both of these wave functions have a simple interpretation. This corresponds to a particle being here. This corresponds to a particle being here. I want to take a superposition of them. So here's my superposition. Oops, let's try that again. And my superposition-- so here is 0 and here is x1 and here is x2-- is going to be some amount times the first one plus some amount times the second one. There's a superposition. Similarly, I could have taken a superposition of the two functions on the second chalkboard. And again I'm taking a superposition of the complex e to the i k1 x and e to the i k2 x and then taking the real part. So that's a particular superposition, a particular linear combination. So now let's go back to this. This was a particle that was here. This is a particle that was there. When we take the superposition, what is the probability distribution? Where is this particle? Well, there's some amplitude that it's here, and there's some amplitude that it's here. And there's rather more amplitude that it's over here, but there's still some probability that it's over here. Where am I going to find the particle? I'm not so sure anymore. It's either going to be here or here, but I'm not positive. It's more likely to be here than it is to be here, but not a whole lot more. So where am I going to find the particle? Well, now we have to define this-- where am I going to find the particle? Look, if I did this experiment a whole bunch of times, it'd be over here more than it would be over here. So the average will be somewhere around here-- it'll be in between the two. So x is somewhere in between. That's where we expect to find it, on average. What's our uncertainty in the position? Well, it's not that small anymore. It's now of order x1 minus x2. Everyone agree with that? Now, what about this guy? Well, does this thing have a single wavelength? No. This is like light that comes at you from the sun. It has many wavelengths. In this case, it has just two-- I've added those two together. So this is a plane wave which is psi is e to the i k1 x plus e to the i k2 x. So in fact, it has two wavelengths associated with it. lambda1 lambda2. And so the probability distribution now, if we take the norm squared of this-- the probability distribution is the norm squared of this guy-- is no longer constant, but there's an interference term. And let's just see how that works out. Let me be very explicit about this. Note that the probability in our superposition of psi1 plus psi2, which I'll call e to the i k1 x plus e to the i k2 x, is equal to the norm squared of the wave function, which is the superposition psi1 plus beta psi2, which is equal to alpha squared psi1 squared plus beta squared psi2 squared plus alpha star psi1 star-- actually, let me write this over here-- beta psi2 plus alpha psi1 beta star psi2 star, where star means complex conjugation. But notice that this is equal to-- that first term is alpha squared times the first probability, or the probability of this thing, of alpha psi1, is equal to probability 1. This term, beta squared psi2 squared, is the probability that the second thing happens. But these terms can't be understood in terms of the probabilities of psi1 or the probability of psi2 alone. They're interference terms. So the superposition principle, together with the interpretation of the probability as the norm squared of the wave function, gives us a correction to the classical addition of probabilities, which is these interference terms. Everyone happy with that? Now, here's something very important to keep in mind. These things are norms squared of complex numbers. That means they're real, but in particular, they're non-negative. So these two are both real and non-negative. But what about this? This is not the norm squared of anything. However, this is its complex conjugate. When you take something and its complex conjugate and you add them together, you get something that's necessarily real. But it's not necessarily positive. So this is a funny thing. The probability that something happens if we add together our two configurations, we superpose two configurations, has a positive probability term. But it's also got terms that don't have a definite sign, that could be negative. It's always real. And you can check but this quantity is always greater than or equal to 0. It's never negative, the total quantity. So remember Bell's inequality that we talked about? Bell's inequality said, look, if we have the probability of one thing happening being P1, and the probability of the other thing happening being P2, the probability of both things happening is just P1 plus P2. And here we see that, in quantum mechanics, probabilities don't add that way. The wave functions add-- and the probability is the norm squared of the wave function. The wave functions add, not the probabilities. And that is what underlies all of the interference effects we've seen. And it's going to be the heart of the rest of quantum mechanics. So you're probably all going, in your head, more or less like I was when I took Intro Quantum, like-- yeah, but I mean, it's just, you know, you're adding complex numbers. But trust me on this one. This is where it's all starting. OK so let's go back to this. Similarly, let's look at this example. We've taken the norm squared. And now we have an interference effect. And now, our probability distribution, instead of being totally trivial and containing no information, our probability distribution now contains some information about the position of the object. It's likely to be here. It is unlikely to be here, likely and unlikely. We now have some position information. We don't have enough to say where it is. But x is-- you have some information. Now, our uncertainty still gigantic. Delta x is still huge. But OK, we just added together two plane waves. Yeah? AUDIENCE: Why is the probability not big, small, small, big, small, small? PROFESSOR: Excellent. This was the real part of the wave function. And the wave function is a complex quantity. When you take e to the i k1, and let me do this on the chalkboard. When we take e to the i k1 x plus e to the i k2 x-- Let me write this slightly differently-- e to the i a plus e to the i b and take its norm squared. So this is equal to-- I'm going to write this in a slightly more suggestive way-- the norm squared of e to the i a times 1 plus e to the i b minus a parentheses norm squared. So first off, the norm squared of a product of things is the product of the norm squareds. So I can do that. And this overall phase, the norm squared of a phase is just 1, so that's just 1. So now we have the norm squared of 1 plus a complex number. And so the norm squared of 1 is going to give me 1. The norm squared of the complex number is going to give me 1. And the cross terms are going to give me the real part-- twice the real part-- of e to the i b minus a, which is going to be equal to cosine of b minus a. And so what you see here is that you have a single frequency in the superposition. So good, our uncertainty is large. So let's look at this second example in a little more detail. By superimposing two states with wavelength lambda1 and lambda2 or k1 and k2 we get something that, OK, it's still not well localized-- we don't know where the particle is going to be-- but it's better localized than it was before, right? What happens if we superpose with three wavelengths, or four, or more? So for that, I want to pull out a Mathematica package. You guys should all have seen Fourier analysis in 18.03, but just in case, I'm putting on the web page, on the Stellar page, a notebook that walks you through the basics of Fourier analysis in Mathematica. You should all be fluent in Mathematica. If you're not, you should probably come up to speed on it. That's not what we wanted. Let's try that again. There we go. Oh, that's awesome-- where by awesome, I mean not. It's coming. OK, good. I'm not even going to mess with the screens after last time. So I'm not going to go through the details of this package, but what this does is walk you through the superposition of wave packets. So here I'm looking at the probability distribution coming from summing up a bunch of plane waves with some definite frequency. So here it's just one. That's one wave, so first we have-- let me make this bigger-- yes, stupid Mathematica tricks. So here we have the wave function and here we have the probability distribution, the norm squared. And it's sort of badly normalized here. So that's for a single wave. And as you see, the probability distribution is constant. And that's not 0, that's 0.15, it's just that I arbitrarily normalized this. So let's add two plane waves. And now what you see is the same effect as we had here. You see a slightly more localized wave function. Now you have a little bit of structure in the probability distribution. So there's the structure in the probability distribution. We have a little more information about where the particle is more likely to be here than it is to be here. Let's add one more. And as we keep adding more and more plane waves to our superposition, the wave function and the probability distribution associated with it become more and more well-localized until, as we go to very high numbers of plane waves that we're superposing, we get an extremely narrow probability distribution-- and wave function, for that matter-- extremely narrow corresponding to a particle that's very likely to be here and unlikely to be anywhere else. Everyone cool with that? What's the expense? Want have we lost in the process? Well we know with great confidence now that the particle will be found here upon observation. But what will its momentum be? Yeah, now it's the superposition of a whole bunch of different momenta. So if it's a superposition of a whole bunch of different momenta, here this is like superposition of a whole bunch of different positions-- likely to be here, likely to be here, likely to be here, likely to be here. What's our knowledge of its position? It's not very good. Similarly, now that we have superposed many different momenta with comparable strength. In fact, here they were all with unit strength. We now have no information about what the momentum is anymore. It could be anything in that superposition. So now we're seeing quite sharply the uncertainty relation. And here it is. So the uncertainty relation is now pretty clear from these guys. So that didn't work? And I'm going to leave it alone. This is enough for the Fourier analysis, but that Fourier package is available with extensive commentary on the Stellar web page. AUDIENCE: Now is that sharp definition in the position caused by the interference between all those waves and all that-- PROFESSOR: That's exactly what it is. Precisely. It's precisely the interference between the different momentum nodes that leads to certainty in the position. That's exactly right. Yeah. AUDIENCE: So as we're certain of the position, we will not be certain of the momentum. PROFESSOR: Exactly. And here we are. So in this example, we have no idea what the position is, but we're quite confident of the momentum. Here we have no idea what the position is, but we have great confidence in the momentum. Similarly here, we have less perfect confidence of the position, and here we have less perfect confidence in the momentum. It would be nice to be able to estimate what our uncertainty is in the momentum here and what our uncertainty is in the position here. So we're going to have to do that. That's going to be one of our next tasks. Other questions? Yeah. AUDIENCE: In this half of the blackboard, you said, obviously, if we do it a bunch of times it'll have more in the x2 than in the x1. PROFESSOR: Yes. AUDIENCE: The average, it will never physically be at that-- PROFESSOR: Yeah, that's right, so, because it's a probability distribution, it won't be exactly at that point. But it'll be nearby. OK, so in order to be more precise-- And so for example for this what we do here's a quick question. How well do you know the position of this particle? Pretty well, right? But how well do you know its momentum? Well, we'd all like to say not very, but tell me why. Why is your uncertainty in the momentum of the particle large? AUDIENCE: Heisenberg's uncertainty principle. PROFESSOR: Yeah, but that's a cheat because we haven't actually proved Heisenberg's uncertainty principle. It's just something we're inheriting. AUDIENCE: I believe it. PROFESSOR: I believe it, too. But I want a better argument because I believe all sorts of crazy stuff. So-- I really do. Black holes, fluids, I mean look, don't get me started. Yeah. AUDIENCE: You can take the Fourier transform of it. PROFESSOR: Yeah, excellent. OK, we'll get to that in just one sec. So before taking the Fourier transform, which is an excellent-- so the answer was, just take a Fourier transform, that's going to give you some information. We're going to do that in just a moment. But before we do Fourier transform, just intuitively, why would de Broglie look at this and say, no, that doesn't have a definite momentum. AUDIENCE: There's no clear wavelength. PROFESSOR: Yeah, there's no wavelength, right? It's not periodic by any stretch of the imagination. It doesn't look like a thing with a definite wavelength. And de Broglie said, look, if you have a definite wavelength then you have a definite momentum. And if you have a definite momentum, you have a definite wavelength. This is not a wave with a definite wavelength, so it is not corresponding to the wave function for a particle with a definite momentum. So our momentum is unknown-- so this is large. And similarly, here, our uncertainty in the momentum is large. So to do better than this, we need to introduce the Fourier transform, and I want to do that now. So you should all have seen Fourier series in 8.03. Now we're going to do the Fourier transform. And I'm going to introduce this to you in 8.04 conventions in the following way. And the theorem says the following-- we're not going to prove it by any stretch of the imagination, but the theorem says-- any function f of x that is sufficiently well-behaved-- it shouldn't be discontinuous, it shouldn't be singular-- any reasonably well-behaved, non-stupid function f of x can be built by superposing enough plane waves of the form e to the ikx. Enough may be infinite. So any function f of x can be expressed as 1 over root 2 pi, and this root 2 pi is a choice of normalization-- everyone has their own conventions, and these are the ones we'll be using in 8.04 throughout-- minus infinity to infinity dk f tilde of k e to the ikx. So here, what we're doing is, we're summing over plane waves of the form e to the ikx. These are modes with a definite wavelength 2 pi upon k. f tilde of k is telling us the amplitude of the wave with wavelengths lambda or wave number k. And we sum over all possible values. And the claim is, any function can be expressed as a superposition of plane waves in this form. Cool? And this is for functions which are non-periodic on the real line, rather than periodic functions on the interval, which is what you should've seen in 8.03. Now, conveniently, if you know f tilde of k, you can compute f of x by doing the sum. But suppose you know f of x and you want to determine what the coefficients are, the expansion coefficients. That's the inverse Fourier transform. And the statement for that is that f tilde of k is equal to 1 over root 2 pi integral from minus infinity to infinity dx f of x e to the minus ikx. OK, that's sometimes referred to as the inverse Fourier transform. And here's something absolutely essential. f tilde of k, the Fourier transform coefficients of f of x, are completely equivalent. If you know f of x, you can determine f tilde of k. And if you know f tilde of k, you can determine f of x by just doing a sum, by just adding them up. So now here's the physical version of this-- oh, I can't slide that out-- I'm now going to put here. Oh. No, I'm not. I'm going to put that down here. So the physical version of this is that any wave function psi of x can be expressed as the superposition in the form psi of x is equal to 1 over root 2 pi integral from minus infinity to infinity dk psi tilde of k e to the ikx of states, or wave functions, with a definite momentum p is equal to h bar k. And so now, it's useful to sketch the Fourier transforms of each of these functions. In fact, we want this up here. So here we have the function and its probability distribution. Now I want to draw the Fourier transforms of these guys. So here's psi tilde of k, a function of a different variable than of x, but nonetheless, it's illuminating to draw them next to each other. And again, I'm drawing the real part. And here, x2-- had I had my druthers about me, I would have put x2 at a larger value. Good, so it's further off to the right there. I'm so loathe to erase the superposition principle. But fortunately, I'm not there yet. Let's look at the Fourier transform of these guys. The Fourier transform of this guy-- this is k. Psi tilde of k well, that's something with a definite value of k. And it's Fourier transform-- this is 0-- there's k1. And for this guy-- there's 0-- k2. And now if we look at the Fourier transforms of these guys, see, this way I don't have to erase the superposition principle-- and the Fourier transform of this guy, so note that there's a sort of pleasing symmetry here. If your wave function is well localized, corresponding to a reasonably well-defined position, then your Fourier transform is not well localized, corresponding to not having a definite momentum. On the other hand, if you have definite momentum, your position is not well defined, but the Fourier transform has a single peak at the value of k corresponding to the momentum of your wave function. Everyone cool with that? So here's a question-- sorry, there was a raised hand. Yeah? AUDIENCE: Are we going to learn in this class how to determine the Fourier transforms of these non-stupid functions? PROFESSOR: Yes, that will be your homework. On your homework is an extensive list of functions for you to compute Fourier transforms of. And that will be the job of problem sets and recitation. So Fourier series and computing-- yeah, you know what's coming-- Fourier series are assumed to have been covered for everyone in 8.03 and 18.03 in some linear combination thereof. And Fourier transforms-- [LAUGHTER AND GROANS] I couldn't help it. So Fourier transforms are a slight embiggening of the space of Fourier series, because we're not looking at periodic functions. AUDIENCE: So when we're doing the Fourier transforms of a wave function, we're basically writing it as a continuous set of different waves. Can we write it as a discrete set? So as a Fourier series? PROFESSOR: Absolutely, so, however, what is true about Fourier series? When you use a discrete set of momenta, which are linear, which are-- It must be a periodic function, exactly. So here what we've done is, we've said, look, we're writing our wave function, our arbitrary wave function, as a continuous superposition of a continuous value of possible momenta. This is absolutely correct. This is exactly what we're doing. However, that's kind of annoying, because maybe you just want one momentum and two momenta and three momenta. What if you want a discrete series? So discrete is fine. But if you make that discrete series integer-related to each other, which is what you do with Fourier series, you force the function f of x to be periodic. And we don't want that, in general, because life isn't periodic. Thank goodness, right? I mean, there's like one film in which it-- but so-- it's a good movie. So that's the essential difference between Fourier series and Fourier transforms. Fourier transforms are continuous in k and do not assume periodicity of the function. Other questions? Yeah. AUDIENCE: So basically, the Fourier transform associates an amplitude and a phase for each of the individual momenta. PROFESSOR: Precisely. Precisely correct. So let me say that again. So the question was-- so a Fourier transform effectively associates a magnitude and a phase for each possible wave vector. And that's exactly right. So here there's some amplitude and phase-- this is a complex number, because this is a complex function-- there's some complex number which is an amplitude and a phase associated to every possible momentum going into the superposition. That amplitude may be 0. There may be no contribution for a large number of momenta, or maybe insignificantly small. But it is indeed doing precisely that. It is associating an amplitude and a phase for every plane wave, with every different value of momentum. And you can compute, before panicking, precisely what that amplitude and phase is by using the inverse Fourier transform. So there's no magic here. You just calculate. You can use your calculator, literally-- I hate that word. OK so now, here's a natural question. So if this is the Fourier transform of our wave function, we already knew that this wave function corresponded to having a definite-- from de Broglie, we know that it has a definite momentum. We also see that its Fourier transform looks like this. So that leads to a reasonable guess. What do you think the probability distribution P of k is-- the probability density to find the momentum to have wave vector h-bar k? AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah, that's a pretty reasonable guess. So we're totally pulling this out of the dark-- psi of k norm squared. OK, well let's see if that works. So psy of k norm squared for this is going to give us a nice, well localized function. And so that makes a lot of sense. That's exactly what we expected, right? Definite value of P with very small uncertainty. Similarly here. Definite value of P, with a very small uncertainty. Rock on. However, let's look at this guy. What is the expected value of P if this is the Fourier transform? Well remember, we have to take the norm squared, and psi of k was e to the i k x1-- the Fourier transform. You will do much practice on taking Fourier transforms on the problem set. Where did my eraser go? There it is. Farewell, principle one. So what does norm squared of psi tilde look like? Well, just like before, the norm squared is constant, because the norm squared of a phase is constant. And again, the norm squared-- this is psi tilde of k norm squared-- we believe, we're conjecturing this is P of k. You will prove this relation on your problem set. You'll prove that it follow from what we said before. And similarly, this is constant-- e to the i k x2. So now we have no knowledge of the momenta. So that also fits. The momenta is, we have no idea. And uncertainty is large. And the momenta is, we have no idea. And the uncertainty is large. So in all these cases, we see that we satisfy quite nicely the uncertainty relation-- small position momentum, large momentum uncertainty. Large position uncertainty, we're allowed to have small momentum uncertainty. And here, it's a little more complicated. We have a little bit of knowledge of position, and we have a little bit of knowledge of the momenta. We have a little bit of knowledge of position, and we have a little bit of knowledge of momenta. So we'll walk through examples with superposition like this on the problem set. Last questions before we get going? OK so I have two things to do before we're done. The first is, after lecture ends, I have clickers. And anyone who wants to borrow clickers, you're welcome to come down and pick them up on a first come first served basis. I will start using the clickers in the next lecture. So if you don't already have one, get one now. But the second thing is-- don't get started yet. I have a demo to do. And last time I told you-- this is awesome. It's like I'm an experimentalist for a day. Last time I told you that one of the experimental facts of lice is-- of lice. One of the experimental facts of lice. One of the experimental facts of life is that there is uncertainty in the world and that there is probability. There are unlikely events that happen with some probability, some finite probability. And a good example of the randomness of the real world involves radiation. So hopefully you can hear this. Apparently, I'm not very radioactive. You'd be surprised at the things that are radioactive. Ah, got a little tick. Shh. This is a plate sold at an Amish county fair. It's called vaseline ware and it's made of local clays. [GEIGER COUNTER CLICKS] It's got uranium in it. But I want to emphasize-- exactly when something goes click, it sounds pretty random. And it's actually a better random number generator than anything you'll find in Mathematica or C. In fact, for some purposes, the decay of radioactive isotopes is used as the perfect random number generator. Because it really is totally random, as far as anyone can tell. But here's my favorite. People used to eat off these. [MUCH LOUDER, DENSER CLICKS] See you next time.
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_4_Expectations_Momentum_and_Uncertainty.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare ocw.mit.edu. PROFESSOR: Anything lingering and disturbing or bewildering? No? Nothing? All right. OK, so the story so far is basically three postulates. The first is that the configuration of a particle is given by, or described by, a wave function psi of x. Yeah? So in particular, just to flesh this out a little more, if we were in 3D, for example-- which we're not. We're currently in our one dimensional tripped out tricycles. In 3D, the wave function would be a function of all three positions x, y and z. If we had two particles, our wave function would be a function of the position of each particle. x1, x2, and so on. So we'll go through lots of details and examples later on. But for the most part, we're going to be sticking with single particle in one dimension for the next few weeks. Now again, I want to emphasize this is our first pass through our definition of quantum mechanics. Once we use the language and the machinery a little bit, we're going to develop a more general, more coherent set of rules or definition of quantum mechanics. But this is our first pass. Two, the meaning of the wave function is that the norm squared psi of x, norm squared, it's complex, dx is the probability of finding the particle- There's an n in their. Finding the particle-- in the region between x and x plus dx. So psi squared itself, norm squared, is the probability density. OK? And third, the superposition principle. If there are two possible configurations the system can be in, which in quantum mechanics means two different wave functions that could describe the system given psi 1 and psi 2, two wave functions that could describe two different configurations of the system. For example, the particles here or the particles over here. It's also possible to find the system in a superposition of those two psi is equal to some arbitrary linear combination alpha psi 1 plus beta psi 2 of x. OK? So some things to note-- so questions about those before we move on? No questions? Nothing? Really? You're going to make he threaten you with something. I know there are questions. This is not trivial stuff. OK. So some things to note. The first is we want to normalize. We will generally normalize and require that the integral over all possible positions of the probability density psi of x norm squared is equal to 1. This is just saying that the total probability that we find the particle somewhere had better be one. This is like saying if I know a particle is in one of two boxes, because I've put a particle in one of the boxes. I just don't remember which one. Then the probability that it's in the first box plus probability that it's in the second box must be 100% or one. If it's less, then the particle has simply disappeared. And basic rule, things don't just disappear. So probability should be normalized. And this is our prescription. So a second thing to note is that all reasonable, or non stupid, functions psi of x are equally reasonable as wave functions. OK? So this is a very reasonable function. It's nice and smooth. It converges to 0 infinity. It's got all the nice properties you might want. This is also a reasonable function. It's a little annoying, but there it is. And they're both perfectly reasonable as wave functions. This on the other hand, not so much. So for two reasons. First off, it's discontinuous. And as you're going to show in your problem set, discontinuities are very bad for wave functions. So we need our wave functions to be continuous. The second is over some domain it's multi valued. There are two different values of the function. That's also bad, because what's the probability? It's the norm squared, but if it two values, two values for the probability, that doesn't make any sense. What's the probability that I'm going to fall over in 10 seconds? Well, it's small, but it's not actually equal to 1% or 3%. It's one of those. Hopefully is much lower than that. So all reasonable functions are equally reasonable as wave functions. And in particular, what that means is all states corresponding to reasonable wave functions are equally reasonable as physical states. There's no primacy in wave functions or in states. However, with that said, some wave functions are more equal than others. OK? And this is important, and coming up with a good definition of this is going to be an important challenge for us in the next couple of lectures. So in particular, this wave function has a nice simple interpretation. If I tell you this is psi of x, then what can you tell me about the particle whose wave function is the psi of x? What can you tell me about it? What do you know? AUDIENCE: [INAUDIBLE]. PROFESSOR: It's here, right? It's not over here. Probability is basically 0. Probability is large. It's pretty much here with this great confidence. What about this guy? Less informative, right? It's less obvious what this wave function is telling me. So some wave functions are more equal in the sense that they have-- i.e. they have simple interpretations. So for example, this wave function continuing on infinitely, this wave function doesn't tell me where the particle is, but what does it tell me? AUDIENCE: Momentum. PROFESSOR: The momentum, exactly. So this is giving me information about the momentum of the particle because it has a well defined wavelength. So this one, I would also say is more equal than this one. They're both perfectly physical, but this one has a simple interpretation. And that's going to be important for us. Related to that is that any reasonable function psi of x can be expressed as a superposition of more equal wave functions, or more precisely easily interpretable wave functions. We saw this last time in the Fourier theorem. The Fourier theorem said look, take any wave function-- take any function, but I'm going to interpret in the language of quantum mechanics. Take any wave function which is given by some complex valued function, and it can be expressed as a superposition of plane waves. 1 over 2pi in our normalization integral dk psi tilde of k, but this is a set of coefficients. e to the ikx. So what are we doing here? We're saying pick a value of k. There's a number associated with it, which is going to be an a magnitude and a phase. And that's the magnitude and phase of a plane wave, e to the ikx. Now remember that e to the ikx is equal to cos kx plus i sin kx. Which you should all know, but just to remind you. This is a periodic function. These are periodic functions. So this is a plane wave with a definite wavelength, 2pi upon k. So this is a more equal wave function in the sense that it has a definite wavelength. We know what its momentum is. Its momentum is h bar k. Any function, we're saying, can be expressed as a superposition by summing over all possible values of k, all possible different wavelengths. Any function can be expressed as a superposition of wave functions with a definite momentum. That make sense? Fourier didn't think about it that way, but from quantum mechanics, this is the way we want to think about it. It's just a true statement. It's a mathematical fact. Questions about that? Similarly, I claim that I can expand the very same function, psi of x, as a superposition of states, not with definite momentum, but of states with definite position. So what's a state with a definite position? AUDIENCE: Delta. PROFESSOR: A delta function, exactly. So I claim that any function psi of x can be expanded a sum over all states with a definite position. So delta of-- well, what's a state with a definite position? x0. Delta of x minus x0. OK? This goes bing when x0 is equal to x. But I want a sum over all possible delta functions. That means all possible positions. That means all possible values of x0, dx0. And I need some coefficient function here. Well, the coefficient function I'm going to call psi of x0. So is this true? Is it true that I can take any function and expand it in a superposition of delta functions? Absolutely. Because look at what this equation does. Remember, delta function is your friend. It's a map from integrals to numbers or functions. So this integral, is an integral over x0. Here we have a delta of x minus x0. So this basically says the value of this integral is what you get by taking the integrand and replacing x by x0. Set x equals x0, that's when delta equals 0. So this is equal to the argument evaluated at x0 is equal to x. That's your psi of x. OK? Any arbitrarily ugly function can be expressed either as a superposition of states with definite momentum or a superposition of states with definite position. OK? And this is going to be true. We're going to find this is a general statement that any state can be expressed as a superposition of states with a well defined observable quantity for any observable quantity you want. So let me give you just a quick little bit of intuition. In 2D, this is a perfectly good vector, right? Now here's a question I want to ask you. Is that a superposition? Yeah. I mean every vector can be written as the sum of other vectors, right? And it can be done in an infinite number of ways, right? So there's no such thing as a state which is not a superposition. Every vector is a superposition of other vectors. It's a sum of other vector. So in particular we often find it useful to pick a basis and say look, I know what I mean by the vector y, y hat is a unit vector in this direction. I know what I mean by the vector x hat. It's a unit vector in this direction. And now I can ask, given that these are my natural guys, the guys I want to attend to, is this a superposition of x and y? Or is it just x or y? Well, that's a superposition. Whereas x hat itself is not. So this somehow is about finding convenient choice of basis. But any given vector can be expressed as a superposition of some pair of basis vectors or a different pair of basis vectors. There's nothing hallowed about your choice of basis. There's no God given basis for the universe. We look out in the universe in the Hubble deep field, and you don't see somewhere in the Hubble deep field an arrow going x, right? So there's no natural choice of basis, but it's sometimes convenient to pick a basis. This is the direction of the surface of the earth. This is the direction perpendicular to it. So sometimes particular basis sets have particular meanings to us. That's true in vectors. This is along the earth. This is perpendicular to it. This would be slightly strange. Maybe if you're leaning. And similarly, this is an expansion of a function as a sum, as a superposition of other functions. And you could have done this in any good space of functions. We'll talk about that more. These are particularly natural ones. They're more equal. These are ones with different definite values of position, different definite values of momentum. Everyone cool? Quickly what's the momentum associated to the plane wave e to the ikx? AUDIENCE: [INAUDIBLE]. PROFESSOR: h bar k. Good. So now I want to just quickly run over some concept questions for you. So whip out your clickers. OK, we'll do this verbally. All right, let's try this again. So how would you interpret this wave function? AUDIENCE: e. PROFESSOR: Solid. How do you know whether the particle is big or small by looking at the wave function? AUDIENCE: [INAUDIBLE]. PROFESSOR: All right. Two particles described by a plane wave of the form e to the ikx. Particle one is a smaller wavelength than particle two. Which particle has a larger momentum? Think about it, but don't say it out loud. And this sort of defeats the purpose of the clicker thing, because now I'm supposed to be able to know without you guys saying anything. So instead of saying it out loud, here's what I'd like you to do. Talk to the person next to you and discuss which one has the larger AUDIENCE: [CHATTER]. All right. Cool, so which one has the larger momentum? AUDIENCE: A. PROFESSOR: How come? [INTERPOSING VOICES] PROFESSOR: RIght, smaller wavelength. P equals h bar k. k equals 2pi over lambda. Solid? Smaller wavelength, higher momentum. If it has higher momentum, what do you just intuitively expect to know about its energy? It's probably higher. Are you positive about that? No, you need to know how the energy depends on the momentum, but it's probably higher. So this is an important little lesson that you probably all know from optics and maybe from core mechanics. Shorter wavelength thing, higher energy. Higher momentum for sure. Usually higher energy as well. Very useful rule of thumb to keep in mind. Indeed, it's particle one. OK next one. Compared to the wave function psi of x, it's Fourier transform, psi tilde of x contains more information, or less, or the same, or something. Don't say it out loud. OK, so how many people know the answer? Awesome. And how many people are not sure. OK, good. So talk to the person next to you and convince them briefly. All right. So let's vote. A, more information. B, less information. C, same. OK, good you got it. So these are not hard ones. This function, which is a sine wave of length l, 0 outside of that region. Which is closer to true? f has a single well defined wavelength for the most part? It's closer to true. This doesn't have to be exact. f has a single well defined wavelengths. Or f is made up of a wide range of wavelengths? Think it to yourself. Ponder that one for a minute. OK, now before we get talking about it. Hold on, hold on, hold on. Since we don't have clickers, but I want to pull off the same effect, and we can do this, because it's binary here. I want everyone close your eyes. Just close your eyes, just for a moment. Yeah. Or close the eyes of the person next to you. That's fine. And now and I want you to vote. A is f has a single well defined wavelength. B is f has a wide range of wavelengths. So how many people think A, a single wavelength? OK. Lower your hands, good. And how many people think B, a wide range of wavelengths? Awesome. So this is exactly what happens when we actually use clickers. It's 50/50. So now you guys need to talk to the person next to you and convince each other of the truth. AUDIENCE: [CHATTER]. All right, so the volume sort of tones down as people, I think, come to resolution. Close your eyes again. Once more into the breach, my friends. So close your eyes, and now let's vote again. f of x has a single, well defined wavelength. And now f of x is made up of a range of wavelengths? OK. There's a dramatic shift in the field to B, it has a wide range of wavelengths, not a single wavelength. And that is, in fact, the correct answer. OK, so learning happens. That was an empirical test. So does anyone want to defend this view that f is made of a wide range of wavelengths? Sure, bring it. AUDIENCE: So, the sine wave is an infinite, and it cancels out past minus l over 2 and positive l over 2, which means you need to add a bunch of wavelengths to actually cancel it out there. PROFESSOR: Awesome, exactly. Exactly. If you only had the thing of a single wavelength, it would continue with a single wavelength all the way out. In fact, there's a nice way to say this. When you have a sine wave, what can you say about it's-- we know that a sine wave is continuous, and it's continuous everywhere, right? It's also differentiable everywhere. Its derivative is continuous and differentiable everywhere, because it's a cosine, right? So if yo you take a superposition of sines and cosines, do you ever get a discontinuity? No. Do you ever get something whose derivative is discontinuous? No. So how would you ever reproduce a thing with a discontinuity using sines and cosines? Well, you'd need some infinite sum of sines and cosines where there's some technicality about the infinite limit being singular, because you can't do it a finite number of sines and cosines. That function is continuous, but its derivative is discontinuous. Yeah? So it's going to take an infinite number of sines and cosines to reproduce that little kink at the edge. Yeah? AUDIENCE: So a finite number of sines and cosines doesn't mean finding-- or an infinite number of sines and cosines doesn't mean infinite [? regular ?] sines and cosines, right? Because over a finite region [INAUDIBLE]. PROFESSOR: That's true, but you need arbitrarily-- so let's talk about that. That's an excellent question. That's a very good question. The question here is look, there's two different things you can be talking about. One is arbitrarily large and arbitrarily short wavelengths, so an arbitrary range of wavelengths. And the other is an infinite number. But an infinite number is silly, because there's a continuous variable here k. You got an infinite number of wavelengths between one and 1.2, right? It's continuous. So which one do you mean? So let's go back to this connection that we got a minute ago from short distance and high momentum. That thing looks like it has one particular wavelength. But I claim, in order to reproduce that as a superposition of states with definite momentum, I need arbitrarily high wavelength. And why do I need arbitrarily high wavelength modes? Why do we need to arbitrarily high momentum modes? Well, it's because of this. We have a kink. And this feature, what's the length scale of that feature? It's infinitesimally small, which means I'm going to have to-- in order to reproduce that, in order to probe it, I'm going to need a momentum that's arbitrarily large. So it's really about the range, not just the number. But you need arbitrarily large momentum. To construct or detect an arbitrarily small feature you need arbitrarily large momentum modes. Yeah? AUDIENCE: Why do you [INAUDIBLE]? Why don't you just say, oh you need an arbitrary small wavelength? Why wouldn't you just phrase that [INAUDIBLE]? PROFESSOR: I chose to phrase it that way because I want an emphasize and encourage-- I emphasize you to think and encourage you to conflate short distance and large momentum. I want the connection between momentum and the length scale to be something that becomes intuitive to you. So when I talk about something with short features, I'm going to talk about it as something with large momentum. And that's because in a quantum mechanical system, something with short wavelength is something that carries large momentum. That cool? Great. Good question. AUDIENCE: So earlier you said that any reasonable wave function, a possible wave function, does that mean they're not supposed to be Fourier transformable? PROFESSOR: That's usually a condition. Yeah, exactly. We don't quite phrase it that way. And in fact, there's a problem on your problem set that will walk you through what we will mean. What should be true of the Fourier transform in order for this to reasonably function. And among other things-- and your intuition here is exactly right-- among other things, being able to have a Fourier transform where you don't have arbitrarily high momentum modes is going to be an important condition. That's going to turn to be related to the derivative being continuous. That's a very good question. So that's the optional problem 8 on problem set 2. Other questions? PROFESSOR: Cool, so that's it for the clicker questions. Sorry for the technology fail. So I'm just going to turn this off in disgust. That's really irritating. So today what I want to start on is pick up on the discussion of the uncertainty principle that we sort of outlined previously. The fact that when we have a wave function with reasonably well defined position corresponding to a particle with reasonably well defined position, it didn't have a reasonably well defined momentum and vice versa. The certainty of the momentum seems to imply lack of knowledge about the position and vice versa. So in order to do that, we need to define uncertainty. So I need to define for you delta x and delta p. So first I just want to run through what should be totally remedial probability, but it's always useful to just remember how these basic things work. So consider a set of people in a room, and I want to plot the number of people with a particular age as a function of the age of possible ages. So let's say we have 16 people, and at 14 we have one, and at 15 we have 1, and at 16 we have 3. And that's 16. And at 20 we have 2. And at 21 we have 4. And at 22 we have 5. And that's it. OK. So 1, 1, 3, 2, 4, 5. OK, so what's the probability that any given person in this group of 16 has a particular age? I'll call it a. So how do we compute the probability that they have age a? Well this is easy. It's the number that have age a over the total number. So note an important thing, an important side note, which is that the sum over all possible ages of the probability that you have age a is equal to 1, because it's just going to be the sum of the number with a particular age over the total number, which is just the sum of the number with any given age. So here's some questions. So what's the most likely age? If you grabbed one of these people from the room with a giant Erector set, and pull out a person, and let them dangle, and ask them what their age is, what's the most likely they'll have? AUDIENCE: 22. PROFESSOR: 22. On the other hand, what's the average age? Well, just by eyeball roughly what do you think it is? So around 19 or 20. It turns out to be 19.2 for this. OK. But if everyone had a little sticker on their lapel that says I'm 14, 15, 16, 20, 21 or 22, how many people have the age 19.2? None, right? So a useful thing is that the average need not be an observable value. This is going to come back to haunt us. Oops, 19.4. That's what I got. So in particular how did I get the average? I'm going to define some notation. This notation is going to stick with us for the rest of quantum mechanics. The average age, how do I compute it? So we all know this, but let me just be explicit about it. It's the sum over all possible ages of the number of the number of people with that age times the age divided by the total number of people. OK? So in this case, I'd go 14,14, 16, 16, 16, 20, 20, 21, 21, 21 21, 22, 22, 22, 22, 22. And so that's all I've written here. But notice that I can write this in a nice way. This is equal to the sum over all possible ages of a times the ratio of Na to N with a ratio of Na to n total. That's just the probability that any given person has a probability a. a times probability of a. So the expected value is the sum over all possible values of the value times the probability to get that value. Yeah? This is the same equation, but I'm going to box it. It's a very useful relation. And so, again, does the average have to be measurable? No, it certainly doesn't. And it usually isn't. So let's ask the same thing for the square of ages. What is the average of a squared? Square the ages. You might say, well, why would I ever care about that? But let's just be explicit about it. So following the same logic here, the average of a squared, the average value of the square of the ages is, well, I'm going to do exactly the same thing. It's just a squared, right? 14 squared, 15 squared, 16 square, 16 squared, 16 squared. So this is going to give me exactly the same expression. So over a of a squared probability of measuring a. And more generally, the expected value, or the average value of some function of a is equal-- and this is something you don't usually do-- is equal to the sum over a of f of a, the value of f given a particular value of a, times the probability that you measure that value of a in the first place. It's exactly the same logic as averages. Right, cool. So here's a quick question. Is a squared equal to the expected value of a squared? AUDIENCE: No. PROFESSOR: Right, in general no, not necessarily. So for example, the average value-- suppose we have a Gaussian centered at the origin. So here's a. Now a isn't age, but it's something-- I don't know. You include infants or whatever. It's not age. Its happiness on a given day. So what's the average value? Meh. Right? Sort of vaguely neutral, right? But on the other hand, if you take a squared, very few people have a squared as zero. Most people have a squared as not a 0 value. And most people are sort of in the middle. Most people are sort of hazy on what the day is. So in this case, the expected value of a, or the average value of a is 0. The average value of a squared is not equal to 0. Yeah? And that's because the squared has everything positive. So how do we characterize-- this gives us a useful tool for characterizing the width of a distribution. So here we have a distribution where its average value is 0, but its width is non-zero. And then the expectation value of a squared, the expected value of a squared, is non-zero. So how do we define the width of a distribution? This is going to be like our uncertainty. How happy are you today? Well, I'm not sure. How unsure are you? Well, that should give us a precise measure. So let me define three things. First the deviation. So the deviation is going to be a minus the average value of a. So this is just take the actual value of a and subtract off the average value of a. So we always get something that's centered at 0. I'm going to write it like this. Note, by the way, just a convenient thing to note. The average value of a minus it's average value. Well, what's the average value of 7? AUDIENCE: 7. PROFESSOR: OK, good. So that first term is the average value of a. And that second term is the average value of this number, which is just this number minus a. So this is 0. Yeah? The average value of a number is 0. The average value of this variable is the average value of that variable, but that's 0. So deviation is not a terribly good thing on average, because on average the deviation is always 0. That's what it means to say this is the average. So the derivation is saying how far is any particular instance from the average. And if you average those deviations, they always give you 0. So this is not a very good measure of the actual width of the system. But we can get a nice measure by getting the deviation squared. And let's take the mean of the derivation squared. So the mean of the derivation squared, mean of a minus the average value of a squared. This is what I'm going to call the standard deviation. Which is a little odd, because really you'd want to call it the standard deviation squared. But whatever. We use funny words. So now what does it mean if the average value of a is 0? It means it's centered at 0, but what does it mean if the standard deviation of a is 0? So if the standard deviation is 0, one then the distribution has no width, right? Because if there was any amplitude away from the average value, then that would give a non-zero strictly positive contribution to this average expectation, and this wouldn't be 0 anymore. So standard deviation is 0, as long as there's no width, which is why the standard deviation is a good useful measure of width or uncertainty. And just as a note, taking this seriously and taking the square, so standard deviation squared, this is equal to the average value of a squared minus twice a times the average value of a plus average value of a quantity squared. But if you do this out, this is going to be equal to a squared minus 2 average value of a average value of a. That's just minus twice the average value of a quantity squared. And then plus average value of a squared. So this is an alternate way of writing the standard deviation. OK? So we can either write it in this fashion or this fashion. And the notation for this is delta a squared. OK? So when I talk about an uncertainty, what I mean is, given my distribution, I compute the standard deviation. And the uncertainty is going to be the square root of the standard deviations squared. OK? So delta a, the words I'm going to use for this is the uncertainty in a given some probability distribution. Different probability distributions are going to give me different delta a's. So one thing that's sort of annoying is that when you write delta a, there's nothing in the notation that says which distribution you were talking about. When you have multiple distributions, or multiple possible probability distributions, sometimes it's useful to just put given the probability distribution p of a. This is not very often used, but sometimes it's very helpful when you're doing calculations just to keep track. Everyone cool with that? Yeah, questions? AUDIENCE: [INAUDIBLE] delta a squared, right? PROFESSOR: Yeah, exactly. Of delta a squared. Yeah. Other questions? Yeah? AUDIENCE: So really it should be parentheses [INAUDIBLE]. PROFESSOR: Yeah, it's just this is notation that's used typically, so I didn't put the parentheses around precisely to alert you to the stupidities of this notation. So any other questions? Good. OK, so let's just do the same thing for continuous variables. Now for continuous variables. I'm just going to write the expressions and just get them out of the way. So the average value of some x, given a probability distribution on x where x is a continuous variable, is going to be equal to the integral. Let's just say x is defined from minus infinity to infinity, which is pretty useful, or pretty typical. dx probability distribution of x times x. I shouldn't use curvy. I should just use x. And similarly for x squared, or more generally, for f of x, the average value of f of x, or the expected value of f of x given this probability distribution, is going to be equal to the integral dx minus infinity to infinity. The probability distribution of x times f of x. In direct analogy to what we had before. So this is all just mathematics. And we define the uncertainty in x is equal to the expectation value of x squared minus the expected value of x quantity squared. And this is delta x squared. If you see me dropping an exponent or a factor of 2, please, please, please tell me. So thank you for that. All of that is just straight up classical probability theory. And I just want to write this in the notation of quantum mechanics. Given that the system is in a state described by the wave function psi of x, the average value, the expected value of x, the typical value if you just observe the particle at some moment, is equal to the integral over all possible values of x. The probability distribution, psi of x norm squared x. And similarly, for any function of x, the expected value is going to be equal to the integral dx. The probability distribution, which is given by the norm squared of the wave function times f of x minus infinity to infinity. And same definition for uncertainty. And again, this notation is really dangerous, because the expected value of x depends on the probability distribution. In a physical system, the expected value of x depends on what the state of the system is, what the wave function is, and this notation doesn't indicate that. So there are a couple of ways to improve this notation. One of which is-- so this is, again, a sort of side note. One way to improve this notation x is to write the expected value of x in the state psi, so you write psi as a subscript. Another notation that will come back-- you'll see why this is a useful notation later in the semester-- is this notation, psi. And we will give meaning to this notation later, but I just want to alert you that it's used throughout books, and it means the same thing as what we're talking about the expected value of x given a particular state psi. OK? Yeah? AUDIENCE: To calculate the expected value of momentum do you need to transform the-- PROFESSOR: Excellent question. Excellent, excellent question. OK, so the question is, how do we do the same thing for momentum? If you want to compute the expected value of momentum, what do you have to do? Do you have to do some Fourier transform to the wave function? So this is a question that you're going to answer on the problem set and that we made a guess for last time. But quickly, let's just think about what it's going to be purely formally. Formally, if we want to know the likely value of the momentum, the likely value the momentum, it's a continuous variable. Just like any other observable variable, we can write as the integral over all possible values of momentum from, let's say, it could be minus infinity to infinity. The probability of having that momentum times momentum, right? Everyone cool with that? This is a tautology, right? This is what you mean by probability. But we need to know if we have a quantum mechanical system described by state psi of x, how do we can get the probability that you measure p? Do I want to do this now? Yeah, OK I do. And we need a guess. Question mark. We made a guess at the end of last lecture that, in quantum mechanics, this should be dp minus infinity to infinity of the Fourier transform. Psi tilde of p up to an h bar factor. Psi tilde of p, the Fourier transform p norm squared. OK, so we're guessing that the Fourier transform norm squared is equal to the probability of measuring the associated momentum. So that's a guess. That's a guess. And so on your problem set you're going to prove it. OK? So exactly the same logic goes through. It's a very good question, thanks. Other questions? Yeah? AUDIENCE: Is that p the momentum itself? Or is that the probability? PROFESSOR: So this is the probability of measuring momentum p. And that's the value p. We're summing over all p's. This is the probability, and that's actually p. So the Fourier transform is a function of the momentum in the same way that the wave function is a function of the position, right? So this is a function of the momentum. It's norm squared defines the probability. And then the p on the right is this p, because we're computing the expected value of p, or the average value of p. That make sense? Cool. Yeah? AUDIENCE: Are we then multiplying by p squared if we're doing all p's? Because we have the dp times p for each [INAUDIBLE]. PROFESSOR: No. So that's a very good question. So let's go back. Very good question. Let me phrase it in terms of position, because the same question comes up. Thank you for asking that. Look at this. This is weird. I'm going to phrase this as a dimensional analysis question. Tell me if this is the same question as you're asking. This is a thing with dimensions of what? Length, right? But over on the right hand side, we have a length and a probability, which is a number, and then another length. That looks like x squared, right? So why are we getting something with dimensions of length, not something with dimensions of length squared? And the answer is this is not a probability. It is a probability density. So it's got units of probability per unit length. So this has dimensions of one over length. So this quantity, p of x dx, tells me the probability, which is a pure number, no dimensions. The probability to find the particle between x and x plus dx. Cool? So that was our second postulate. Psi of x dx squared is the probability of finding it in this domain. And so what we're doing is we're summing over all such domains the probability times the value. Cool? So this is the difference between discrete, where we didn't have these probability densities, we just had numbers, pure numbers and pure probabilities. Now we have probability densities per unit whatever. Yeah? AUDIENCE: How do you pronounce the last notation that you wrote? PROFESSOR: How do you pronounce? Good, that's a good question. The question is, how do we pronounce these things. So this is called the expected value of x, or the average value of x, or most typically in quantum mechanics, the expectation value of x. So you can call it anything you want. This is the same thing. The psi is just to denote that this is in the state psi. And it can be pronounced in two ways. You can either say the expectation value of x, or the expectation of x in the state psi. And this would be pronounced one of two ways. The expectation value of x in the state psi, or psi x psi. Yeah. That's a very good question. But they mean the same thing. Now, I should emphasize that you can have two ways of describing something that mean the same thing, but they carry different connotations, right? Like have a friend who's a really nice guy. He's a mensch. He's a good guy. And so I could see he's a nice guy, I could say he's [? carinoso ?], and they mean different things in different languages. It's the same idea, but they have different flavors, right? So whatever your native language is, you've got some analog of this. This means something in a particular mathematical language for talking about quantum mechanics. And this has a different flavor. It carries different implications, and we'll see what that is later. We haven't got there yet. Yeah? AUDIENCE: Why is there a double notation of psi? PROFESSOR: Why is there a double notation of psi? Yeah, we'll see later. Roughly speaking, it's because in computing this expectation value, there's a psi squared. And so this is to remind you of that. Other questions? Terminology is one of the most annoying features of quantum mechanics. Yeah? AUDIENCE: So it seems like this [INAUDIBLE] variance is a really convenient way of doing it. How is it the Heisenberg uncertainty works exactly as it does for this definition of variance. PROFESSOR: That's a very good question. In order to answer that question, we need to actually work out the Heisenberg uncertainty relation. So the question is, look, this is some choice of uncertainty. You could have chosen some other definition of uncertainly. We could have considered the expectation value of x to the fourth minus x to the fourth and taken the fourth root of that. So why this one? And one answer is, indeed, the uncertainty relation works out quite nicely. But then I think important to say here is that there are many ways you could construct quantities. This is a convenient one, and we will discover that it has nice properties that we like. There is no God given reason why this had to be the right thing. I can say more, but I don't want to take the time to do it, so ask in office hours. OK, good. The second part of your question was why does the Heisenberg relation work out nicely in terms of these guys, and we will study that in extraordinary detail. We'll see that. So we're going to derive it twice soon and then later. The later version is better. So let me work out some examples. Or actually, I'm going to skip the examples in the interest of time. They're in the notes, and so they'll be posted on the web page. By the way, the first 18 lectures of notes are posted. I had a busy night last night. So let's come back to computing expectation values for momentum. So I want to go back to this and ask a silly-- I want to make some progress towards deriving this relation. So I want to start over on the definition of the expected value of momentum. And I'd like to do it directly in terms of the wave function. So how would we do this? So one way of saying this is what's the average value of p. Well, I can phrase this in terms of the wave function the following way. I'm going to sum over all positions dx. Expectation value of x squared from minus infinity to infinity. And then the momentum associated to the value x. So it's tempting to write something like this down to think maybe there's some p of x. This is a tempting thing to write down. Can we? Are we ever in a position to say intelligently that a particle-- that an electron is both hard and white? AUDIENCE: No. PROFESSOR: No, because being hard is a superposition of being black and white, right? Are we ever in a position to say that our particle has a definite position x and correspondingly a definite momentum p. It's not that we don't get too. It's that it doesn't make sense to do so. In general, being in a definite position means being in a superposition of having different values for momentum. And if you want a sharp way of saying this, look at these relations. They claim that any function can be expressed as a superposition of states with definite momentum, right? Well, among other things a state with definite position, x0, can be written as a superposition, 1 over 2pi integral dk. I'll call this delta tilde of k. e to the ikx. If you haven't played with delta functions before and you haven't seen this, then you will on the problem set, because we have a problem that works through a great many details. But in particular, it's clear that this is not-- this quantity can't be a delta function of k, because, if it were, this would be just e to the ikx. And that's definitely not a delta function. Meanwhile, what can you say about the continuity structure of a delta function. Is it continuous? No. Its derivative isn't continuous. Its second derivative. None of its derivatives are in any way continuous. They're all absolutely horrible, OK? So how many momentum modes am I going to need to superimpose in order to reproduce a function that has this sort of structure? An infinite number. And it turns out it's going to be an infinite number with the same amplitude, slightly different phase, OK? So you can never say that you're in a state with definite position and definite momentum. Being in a state with definite position means being in a superposition of being in a superposition. In fact, I'm just going right down the answer here. e to the ikx0. Being in a state with definite position means being in a superposition of states with arbitrary momentum and vice versa. You cannot be in a state with definite position, definite momentum. So this doesn't work. So what we want is we want some good definition. So this does not work. We want some good definition of p given that we're working with a wave function which is a function of x. What is that good definition of the momentum? We have a couple of hints. So hint the first. So this is what we're after. Hint the first is that a wave-- we know that given a wave with wave number k, which is equal 2pi over lambda, is associated, according to de Broglie and according to Davisson-Germer experiments, to a particle-- so having a particle-- a wave, with wave number k or wavelength lambda associated particle with momentum p is equal to h bar k. Yeah? But in particular, what is a plane with wavelength lambda or wave number k look like? That's e to the iks. And if I have a wave, a plane wave e to the iks, how do I get h bar k out of it? Note the following, the derivative with respect to x. Actually let me do this down here. Note that the derivative with respect to x of e to the ikx is equal to ik e to the ikx. There's nothing up my sleeves. So in particular, if I want to get h bar k, I can multiply by h bar and divide by i. Multiply by h bar, divide by i, derivative with respect to x e to the ikx. And this is equal to h bar k e to the ikx. That's suggestive. And I can write this as p e to the ikx. So let's quickly check the units. So first off, what are the units of h bar? Here's the super easy to remember the units of-- or dimensions of h bar are. Delta x delta p is h bar. OK? If you're ever in doubt, if you just remember, h bar has units of momentum times length. It's just the easiest way to remember it. You'll never forget it that way. So if h bar has units of momentum times length, what are the units of k? 1 over length. So does this dimensionally make sense? Yeah. Momentum times length divided by length number momentum. Good. So dimensionally we haven't lied yet. So this makes it tempting to say something like, well, hell h bar upon i derivative with respect to x is equal in some-- question mark, quotation mark-- p. Right? So at this point it's just tempting to say, look, trust me, p is h bar upon idx. But I don't know about you, but I find that deeply, deeply unsatisfying. So let me ask the question slightly differently. We've followed the de Broglie relations, and we've been led to the idea that using wave functions that there's some relationship between the momentum, the observable quantity that you measure with sticks, and meters, and stuff, and this operator, this differential operator, h bar upon on i derivative with respect to x. By the way, my notation for dx is the partial derivative with respect to x. Just notation. So if this is supposed to be true in some sense, what is momentum have to do with a derivative? Momentum is about velocities, which is like derivatives with respect to time, right? Times mass. Mass times derivative with respect to time, velocity. So what does it have to do with the derivative with respect to position? And this ties into the most beautiful theorem in classical mechanics, which is the Noether's theorem, named after the mathematician who discovered it, Emmy Noether. And just out of curiosity, how many people have seen Noether's theorem in class. Oh that's so sad. That's a sin. OK, so here's a statement of Noether's theorem, and it underlies an enormous amount of classical mechanics, but also of quantum mechanics. Noether, incidentally, was a mathematician. There's a whole wonderful story about Emmy Noether. Ville went to her and was like, look, I'm trying to understand the notion of energy. And this guy down the hall, Einstein, he has a theory called general relativity about curved space times and how that has something to do with gravity. But it doesn't make a lot of sense to me, because I don't even know how to define the energy. So how do you define momentum and energy in this guy's crazy theory? And so Noether, who was a mathematician, did all sorts of beautiful stuff in algebra, looked at the problem and was like I don't even know what it means in classical mechanics. So what is a mean in classical mechanics? So she went back to classical mechanics and, from first principles, came up with a good definition of momentum, which turns out to underlie the modern idea of conserved quantities and symmetries. And it's had enormous far reaching impact, and say her name would praise. So Noether tells us the following statement, to every symmetry-- and I should say continuous symmetry-- to every symmetry is associated a conserved quantity. OK? So in particular, what do I mean by symmetry? Well, for example, translations. x goes to x plus some length l. This could be done for arbitrary length l. So for example, translation by this much or translation by that much. These are translations. To every symmetry is associated a conserved quantity. What symmetry is associated to translations? Conservation of momentum, p dot. Time translations, t goes to t plus capital T. What's a conserved quantity associated with time translational symmetry? Energy, which is time independent. And rotations. Rotational symmetries. x, as a vector, goes to some rotation times x. What's conserved by virtue of rotational symmetry? AUDIENCE: Angular momentum. PROFESSOR: Angular momentum. Rock on. OK So quickly, I'm not going to prove to you Noether's theorem. It's one of the most beautiful and important theorems in physics, and you should all study it. But let me just convince you quickly that it's true in classical mechanics. And this was observed long before Noether pointed out why it was true in general. What does it mean to have transitional symmetry? It means that, if I do an experiment here and I do it here, I get exactly the same results. I translate the system and nothing changes. Cool? That's what I mean by saying I have a symmetry. You do this thing, and nothing changes. OK, so imagine I have a particle, a classical particle, and it's moving in some potential. This is u of x, right? And we know what the equations of motion are in classical mechanics from f equals ma p dot is equal to the force, which is minus the gradient of u. Minus the gradient of u. Right? That's f equals ma in terms of the potential. Now is the gradient of u 0? No. In this case, there's a force. So if I do an experiment here, do I get the same thing as doing my experiment here? AUDIENCE: No. PROFESSOR: Certainly not. The [? system ?] is not translationally invariant. The potential breaks that translational symmetry. What potential has translational symmetry? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah, constant. The only potential that has full translational symmetry in one dimension is translation invariant, i.e. constant. OK? What's the force? AUDIENCE: 0. PROFESSOR: 0. 0 gradient. So what's p dot? Yep. Noether's theorem. Solid. OK. Less trivial is conservation of energy. I claim and she claims-- and she's right-- that if the system has the same dynamics at one moment and a few moments later and, indeed, any amount of time later, if the laws of physics don't change in time, then there must be a conserved quantity called energy. There must be a conserved quantity. And that's Noether's theorem. So this is the first step, but this still doesn't tell us what momentum exactly has to do with a derivative with respect to space. We see that there's a relationship between translations and momentum conservation, but what's the relationship? So let's do this. I'm going to define an operation called translate by L. And what translate by L does is it takes f of x and it maps it to f of x minus L. So this is a thing that affects the translation. And why do I say that's a translation by L rather than minus L. Well, the point-- if you have some function like this, and it has a peak at 0, then after the translation, the peak is when x is equal to L. OK? So just to get the signs straight. So define this operation, which takes a function of x and translates it by L, but leaves it otherwise identical. So let's consider how translations behave on functions. And this is really cute. f of x minus L can be written as a Taylor expansion around the point x-- around the point L equals 0. So let's do Taylor expansion for small L. So this is equal to f of x minus L derivative with respect to x of f of x plus L squared over 2 derivative squared, two derivatives of x, f of x plus dot, dot, dot. Right? I'm just Taylor expanding. Nothing sneaky. Let's add the next term, actually. Let me do this on a whole new board. All right, so we have translate by L on f of x is equal to f of x minus L is equal to f of x. Now Taylor expanding minus L derivative with respect to x of f plus L squared over 2-- I'm not giving myself enough space. I'm sorry. f of x minus L is equal to f of x minus L with respect to x of f of x plus L squared over 2 to derivatives of x f of x minus L cubed over 6-- we're just Taylor expanding-- cubed with respect to x of f of x and so on. Yeah? But I'm going to write this in the following suggestive way. This is equal to 1 times f of x minus L derivative with respect to x f of x plus L squared over 2 derivative with respect to x squared times f of x minus L cubed over 6 derivative cubed with respect to x plus dot, dot, dot. Everybody good with that? But this is a series that you should recognize, a particular Taylor series for a particular function. It's a Taylor expansion for the AUDIENCE: Exponential. PROFESSOR: Exponential. e to the minus L derivative with respect to x f of x. Which is kind of awesome. So let's just check to make sure that this makes sense from dimensional grounds. So that's a derivative with respect to x as units of 1 over length. That's a length, so this is dimensionless, so we can exponentiate it. Now you might look at me and say, look, this is silly. You've taken an operation like derivative and exponentiated it. What does that mean? And that is what it means? [LAUGHTER] OK? So we're going to do this all the time in quantum mechanics. We're going to do things like exponentiate operations. We'll talk about it in more detail, but we're always going to define it in this fashion as a formal power series. Questions? AUDIENCE: Can you transform operators from one space to another? PROFESSOR: Oh, you totally can. But we'll come back to that. We're going to talk about operators next time. OK, so here's where we are. So from this what is a derivative with respect to x mean? What does a derivative with respect to x do? Well a derivative with respect to x is something that generates translations with respect to x through a Taylor expansion. If we have L be arbitrarily small, right? L is arbitrarily small. What is the translation by an arbitrarily small amount of f of x? Well, if L is arbitrarily small, we can drop all the higher order terms, and the change is just Ldx. So the derivative with respect to x is telling us about infinitesimal translations. Cool? The derivative with respect to a position is something that tells you, or controls, or generates infinitesimal translations. And if you exponentiate it, you do it many, many, many times in a particular way, you get a macroscopic finite translation. Cool? So this gives us three things. Translations in x are generated by derivative with respect to x. But through Noether's theorem translations, in x are associated to conservation of momentum. So you shouldn't be so shocked-- it's really not totally shocking-- that in quantum mechanics, where we're very interested in the action of things on functions, not just in positions, but on functions of position, it shouldn't be totally shocking that in quantum mechanics, the derivative with respect to x is related to the momentum in some particular way. Similarly, translations in t are going to be generated by what operation? Derivative with respect to time. So derivative with respect to time from Noether's theorem is associated with conservation of energy. That seems plausible. Derivative with respect to, I don't know, an angle, a rotation. That's going to be associated with what? Angular momentum? But angular momentum around the axis for whom this is the angle, so I'll call that z for the moment. And we're going to see these pop up over and over again. But here's the thing. We started out with these three principles today, and we've let ourselves to some sort of association between the momentum and the derivative like this. OK? And I've given you some reason to believe that this isn't totally insane. Translations are deeply connected with conservation of momentum. Transitional symmetry is deeply connected with conservation momentum. And an infinitesimal translation is nothing but a derivative with respect to position. Those are deeply linked concepts. But I didn't derive anything. I gave you no derivation whatsoever of the relationship between d dx and the momentum. Instead, I'm simply going to declare it. I'm going to declare that, in quantum mechanics-- you cannot stop me-- in quantum mechanics, p is represented by an operator, it's represented by the specific operator h bar upon I derivative with respect to x. And this is a declaration. OK? It is simply a fact. And when they say it's a fact, I mean two things by that. The first is it is a fact that, in quantum mechanics, momentum is represented by derivative with respect to x times h bar upon i. Secondly, it is a fact that, if you take this expression and you work with the rest of the postulates of quantum mechanics, including what's coming next lecture about operators and time evolution, you reproduce the physics of the real world. You reproduce it beautifully. You reproduce it so well that no other models have even ever vaguely come close to the explanatory power of quantum mechanics. OK? It is a fact. It is not true in some epistemic sense. You can't sit back and say, ah a priori starting with the integers we derive that p is equal to-- no, it's a model. But that's what physics does. Physics doesn't tell you what's true. Physics doesn't tell you what a priori did the world have to look like. Physics tells you this is a good model, and it works really well, and it fits the data. And to the degree that it doesn't fit the data, it's wrong. OK? This isn't something we derive. This is something we declare. We call it our model, and then we use it to calculate stuff, and we see if it fits the real world. Out, please, please leave. Thank you. [LAUGHTER] I love MIT. I really do. So let me close off at this point with the following observation. [LAUGHTER] We live in a world governed by probabilities. There's a finite probability that, at any given moment, that two pirates might walk into a room, OK? [LAUGHTER] You just never know. [APPLAUSE] But those probabilities can be computed in quantum mechanics. And they're computed in the following ways. They're computed the following ways as we'll study in great detail. If I take a state, psi of x, which is equal to e to the ikx, this is a state that has definite momentum h bar k. Right? We claimed this. This was de Broglie and Davisson-Germer. Note the following, take this operator and act on this wave function with this operator. What do you get? Well, we already know, because we constructed it to have this property. P hat on psi of x-- and I'm going to call this psi sub k of x, because it has a definite k-- is equal to h bar k psi k of x. A state with a definite momentum has the property that, when you hit it with the operation associated with momentum, you get back the same function times a constant, and that constant is exactly the momentum we ascribe to that plane wave. Is that cool? Yeah? AUDIENCE: Question. Just with notation, what does the hat above the p [INAUDIBLE]? PROFESSOR: Good. Excellent. So the hat above the P is to remind you that P is on a number. It's an operation. It's a rule for acting on functions. We'll talk about that in great detail next time. But here's what I want to emphasize. This is a state which is equal to all others in the sense that it's a perfectly reasonable wave function, but it's more equal because it has a simple interpretation. Right? The probability that I measure the momentum to be h bar k is one, and the probability that I measure it to be anything else is 0, correct? But I can always consider a state which is a superposition. Psi is equal to alpha, let's just do 1 over 2 e to the ikx. k1 x plus 1 over root 2 e to the minus ikx. Is this state a state with definite momentum? If I act on this state-- I'll call this i sub s-- if I act on this state with the momentum operator, do I get back this state times a constant? No. That's interesting. And so it seems to be that if we have a state with definite momentum and we act on it with momentum operator, we get back its momentum. If we have a state that's a superposition of different momentum and we act on it with a momentum operator, this gives us h bar k 1, this gives us h bar k2. So it changes which superposition we're talking about. We don't get back our same state. So the action of this operator on a state is going to tell us something about whether the state has definite value of the momentum. And these coefficients are going to turn out to contain all the information about the probability of the system. This is the probability when norm squared that will measure the system to have momentum k1. And this coefficient norm squared is going to tell us the probability that we have momentum k2. So I think the current wave function is something like a superposition of 1/10 psi pirates plus 1 minus is 1/100 square root. To normalize it properly psi no pirates. And I'll leave you with pondering this probability. See you guys next time. [APPLAUSE] CHRISTOPHER SMITH: We've come for Prof. Allan Adams. PROFESSOR: It is I. CHRISTOPHER SMITH: When in the chronicles of wasted time, I see descriptions of fairest rights, and I see lovely shows of lovely dames. And descriptions of ladies dead and lovely nights. Then in the bosom of fair loves depths. Of eyes, of foot, of eye, of brow. I see the antique pens do but express the beauty that you master now. So are all their praises but prophecies of this, our time. All you prefiguring. But though they had but diving eyes-- PROFESSOR: I was wrong about the probabilities. [LAUGHTER] CHRISTOPHER SMITH: But though they had but diving eyes, they had not skill enough you're worth to sing. For we which now behold these present days have eyes to behold. [LAUGHTER] But not tongues to praise. [APPLAUSE] It's not over. You wait. ARSHIA SURTI: Not marbled with gilded monuments of princes shall outlive this powerful rhyme. But you shall shine more bright in these contents that unswept stone besmear its sluttish tide. When wasteful war shall statues overturn and broils root out the work of masonry. Nor Mars his sword. Nor war's quick fire shall burn the living record of your memory. Gainst death and all oblivious enmity shall you pace forth. Your praise shall still find room, even in the eyes of all posterity. So no judgment arise till you yourself judgment arise. You live in this and dwell in lover's eyes. [APPLAUSE] CHRISTOPHER SMITH: Verily happy Valentine's day upon you. May your day be filled with love and poetry. Whatever state you're in, we will always love you. [LAUGHTER] [APPLAUSE] Signed, Jack Florian, James [INAUDIBLE]. [LAUGHTER] PROFESSOR: Thank you, sir. Thank you. CHRISTOPHER SMITH: Now we go. [APPLAUSE]
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_8_Quantum_Harmonic_Oscillator.txt
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PRESENTER: The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Today we begin with the harmonic oscillator. And before we get into the harmonic oscillator, I want to touch on a few concepts that have been mentioned in class and just elaborate on them. It is the issue of nodes, and how solutions look at, and why solutions have more and more nodes, why the ground state has no nodes. This kind of stuff. So these are just a collection of remarks and an argument for you to understand a little more intuitively why these properties hold. So one first thing I want to mention is, if you have a Schrodinger equation for an energy eigenstate. Schrodinger equation for an energy eigenstate. You have an equation of the from minus h squared over 2m. d second dx squared psi of x plus v of x psi of x equal e times psi of x. Now the issue of this equation is that you're trying to solve for two things at the same time. If you're looking at what we call bound states, now what is a bound state? A bound state is something that is not extended all that much. So a bound state will be a wave function that goes to 0 as the absolute value of x goes to infinity. So it's a probability function that certainly doesn't extend all the way to infinity. It just collapses. It's normalizable. So these are bound states, and you're looking for bound states of this equation. And your difficulty is that you don't know psi, and you don't know E either. So you have to solve a problem in which, if you were thinking oh this is just a plain differential equation, give me the value of E. We know the potential, just calculate it. That's not the way it works in quantum mechanics, because you need to have normalizable solutions. So at the end of the day, as will be very clear today, this E gets fixed. You cannot get arbitrary values of E's. So I want to make a couple of remarks about this equation. Is that there's this thing that can't happen. Certainly, if v of x is a smooth potential, then if you observer that the wave function vanishes at some point, and the derivative of the wave function vanishes at that same point, these two things imply that psi of x is identically 0. And therefore it means that you really are not interested in that. That's not a solution of the Schrodinger equation. Psi equals 0 is obviously solves this, but it's not interesting. It doesn't represent the particle. So what I claim here is that, if it happens to be that you're solving the Schrodinger problem with some potential that is smooth, you can take derivatives of it. And then you encounter that the wave function vanishes at some point, and its slope vanishes at that same point. Then the wave function vanishes completely. So you cannot have a wave function, a psi of x that does the following. Comes down here. It becomes an inflection point and goes down. This is not allowed. If the wave function vanishes at some point, then the wave function is going to do this. It's going to hit at an angle, because you cannot have that the wave function is 0 and needs the derivative 0 at the same point. And the reason is simple. I'm not going to prove it now here. It is that you have a second order differential equation, and a second order differential equation is completely determined by knowing the function at the point and the the derivative at the point. And if both are 0s, like the most trivial kind of initial condition, the only solution consistent with this is psi equals 0 everywhere. So this can't happen. And it's something good for you to remember. If you have to do a plot of a wave function, you should never have this. So this is what we call the node in a wave function. It's a place where the wave function vanishes, and the derivative of the wave function better not vanish at that point. So this is one claim that it's not hard to prove, but we just don't try to do it. And there's another claim that I want you to be aware of. That for bound states in one dimension, in the kind of thing that we're doing now in one dimension, no degeneracy is possible. What do I mean by that? You will never find two bound states of a potential that are different that have the same energy. It's just absolutely impossible. It's a very wonderful result, but also I'm not going to prove it here. Maybe it will be given as an exercise later in the course. And it's discussed in 805 as well. But that's another statement that is very important. There's no degeneracy. Now you've looked at this simple potential, the square well infinite one. And how does it look? You have x going from 0 to a. And the potential is 0. From 0 to a is infinite otherwise. The particle is bound to stay inside the two walls of this potential. So in here we've plotted the potential as a function of x. Here is x. And the wave functions are things that you know already. Yes? AUDIENCE: Is it true if you have two wells next to each other that there's still no degeneracy if it's an infinite barrier? PROFESSOR: If there's two wells and there's an infinite barrier between them, it's like having two universes. So it's not really a one dimensional problem. If you have an infinite barrier like two worlds that can talk to each other. So yes, then you would have degeneracy. It's like saying you can have here one atom of hydrogen or something in one energy level, here another one. They don't talk to each other. They're degenerate states. But in general, we're talking about normal potentials that are preferably smooth. Most of these things are true even if they're not smooth. But it's a little more delicate. But certainly two potentials that are separated by an infinite barrier is not part of what we really want to consider. OK so these wave functions start with m equals 0, 1, 2, 3, and psi ends of x are square root of 2 over a sin n plus 1 pi x over a. Things that you've seen already. And En, the energies are ever growing as a function of the integer n that characterizes them. 2ma squared. And the thing that you notice is that psi 0 has technically no nodes. That is to say these wave functions have to varnish at the end, because the potential becomes infinite, which means a particle really can go through. The wave function has to be continuous. There cannot be any wave function to the left. So it has to vanish here. These things don't count as nodes. It is like a bound state has to vanish at infinity. And that's not what we count the node. A node is somewhere in the middle of the range of x where the wave function vanishes. So this is the ground state. This is psi zero has no nodes. Psi one would be something like that, has one node. And try the next ones. They have more and more nodes. So psi n has n nodes. And the interesting thing is that this result actually is true for extremely general potentials. You don't have to just do the square well to see that the ground state has no nodes. That first excited state was one node, and so on and so forth. It's true in general. This is actually a very nice result, but its difficult to prove. In fact, it's pretty hard to prove. So there is a nice argument. Not 100% rigorous, but thoroughly nice and really physical that I'm going to present to you why this result is true. So let's try to do that. So here is the general case. So I'm going to take a smooth v of x. That will be of this kind. The potential, here is x, and this potential is going to be like that. Smooth all over. And I will actually have it that it actually goes to infinity as x goes to infinity. Many of these things are really not necessary, but it simplifies our life. OK, so here is a result that it's known to be true. If this thing grows to infinity, the potential never stops growing, you get infinite number of bound states at fixed energies. One energy, two energy, three energy, infinite number of bound states. Number of bound states. That's a fact. I will not try to prove it. We'll do it for the harmonic oscillator. We'll see those infinite number of states, but here we can't prove it easily. Nevertheless, what I want to argue for you is that these states, as the first one, will have no nodes. The second state, the first excited state, will have one node. Next will have two nodes, three nodes, four nodes. We want to understand what is this issue of the nodes. OK. You're not trying to prove everything. But we're trying to prove or understand something that is very important. So how do we prove this? Or how do we understand that the nodes-- so there will be an infinite number of bound states, psi 0, psi 1, psi 2, up to psi n, and it goes on. And psi n has n nodes. All right. So what I'm going to do, in order to understand this, is I'm going to produce what we will call screened potentials. Screened potentials. I'm going to select the lowest point of the potential here for convenience. And I'm going to call it x equals 0 is the lowest point. And the screen potential will have a parameter a. It's a potential which is equal to v of x if the absolute value of x is less than a. And its infinity if the absolute value of x is greater than a. So I come here, and I want to see this is this potential, v of x, what is the screened potential for sum a? Well, I wanted colored chalk, but I don't have it. I go mark a here minus a. Here are the points between x and a. Absolute value of x less than a. Throughout this region the screened potential is the potential that you have. Nevertheless, for the rest, its infinite. So the screened potential is this thing. Is infinite there, and it's here this thing. So it's just some potential. You take it a screen and you just see one part of the potential, and let it go to infinity. So that's a screen potential. So now what I'm going to do is that I'm going to try to argue that you could try to find the bound state of the screen potential. Unless you remove the screen, you will find, as you let a go to infinity, you will find the bound states of the original potential. It's reasonable that that's true, because as you remove the screen, you're letting more of the potential be exposed, and more of the potential be exposed. And the wave functions eventually die, so as the time that you're very far away, you affect the wave functions less and less. So that's the argument. We're going to try to argue that we're going to look at the bound states of the screened potentials and see what happened, whether they tell us about the bound states the original potential. So for this, I'm going to begin with a screen potential in which a goes to 0 and say that a is equal to epsilon, very small. So what potential so do I have? A very tiny potential here from epsilon to minus epsilon. Now I chose the original point down here to be the minimum. So actually, the bottom part of the potential is really flat. And if you take epsilon going to 0, well, the potential might do this, but really at the bottom for sufficiently small epsilon, this is an infinite square well with psis to epsilon. I chose the minimum so that you don't get something like this. If it would be a point with a slope, this would be an ugly thing. So let's choose the minimum. And we have the screen potential here, and that's it. Now look what we do. We say all right, here there is a ground state. Very tiny. Goes like that. Vanishes here. Vanishes there. And has no nodes. Very tiny. You know the two 0s are very close to each other. And now I'm going to try to increase the value of the screen a. So suppose we've increased the screen, and now the potential is here. And now we have a finite screen. Here is the potential. And I look at the wave function. How it looks. Here is psi 0. This ground state psi 0. Well, since this thing in here, the potential becomes infinite, the wave function still must vanish here and still must vanish here. Now just for your imagination, think of this. At this stage, it still more or less looks like this. Maybe. Now I'm going to ask, as I increase, can I produce a node? And look what's going to happen. So suppose it might happen that, as you increase, suddenly you produce a node. So here's what I'm saying here. I'm going to show it here. Suppose up to this point, there is no node. But then when I double it, when I increase it to twice the size, when I go to screen potential like that, suddenly there is a node in the middle. So if there is a node in the middle, one thing that could have happened is that you have this. And now look what must have happened then. As I stretch this, this slope must have been going down, and down, and down, until it flips to the other side to produce a node here. It could have happened on this side, but it's the same, so the argument is just done with this side. To produce a node you could have done somehow the slope here must have changed sine. But for that to happen continuously, at some point the this slope must have been 0. But you cannot have a 0 and 0 slope. So this thing can't flip, can't do this. Another thing that could have happened is that when we are here already, maybe the wave function looks like that. It doesn't flip at the edges, but produces something like that. But the only way this can happen continuously, and this potential is changing continuously, is for this thing at some intermediate stage, as you keep stretching the screen, this sort of starts to produce a depression here. And at some point, to get here it has to do this. But it can't do this either. It cannot vanish and have derivative like that. So actually, as you stretch the screen, there's no way to produce a node. That property forbids it. So by the time you go and take the screen to infinity, this wave function has no nodes. So that proves it that the ground state has no nodes. You could call this a physicist proof, which means-- not in the pejorative way. It means that it's reasonable, it's intuitive, and a mathematician working hard could make it rigorous. A bad physicist proof is one that is a little sloppy and no mathematician could fix it and make it work. So I think this is a good physics proof in that sense. Probably you can construct a real proof, or based on this, a very precise proof. Now look at excited states. Suppose you take now here this screen very little, and now consider the third excited state, psi three. I'm sorry, we'll call this psi 2 because it has two nodes. Well, maybe I should do psi 1. Psi 1. One node. Same thing. As you increase it, there's no way to create another node continuously. Because again, you have to flip at the edges, or you have to depress in the middle. So this one will evolve to a wave function that will have one node in the whole big potential. Now stayed does that state have more energy than the ground state? Well, it certainly begins with a small screen with more energy, because in the square well psi 1 has more energy. And that energy should be clear that it's not going to go below the energy of the ground state. Why? Because if it went below the energy of the ground state slowly, at some point for some value of the screen, it would have the same energy as the ground state. But no degeneracy is possible in one dimensional problems. So that can't happen. Cannot have that. So it will always stay a little higher. And therefore with one node you will be a little higher energy. With two nodes will be higher and higher. And that's it. That's the argument. Now, we've argued by this continuous deformation process that this potential not only has these bound states, but this is n nodes and En is greater than En prime for n greater than n prime. So the more nodes, the more energy. Pretty nice result, and that's really all I wanted to say about this problem. Are there any questions? Any? OK. So what we do now is the harmonic oscillator. That's going to keep us busy for the rest of today's lecture. It's a very interesting problem. And it's a most famous quantum mechanics problem in a sense, because it happens to be useful in many, many applications. If you have any potential-- so what is the characteristic of the harmonic oscillator? Harmonic oscillator. Oscillator. Well, the energy operator is p squared over 2m plus, we write, one half m omega squared x squared where omega is this omega that you always think of angular velocity, or angular frequency. It's more like angular frequency. Omega has units of 1 over time. It's actually put 2pi over the period of an oscillation. And this you know from classical mechanics. If you have a harmonic oscillator of this form, yeah, it actually oscillates with this frequency. And E is the energy operator, and this is the energy of the oscillator. So what defines an oscillator? It's something in which the potential energy, this term is v of x. v of x is quadratic in x. That is a harmonic oscillator. Then you arrange the constants to make sense. This has units of energy, because this has units of length squared. 1 over time squared. Length over time is velocity squared times mass is kinetic energy. So this term has the units of energy. And you good with that. And why is this useful? Because actually in any sort of arbitrary potential, rather general potential at least, whenever you have a minimum where the derivative vanishes, then the second derivative need not vanish. Then it's a good approximation to think of the potential at the minimum as a quadratic potential. It fits the potential nicely over a good region. And therefore when you have two molecules with a bound or something oscillating, there is a potential. It has a minimum at the equilibrium position. And the oscillations are governed by some harmonic oscillator. When you have photons in space time traveling, there is a set of harmonic oscillators that correspond to photons. Many, many applications. Endless amount of applications for the harmonic oscillator. So we really want to understand this system quantum mechanically. And what does that mean? Is that we really want to calculate and solve the Schrodinger equation. This is our first step in understanding the system. There's going to be a lot of work to be done even once we have the solutions of the Schrodinger equation. But the first thing is to figure out what are the energy eigenstates or the solutions of the Schrodinger equation for this problem. So notice that here in this problem there's an energy quantity. Remember, when you have a harmontonian like that, and people say so what is the ground state energy? Well, have to find the ground state wave function. Have to do things. Give me an hour, I'll find it. And all that. But if you want an approximate value, dimensional analysis will do it, roughly what is it going to be. Well, with this constant how do you produce an energy? Well, you remember what Einstein did, and you know that h bar omega has units of energy. So that's an energy associated with Lagrangian energy like quantity. And we expect that that energy is going to be the relevant energy. And in fact, we'll find that the ground state energy is just one half of that. There's another quantity that may be interesting. How about the length? How do you construct a length from these quantities? Well, you can start doing m omega h bar and put powers and struggle. I hate doing that. I always try to find some way of doing it and avoiding that thing. So I know that energies go like h over h squared over m length squared. So I'm going to call the length a quantity a. So ma squared. That has units of energy. And you should remember that because energy is b squared over 2m, and b by De Broglie is h bar over sub lamda. So h bar squared, lambda squared, and m here, that's units of energy. So that's a length. On the other hand, we have another way to construct an energy is with this thing, m omega squared length squared. So that's also m omega squared a squared. That's another energy. So from this equation I find that a to the fourth is h squared over m squared omega squared. And it's a little complicated, so a squared is h bar over m omega. So that's a length. Length squared. I don't want to take the square root. We can leave it for a moment there. But that's important because of that's a length scale. And if somebody would ask you in the ground state, how far is this particle oscillating, you would say probably about a square root of this. Would be a natural answer and probably about right. So OK, energy and units is very important to begin your analysis. So what is the Schrodinger equation? The Schrodinger equation for this thing is going to be minus h squared over 2m, d second psi, dx squared plus the potential, one half m omega squared x squared psi is equal E psi. And the big problem is I don't know psi and I don't know E. Now there's so many elegant ways of solving the harmonic oscillator. You will see those next lecture. Allan Adams will be back here. But we all have to go through once in your life through the direct, uninspired method of solving it. Because most of the times when you have a new problem, you will not come up with a beautiful, elegant method to avoid solving the differential equation. You will have to struggle with the differential equation. So today we struggle with the differential equation. We're going to just do it. And I'm going to do it slow enough and in detail enough that I hope you follow everything. I'll just keep a couple of things, but it will be one line computations that I will skip. So this equation is some sort of fairly difficult thing. And it's complicated and made fairly unpleasant by the presence of all these constants. What kind of equation is that with all these constants? They shouldn't be there, all this constants, in fact. So this is the first step, cleaning up the equation. We have to clean it up. Why? Because the nice functions in life like y double prime is equal to minus y have no units. The derivatives create no units. y has the same units of that, and the solution is sine of x, where x must have no units, because you cannot find the sine of one centimeter. So this thing, we should have the same thing here. No units anywhere. So how can we do that? This is an absolutely necessary first step. If you're going to be carrying all these constants, you'll get nowhere. So we have to clean it up. So what I'm going to try to see is that look, here is psi, psi, and psi. So suppose I do the following thing, that I will clean up the right hand side by dividing by something with units of energy. So I'm going to do the following way. I'm going to divide all by 1 over h bar omega. And this 2 I'm going to multiply by 2. So multiply by 2 over h bar omega. So what do I achieve with that first step? I achieve that these 2s disappear. Well, that's not too bad. Not that great either, I think. But in the right hand side, this has units of energy. And the right hand side will not have units of energy. So what do we get here? So we get minus. The h becomes an h alone over-- the m disappears-- so m omega. The second psi the x squared. The 1/2 disappeared, so m omega over h bar x squared psi equals 2 E over h bar omega psi. It looks actually quite better already. Should agree with that. It looks a lot nicer Now I can use a name for this. I want to call this the dimensionless value of the energy. So a calligraphic e. It has no units. It's telling me if I find some energy, that that energy really is this number, this pure number is how many times bigger is e with respect to h omega over 2. So I'll write this now as e psi. And look what I have. I have no units here. And I have a psi. And I have a psi. But things have worked out already. Look, the same factor here, h over m omega is upside down here. And this factor has units of length squared. Length squared times d d length squared has no units. And here's 1 over length squared. 1 over length squared times length squared. So things have worked out. And we can now simply say x is going to be equal to au, a new variable. This is going to be your new variable for your differential equation in which is this thing. And then this differential equation really has cleaned up perfectly well. So how does it look now? Well, it's all gone actually, because if you have x equals au, d dx by chain rule is 1 over a d du. And to derivatives this with respect to x it's 1 over a squared times the d second du squared. And this thing is a squared. So actually you cancel this factor. And when I write x equals to au, you get an a squared times this. And a squared times this is 1. So your differential equations has become minus the second psi du squared, where u is a dimensionless quantity, because this has units of length, this has units of length. No units here. You have no units. So minus d second du squared plus u squared psi is equal to e psi. Much nicer. This is an equation we can think about without being distracted by this endless amount of little trivialities. But still we haven't solved it, and how are we going to solve this equation? So let's again think of what should happen. Somehow it should happen that these e's get fixed. And there is some solution just for some values of e's. It's not obvious at this stage how that is going to happen. Yes? AUDIENCE: [INAUDIBLE]. PROFESSOR: Here for example, let me do this term. h bar over m omega is minus, from that equation, a squared. But dx squared is 1 over a squared d du squared. So a squared cancels. And here the x is equal a squared times u, so again cancels. OK so what is the problem here? The problem is that most likely what is going to go wrong is that this solution for arbitrary values of e's is going to diverge at infinity, and you're never going to be able to normalize it. So let's try to understand how the solution looks as we go to infinity. So this is the first thing you should do with an equation like that. How does this solution look as u goes to infinity? Now we may not be able to solve it exactly in that case either, but we're going to gain insight into what's happening. So here it is. When u goes to infinity, this term, whatever psi is, this term is much bigger than that, because we're presumably working with some fixed energy that we still don't know what it is, but it's a fixed number and, for you, sufficiently large. This is going to dominate. So the equation that we're trying to solve as u goes to infinity, the equation sort of becomes psi double prime-- prime is for two derivatives-- is equal to u squared psi. OK, so how do we get an idea what solves this is not all that obvious. It's certainly not a power of u, because when you differentiate the power of u, you lower the degree rather than increase the degree. So what function increases degree as you differentiate? It's not the trivial function. Cannot be a polynomial. If it could be even a polynomial, if you take two derivatives, it kind cannot be equal to x squared times a polynomial. It's sort of upside down. So if you think about it for a little while, you don't have an exact solution, but you would imagine that something like this would do it, an e to the u squared. Because an e to the u squared, when you differentiate with respect to us, you produce a u down. When you one derivative. When you take another derivative, well, it's more complicated, but one term you will produce another u down. So that probably is quite good. So let's try that. Let's try to see if we have something like that. So I will try something. I'll try psi equals 2. I'm going to try the following thing. e to the alpha u squared over 2 where alpha is a number. I don't know how much it is. Alpha is some number. Now could try this alone, but I actually want to emphasize to you that if this is the behavior near infinity, it won't make any difference if you put here, for example, something like u to the power k. It will also be roughly a solution. So let's see that. So for that I have to differentiate. And let's see what we get. So we're trying to see how the function behaves far, far away. You might say well look, probably that alpha should be negative. But let's see what the equation tells us before we put anything in there. So if I do psi prime, you would get what? You would get one term that would be alpha u times this u to the k into the alpha u squared over 2. I differentiated the exponential. I differentiated the exponential. And then you would get a term where you differentiate the power. So you get ku to the k minus 1 into the alpha u squared over 2. If you take a second derivative, well, I can differentiate the exponential again, so I will get alpha u now squared, because each derivative of this exponent produces a factor of alpha u. u to the k into the alpha u squared over 2. And a couple more terms that they all have less powers of u, because this term has u to the k plus-- already has u to the k plus 1. And this has u to the k minus 1. They differ by two powers of u. So for illustration, please, if you want, do it. Three lines, you should skip three lines in your notebook if you're taking notes and get the following. No point in me doing this algebra here. Alpha u squared over 2. Because actually it's not all that important. Over alpha 1 over u squared plus k minus 1 over alpha squared 1 over u to the fourth. That's all you get. Look, this is alpha squared u squared times psi times these things. 1 plus 2 k plus 1 over alpha 1 over u squared. So when u goes to infinity, your solution works, because these thing's are negligible. So you get a number times u squared. That is the equation you are trying to solve up there. And therefore, you get that the equation if alpha squared is equal to 1. And that means and really that alpha can be plus minus 1. And roughly this solution near infinity, probably there's two solutions. This is a second order differential equation, so even near infinity there should be two solutions. So we expect as u goes to infinity psi of u will be some constant A times u to the k times e to the minus u squared over 2. That's where alpha equal minus 1. Plus Bu to the k into the plus u squared over 2. And what is k? Well, we don't know what is k. It seems to work for all k. That may seem a little confusing now, but don't worry. We'll see other things happening here very soon. So look at what has happened. We've identified that most likely your wave function is going to look like this at infinity. So we're going to want to this part not to be present. So presumably we're going to want a solution that just has this, because this is normalizable. The integral of any power times a Gaussian is convergence. So this can be normalized. The Gaussian falls so fast that any power can be integrated against a Gaussian. Any power however big doesn't grow big enough to compensate a Gaussian. It's impossible to compensate a Gaussian. So we hope for this. But we want to translate what we've learned into some technical advantage in solving the differential equation, because, after all, we wanted be insight how it looks far way, but we wanted to solve the differential equation. So how can we use this insight we now have to simplify the solution of the differential equation? The idea is to change variables a little bit. So write psi of u to be equal to h of u times e to the minus u squared over 2. Now you're going to say wait, what are you doing? Are you making an approximation now that this is what is going to look far away? Or what are you putting there? I'm not making any approximation. I'm just saying whatever pis is, it can always be written in this way. Why? Because if you have a psi of u, you can write it as psi of u over e to the minus u squared over 2 times e minus u squared over 2. Very trivially this can always be done. As long as we say that h is arbitrary, there's nothing, no constraint here. I have not assume anything, nothing. I'm just hoping that I have a differential equation for psi. That because this is a very clever factor, the differential equation for h will be simpler. Because part of the dependence has been taken over. So maybe h, for example, could be now a polynomial solution, because this product has been taken care. So the hope is that by writing this equation it will become an equation for h of u, and that equation will be simpler. So will it be simpler? Well, here again this is not difficult. You're supposed to plug into equation one-- this is the equation one-- plug into one. I won't do it. It's three lines of algebra to plug into one and calculate the equation for h of u. You should do it. It's the kind of thing that one should do at least once. So please do it. It's three, four lines. It's not long. But I'll just write the answer. So by the time you substitute, of course, the e to the minus u squared over 2 is going to cancel from everywhere. It's very here. You just need to take two derivatives, so it becomes a second order differential equation. And indeed, it becomes a tractable differential equation. The second h, du squared minus 2u dh du plus e minus 1 h equals 0. OK, that is our equation now. So now we face the problem finally solving this equation. So before we start, maybe there's some questions of what we've done so far. Let's see. Any questions? Yes? AUDIENCE: Do you have right there in the middle would be-- this equation is linear, so can we just [INAUDIBLE] minus u squared over 2 and you stuck it to that u to the k. PROFESSOR: It's here? This thing? AUDIENCE: Yeah. Could you then just power series what's going on at 0 with those u to the k terms [INAUDIBLE]? PROFESSOR: No. This is the behavior as u goes to infinity. So I actually don't know that the function near 0 is going to behave like u to the k. We really don't know. It suggest to you that maybe the solution is going to be near 0 u to the k times some polynomial or something like that. But it's not that, because this analysis was just done at infinity. So we really have no information still what's going on near 0. Other questions? Yes? AUDIENCE: So is k some arbitrary number or is it an integer? PROFESSOR: At this moment, actually, it doesn't matter. Is that right? Doesn't matter. The analysis that we did here suggests it could be anything. That's why I just didn't put it into h or u. I didn't put it because would be strange to put here a u to the k. I wouldn't know what to make of it. So at this moment, the best thing to say is we don't know what it is, and maybe we'll understand it. And we will. In a few seconds, we'll sort of see what's going on. OK, so how does one solve this equation? Well, it's not a trivial equation, again. But it can be solved by polynomials, and we'll see that. But the way we solve this equation is by a power series expansion. Now you could do it by hand first, and I did it when I was preparing the lecture yesterday. I said I'm going to just write h of u equal a constant a0 plus a1u plus a2u squared plus a3u cubed. And I plugged it in here. And I just did the first few terms and start to see what happened. And I found after a little thinking that a2 is determined by a0, and a3 is determined by a1 once you substitute. It's not the obvious when you look at this, but that happens. So when you face a problem like that, don't go high power to begin with. Just try a simple series and see what happens. And you see a little pattern. And then you can do a more sophisticated analysis. So what would be a more sophisticated analysis? To write h of u equal the sum from j equals 0 to infinity aju to the j. Then if you take a derivative, because we're going to need the derivative, dh du would be the sum from j equals 0 to infinity. j times aju to the j minus 1. You would say that doesn't look very good because for j equals 0 you have 1 over u. That's crazy. But indeed for j equals 0, the j here multiplies it and makes it 0. So this is OK. Now the term that we actually need is minus 2u dh du. So here minus 2u dh du would be equal to the sum from j equals 0 to infinity, and I will have minus 2jaju to the j. The u makes this j minus 1 j, and the constant went there. So here is so far h. Here is this other term that we're going to need for the differential equation. And then there's the last term that we're going to need for the differential equation, so I'm going to go here. So what do we get for this last term. We'll have to take a second derivative. So we'll take-- h prime was there, so d second h du squared will be the sum from j equals 0 of j times j minus 1 aju to the j minus 2. Now you have to rewrite this in order to make it tractable. You want everything to have u to the j's. You don't want actually to have u to the j minus 2. So the first thing that you notice is that this sum actually begins with 2, because for 0 and 1 it vanishes. So I can write j times j minus 1 aj u to j minus 2. Like that. And then I can say let j be equal to j prime plus 2. Look, j begins with 2 in this sum. So if j is j prime plus 2, j prime will begin with 0. So we've shifted the sum so it's j prime equals 0 to infinity. And whenever I have a j I must put j prime plus 2. So j prime plus 2. j prime plus 1 aj prime plus 2 u to the j prime. Wherever I had j, I put j prime plus 2. And finally you say j or j prime is the same name, so let's call it j. j equals 0. j plus 2. j plus 1. aj plus 2 uj. So we got the series expansion of everything, so we just plug into the differential equation. So where is the differential equation? It's here. So I'll plug it in. Let's see what we get. We'll get some from j equals 0 to infinity. Let's see the second derivative is here. j plus 2 times j plus 1 aj plus 2 uj, so I'll put it here. So that's this second derivative term. Now this one. It's easy. Minus 2j aj and the uj is there. So minus 2jaj. Last term is just e minus 1, because it's the function this times aj as well. That's h. And look, this whole thing must be 0. So what you learn is that this coefficient must be 0 for every value of j. Now it's possible to-- here is aj and aj, so it's actually one single thing. Let me write it here. j plus 2 times j plus 1 aj plus 2 minus 2j plus 1 minus e aj uj. I think I got it right. Yes. And this is the same sum. And now, OK, it's a lot of work, but we're getting there. This must be 0. So actually that solves for aj plus 2 in terms of aj. What I had told you that you can notice in two minutes if you try it a little. That a2 seems to be determined by a0. And a3 seems to be determined by a2. So this is saying that aj plus 2 is given by 2j plus 1 minus e over j plus 2 j plus 1 aj. A very nice recursive relation. So indeed, if you put the value of a0, it will determine for you a2, a4, a6, a8, all the even ones. If you put the value of a1, it will determine for you a3, a5. So a solution is determined by you telling me how much is a0, and telling me how much is a1. Two constants, two numbers. That's what you expect from a second order differential equation. The value of the function at the point, the derivative at a point. In fact, you are looking at a0 and a1 as the two constants that will determine a solution. And this is the value of h at 0. This is the derivative of h at 0. So we can now write the following facts about the solution that we have found. So what do we know? That solutions fixed by giving a0 and a1. That correspond to the value of the function at 0 and the derivative of the function at 0. And this gives one solution. Once you fix a0, you get a2, a4. And this is an even solution, because it has only even powers. And then from a1, you fixed a3, a5, all the other ones with an odd solution. OK. Well, we solve the differential equation, which is really, in a sense, bad, because we were expecting that we can only solve it for some values of the energy. Moreover, you have a0, you get a2, a4, a6, a8. This will go on forever and not terminate. And then it will be an infinite polynomial that grows up and doesn't ever decline, which is sort of contradictory with the idea that we had before that near infinity the function was going to be some power, some fixed power, times this exponential. So this is what we're looking for, this h function now. It doesn't look like a fixed power. It looks like it goes forever. So let's see what happens eventually when the coefficient, the value of the j index is large. For large j. aj plus 2 is roughly equal to, for large a, whatever the energy is, sufficiently large, the most important here is the 2j here, the j and the j. So you get 2 over j aj. So roughly for large j, it behaves like that. And now you have to ask yourself the question, if you have a power series expansion whose coefficients behave like that, how badly is it at infinity? How about is it? You know it's the power series expansion because your h was all these coefficients. And suppose they behave like that. They grow in that way or decay in this way, because they're decaying. Is this a solution that's going to blow up? Or is it not going to blow up? And here comes an important thing. This is pretty bad behavior, actually. It's pretty awful behavior. So let's see that. That's pretty bad. How do we see that? Well you could do it in different ways, depending on whether you want to derive that this is a bad behavior or guess it. I'm going to guess something. I'm going to look at how does e to the u squared behave as a power series. Well, you know as a power series exponential is 1 over n u squared to the n. Here's n factorial. n equals 0 to infinity. Now these two n's, u to the 2n, these are all even powers. So I'm going to change letters here, and I'm going to work with j from 0, 2, 4, over the evens. So I will write u to the j here. And that this correct, because you produce u to the 0, u to the 2, u to the fourth, these things. And j is really 2n, so here you will have one over j over 2 factorial. Now you might say, j over 2, isn't that a fraction? No, it's not a fraction, because j is even. So this is a nice factorial. Now this is the coefficient, cj u to the j. And let's see how this coefficients vary. So this cj is 1 over j over 2 factorial. What is cj plus 2 over cj? Which is the analogue of this thing. Well, this would be 1 over j plus 2 over 2 factorial. And here is up there, so j over 2 factorial. Well, this has one more factor in the denominator than the numerator. So this is roughly one over j over 2 plus 1, the last value of this. This integer is just one bigger than that. Now if j is large, this is roughly 1 over j over 2, which is 2 over j. Oh, exactly that stuff. So it's pretty bad. If this series goes on forever, it will diverge like e to the u squared. And your h will be like e to the u squared with e to the minus u squared over 2 is going to be like e to the plus. u squared over 2 is going to go and behave this one. So it's going to do exactly the wrong thing. If this series doesn't terminate, we have not succeeded. But happily, the series may terminate, because the j's are integers. So maybe for some energies that are integers, it terminates, and that's a solution. The only way to get a solution is if the series terminates. The only way it can terminate is that the e is some odd number over here. And that will solve the thing. So we actually need to do this. This shows the energy. You found why it's quantized. So let's do it then. We're really done with this in a sense. This is the most important point of the lecture, is that the series must terminate, otherwise it will blow up horrendously. If it terminates as a polynomial, then everything is good. So to terminate you can choose 2j plus 1 minus e to be 0. This will make aj plus 2 equal to 0. And your solution, your h of u, will begin. aj will be the last one that is non-zero, so it will be aj times u to the j, and it will go down like aj minus 2 u to the j minus 2. It will go down in steps of 2, because this recursion is always by steps of two. So that's it. That's going to be the solution where these coefficients are going to be fixed by the recursive relation, and we have this. Now most people here call j equal n. So let's call it n. And then we have 2n plus 1 minus e equals 0. And h of u would be an u to the n plus all these things. That's the h. The full solution is h times e to the minus u squared over 2 as we will see. But recall what e was. e here is 2n plus 1. But he was the true energy divided by h omega over two. That was long ago. It's gone. Long gone. So what have you found therefore? That the energy, that' we'll call en, the energy of the nth solution is going to be h omega over 2 2n plus 1. So it's actually h omega, and people write it n plus 1/2. Very famous result. The nth level of the harmonic oscillator has this energy. And moreover, these objects, people choose these-- you see the constants are related by steps of two. So just like you could start with a0, or a1 and go up, you can go down. People call these functions Hermite functions. And they fix the notation so that this an is 2 to the n. They like it. It's a nice normalization. So actually h of n is what we call the Hermite function of u sub n. And it goes like 2 to the n u to the n plus order u to the n minus 2 plus n minus 4, and it goes on and on like that. OK, a couple things and we're done. Just for reference, the Hermite polynomial, if you're interested in it, is the one that solves this equation. And the Hermite sub n corresponds to e sub n, which is 2n plus 1. So the Hermite solution from that the equation is that the Hermite polynomial satisfies this minus 2u d Hn du plus 2n. Because en is 2n plus 1. So it's 2n Hn equals 0. That's the equation for the Hermite polynomial, and interesting thing to know. Actually, if you want to generate the efficiently the Hermite polynomials, there's something called the generating function. e to the minus z squared plus 2zu. If you expand it in a power series of z, it actually gives you n equals 0 to infinity. If it's a power series of z, it will be some z to the n's. You can put a factor here n, and here is Hn of u. So you can use your mathematic program and expand this in powers of z. Collect the various powers of u that appear with z to the n, and that's Hn It's the most efficient way of generating Hn And moreover, if you want to play in mathematics, you can show that such definition of Hn satisfies this equation. So it produces the solution. So what have we found? Our end result is the following. Let me finish with that here. We had this potential, and the first energy level is called E0 and has energy h omega over 2. The next energy is E1. It has 3/2 h omega. Next one is E2 5/2 h omega. This polynomial is nth degree polynomial. So it has n zeros, therefore n nodes. So these wave functions will have the right number of nodes. E0, the psi 0, will have no nodes. When you have psi 0, the Hn becomes a number for n equals zero. And the whole solution is the exponential of u squared over 2. The whole solution, in fact, is, as we wrote, psi n Hn of u e to the minus u squared over 2. In plain English, if you use an x, it will be Hn u with x over that constant a we had. And you have minus x squared over 2a squared. Those are your eigenfunctions. These are the solutions. Discrete spectrum, evenly spaced, the nicest spectrum possible. All the nodes are there. You will solve this in a more clever way next time. [APPLAUSE]
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_10_Clicker_Bonanza_and_Dirac_Notation.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Hi everyone. I'm in a very good mood today. It's nothing to do with the class, but I'm having a baby. [APPLAUSE] PROFESSOR: So that's kind of exciting. So if I just started giggling, you'll know why. And in six months if I am just weeping and on the ground, you'll also know why. So today we're going to do two things. The first is I'm going to give you-- well, the first is review a little bit of practice for the exam we're going to have on Thursday. So let me tell you a little bit more about the exam. The exam, by the way, has been rescheduled to be in the 6120, not in Walker Gym. So it's going to be the usual place, we're not moving. And the reason is I'm changing the format on this exam, in part to make it a little less of a burden to everyone. But also in part because I've been struggling with the question of how to make the exam most useful. The purpose of an exam like this is not to get grades for you guys, although that's an incidental byproduct. The purpose is to give you some feedback on how you're doing, how your command of the material has evolved. And also to help you learn some of the things that you might not have mastered. So the way the exam is going to be structured is going about 15 minutes of short answer questions-- a couple of very short computations but mostly short answer questions-- on paper. You'll hand those back, and then we'll go over those questions in class afterwards. So it's a relatively low pressure exam and it's mostly conceptual. It will cover everything we've done through this last problem set to the degree that we get to the lecture part of today after questions. Today's lecture will not be directly covered, however it will be fair game for the next midterm-- which will be more of a traditional midterm-- and that's coming in April. So the structure of today is, I'm going to give you a whole bunch of clicker questions. So make sure you've got your clickers out. And those clicker questions are going to give you a sense for the level and scope of the exam. The exam will be a little harder than the clicker questions, but not a whole lot. And the difference is just that it's going to be on paper in front of you instead of clickered. And thus, that gives you a little more room to do calculations on a piece of paper, short calculations. And then after that we'll come on to-- in some sense a review, but also an introduction to the Dirac Bra-Ket notation that many of our textbooks use, but that we haven't introduced in lectures so far. Any questions before we get started? AUDIENCE: What channel are we? PROFESSOR: 41. Other questions? AUDIENCE: So no practice exam? PROFESSOR: I think there will probably not be a practice exam because of the shift in format. Today will basically be your practice exam. If I can get my work together and get you guys a practice exam of the right format, then maybe. But it wouldn't be all that useful is the problem. So watch what happens today. AUDIENCE: Will you post the clicker questions from today and the last time on the website? PROFESSOR: I will post the questions from today on the website, yes. AUDIENCE: So it's 50 minutes, short answer. Is there a certain number of questions that's going to be on it? PROFESSOR: I'm not going to tell you that. But it's not going to be a time trial. You're not going to be racing to get the, you know. AUDIENCE: Will it still be worth as much as an exam? PROFESSOR: It will. Because to my mind, part of the reason to design the exam this way is that it's testing your conceptual understanding, which is the important thing. So it will be. Other questions? All right. So let's get started with the clicker part of today. So we're on channel 41. So consider this eigenvalue equation, two derivatives on f-- it's constant-- times f of x. How many of these are eigenfunctions with the corresponding eigenvalue? You've got about 10 seconds. So go ahead and put in your clicks. And so your-- whoops, oh sorry. That just cleared your responses, unfortunately. Don't worry, they're saved in memory. It just cleared them off my screen. So the responses-- here we go. Oops, that didn't work. Wow, it just totally disappeared out of that app. Wow, that's so weird. Oh no, that didn't work. Let's try this. Where did it go? Ah, there it is. Whoa, this is all very confusing. So that was your response, B and C, 63% and 38%. So let's go back to this. Quickly discuss this with your neighbor. And now go ahead and click in your new answers if you-- you can keep talking, that's fine. Another five seconds. Awesome. OK, that's it. So the answer is B. And 99% of you all got that. I suspect someone didn't click. OK good, next question. Psi I and Psi II are two solutions of the Schrodinger equation. Is the sum of the two of them with coefficients A and B also a solution of the Schrodinger equation? Oh, and I forgot to start the clicky thingy. So click now. AUDIENCE: Did you mean to say B Psi 2 on that thing? PROFESSOR: Oh yeah, it is supposed to say B Psi 2. It says B Psi 2 on the next-- sorry, it's A Psi I plus B Psi I. It should say I Psi I plus B Psi II. Thank you. AUDIENCE: Does it mean that the same Schrodinger equation with the same potential? PROFESSOR: Yeah, with the same potential. Yeah, Yeah. [LAUGHTER] PROFESSOR: Wow, Yeah. So you know Einstein said, God didn't play dice. And let me paraphrase that as, God doesn't mess with you in clicker questions. And you guys have effectively universally that the answer is A, yes, superposition principle. OK next question is going to be for answer-- here we go. Consider an infinite square well with width a. And compared to the infinite square well with A, the ground state of a finite well is lower, higher, same energy, or undetermined. You've got 10 seconds, so continue thinking through this. 5 seconds. OK, and you mostly got it, but have a quick chat with the person next to you. All right, let's try again. Such a good technique. All right, another five seconds to put in your answer. 4-3-2-1. And you virtually all got it right, a lower energy. And let's just think about this intuitively. Intuitively, the gradient of the potential is the force, right? So in the second case, you've got less force cramming the article inside the box so it's being squeezed less tightly. More physically, you see that there is an evanescent tail on the outside. What that tells you is the wave function didn't have to go to zero at the ends. It just had to get small and latch onto a dying exponential. That's from the qualitative analysis of wave functions. But meanwhile, what that tells you is, it has to curve less inside in order-- it doesn't have to get to zero, it just has to get to a small value where it matches to the decaying exponential. So if the curvature is less, then the energy is less. Cool? OK, next question. So any questions on that before I? Good. Time zero wave function infinite well with a is this, sine squared with a normalization. What's the wave function at a subsequent time t? I will remind you that you have solved the problem of the infinite square well, and you know what the eigenenergies and eigenfunctions are of the energy operator in the infinite square well. So remember back to what those are. All right, and you have five seconds. OK, we are at about 50-50 correct. So chat with the person next to you. All right. And now, any moment, go ahead and vote again. Good, five more seconds and then put in your final vote. OK, that's it for now. So what's the answer? D is the answer, but a lot of people still had some doubts. So who wants to give an explanation for why it's D? AUDIENCE: So sine squared is not an eigenfunction. PROFESSOR: Fantastic. AUDIENCE: So in some way it hast to be a summation of eigenfunctions. So not even having to know what the eigenfunctions are, there's only one summation in it. PROFESSOR: Excellent, excellent. AUDIENCE: And if you do know what the eigenfunctions are, now you know that [INAUDIBLE]. PROFESSOR: Brilliant, so I'm going to restate that. That was exactly correct in every step. So the first thing is, that wave function at time 0, sine squared of x, is not an eigenfunction of the energy operator for the system. In fact, we've computed the eigenfunctions for the energy operator in the infinite square well, and they're sines where they get a zero at the ends. On the other hand, any function satisfying the boundary conditions-- normalizable, hits zero at the boundaries-- is a superposition of energy eigenfunctions, and we can use that to determine the time evolution we take that superposition and add on a phase-- e to the minus i et upon h bar for each of the energy eigenfunctions. And in part D we express that wave function as a superposition with the coefficient cn determined from the overlap of our original function and the energy eigenfunctions. Everyone cool with that? So this is literally just a transcription of one of our postulates. OK, questions? If you have any questions at all, ask them. This is the time to ask them. Yeah. AUDIENCE: So the way get cn [INAUDIBLE]?? PROFESSOR: Exactly. So the way you get cn is by saying, look I have my wave function, psi of x is equal to sum over n cn phi n of z, where these phi n's energy eigenfunctions. E phi n is equal to en phi m of x. And we also know that the integral of phi n's complex conjugate phi m is equal to delta mn. This is the statement that they're orthogonal and properly normalized. We also write this as equal to phi n phi m. And we can use this to determine the cn is equal to the inner product of phi m with our wave function. This is equal to the integral dx phi star phi n star psi, which is equal to the integral dx phi n star sum over m of cm phi m. But this sum over m of cm can be pulled out because this is just an integral over a sum of terms, which is the same as the sum over m cm integral phi m-- complex conjugate-- phi m. And that's delta mn, which is 0 unless n is equal to m, because these guys are orthogonal and properly normalized. So this is zero unless n is equal to m. So in the sum, the only term that contributes is when m is equal to n, this is equal to cn. So cn is given by the overlap of our wave function with the corresponding eigenfunction. This allows us to take our function, a known function-- for example, sine squared-- and express the coefficients in terms of an overlap of our wave function sine squared with the wave functions sine. And that's exactly the expression you see below, cn is equal to the integral of sine squared-- our wave function-- times sine, which is the energy eigenfunction. Other questions? OK, great. Next problem. Come to me computer. Why can they just be written in Python. OK, good. the eigenstates phi n-- which we usually call psi n, but there it is-- form an orthonormal set. Meaning integral of phi m star phi n is delta mn. What is the value of integral psi m against the sum cn phi n? You've got five seconds. I'll give you a little extra time, because people are clicking away. OK you now have seven seconds, because time is non-linear. OK, so quickly discuss, because there's still some ambiguity here. All right, you have 10 seconds to modify your clicks. Click. All right, yes and the answer is, C, excellent. Right? Because in fact, we just did that. That won't happen on the actual exam. So here's the next question. Let the Hamiltonian on a dagger Un equal en plus h omega-- that should be an h bar-- a dagger Un. What can you say about a dagger Un? Here we should probably say, what can one say, because it's possible-- OK, what can one say? Here the assumption-- just to say it out loud-- the assumption is that Un is an eigenfunction of the energy operator h with eigenvalue en. You've got five seconds. All right, we are at 50/50. So discuss amongst yourselves. All right. OK, go ahead and enter again. Enter your modified guesses. And you have 10 seconds to do so. This is a lot better. OK, this is great. So this is one of those really satisfying moments. It's improved, but there's still some real doubt here. So I would like to get one person to argue for b and one person to argue for c. So who wants to volunteer for each of those? AUDIENCE: I'll argue for c. PROFESSOR: Who's going to argue for b? Someone's got to argue for b, come on. AUDIENCE: You can argue. PROFESSOR: I'm not going to argue for b. I'm not going to argue for c either. That defeats the purpose. I'm the professor. I have to say this all the time. So who's going to argue for b? OK, you argue for b. Who's going to argue for c? All right, yeah, that worked out well. Great. Argue for b. You can do it. AUDIENCE: So I originally accidentally mis-clicked b, so I guess I can do this. So originally I didn't read the question, and I thought since you were acting the letter operator on the eigenstate that you'd get a proportionality constant times some eigenstate. So that's why it could potentially be b. But I disbelieve. PROFESSOR: It's not the best argument imaginable for b, but we'll take it. So thank you. When I put someone in an impossible position. So c. AUDIENCE: All right. I argue for c. So we can see just from the first line here that this is clearly going to be a stationary [INAUDIBLE].. However, we can also see that it is a distinctly different energy from the state that you get an eigenfunction [INAUDIBLE]. So I said c, because it was not the same state. PROFESSOR: Fantastic. That's exactly right. So let me walk through that. Let me say that allowed. So from the first line, it is clear that the object a Un is an eigenfunction of the energy operator. And it's got eigenvalue en plus h omega. So it is a stationary state. It's an eigenfunction of the energy [INAUDIBLE].. However, it is not the same one, because it has a different eigenvalue. So what it means to be the same energy eigenstate is it has the same eigenvalue. If it has a different energy eigenvalue, it is a different state and they're orthogonal. We proved this before. Two states with different energy are orthogonal. So not only are they not the same, they don't even have any overlap. So it is a stationary state, but it's not proportional with the state Un. It's an important one. Questions about that one before we move on to the next? AUDIENCE: How did you tell that it's proportional to the state Un? PROFESSOR: If it were proportional to the state Un, then it would be some constant times Un. That's what we mean by saying it's proportional. But then if we acted on it with the energy operator, what would the eigenvalue be? Let me say this again. Suppose I have a state the Un, which I know that if I act with the energy operator on it-- or this is sometimes called h Un is equal to En Un. If I act with E on alpha Un, where alpha's a constant, what is this equal to? Alpha EnUn. So the eigenvalue is the same. It's En, because it can divide through by [INAUDIBLE].. So if you have the same state-- meaning proportional to it-- then we'll have the same eigenvalue. But this manifestly has a different eigenvalue. Cool? Awesome, OK other questions? Yeah. AUDIENCE: I must have missed the-- this is [INAUDIBLE].. What is the [INAUDIBLE]. PROFESSOR: Ah, good. So H is another name. So some books use H and some books use E. And the reason goes back to classical mechanics. There's an object called the Hamiltonian, which is sort of like saying-- I don't know. It's like saying bread and "pain" are the same thing. But they evoke different things. You say bread and you think like Dixie. And you say "pain" and you think, the me of some gorgeous baguette. It has a different feel. It has a different feel. So this evokes different things. In particular, to a classical mechanic the Hamiltonian is the generator of time translations. Now, we don't usually-- that wasn't the classic way of thinking about the energy operator in, say, Newton's time. But it is how we think about energy now. So that's just terminology. For anyone who didn't get that, sorry about that. This is something that we've run across before, but I should emphasize. So when you hear people say the Hamiltonian, that means the energy operator. The reason I say energy operator rather than saying Hamiltonian-- which is the more common thing in the Lingua Franca of quantum mechanics-- is because I want to impress upon you guys that it's just energy. It is the energy, it's nothing else. It's not some strange beast. It's just energy. Other questions? AUDIENCE: Which one's the bread? PROFESSOR: Bread. Other questions? Next question. Which of these graphs shows the curvature of the wave function in a classically allowed region? OK, five seconds left. All right, fantastic. And the answer is a, because the curvature has to be towards the axis. In the classically allowed region, the wave function is sinusoidal. it's oscillatory. All right, and-- whoops. And the partner of that question is, what about in the classically disallowed region? Five seconds. OK, awesome, and basically everyone got this. B, great. Questions? Yeah, sorry. AUDIENCE: [INAUDIBLE]. PROFESSOR: No. No, that's a very good question. So C and D, let me be a little bit more precise about it. It's a very reasonable question, so let me give you an answer. Suppose you saw a wave function that had the curvature structure of C. It was curved oscillatory over there and curved exponential over there on the right. What would you say about the potential relative to the energy? Say again? Yeah, good. So on one side we're allowed, and on the other side we're disallowed. So if you saw a wave function that had that curvature structure, you'd immediately know that it was an allowed region on the left. But then on the right, it was classically disallowed, because it had curvature away from the axis. That cool? So if you saw a wave function-- so for example, you might say look, you never get a wave function that does this. And that's not true. You could do that. So this would be classically disallowed, because it's curvature down. This would be classically allowed. So the potential, for example, if this is the energy, the potential could be. So a potential like this could lead to a function like this. So it's a very good question. I'm sorry, I should have seen what you were asking more immediately. Other questions following up on that? No one ever asked that question before. And I just want to emphasize that you guys should just ask questions when you don't-- when you've confused about something, someone else is going to be confused about it. I wouldn't even necessarily know that the question is a meaningful question. I get questions I've never heard before all the time in this class. Just don't hesitate to ask. It's always a good idea. OK, next one. Wave function psi has been expressed as a sum over energy eigenfunction Un. Compared to the original wave function, is a function of x a set of coefficients, C1 dot dot dot contains more or less, same information? Or it can't be determined? Or depends? Five seconds. And the answer is. AUDIENCE: C. PROFESSOR: Same information. Good questions. Some of these you've seen before. f and g are wave functions and c is a constant. Then the inner product cf with g is equal to? Click away, 10 seconds. All right. OK, you've got about five seconds. Still a lot of fluctuations here. Wow, I'm surprised by this one. Yeah, question. AUDIENCE: Sorry, quick question on the previous one. PROFESSOR: Oh, let's come back to the previous one after we do this one. Thanks. But remind me too after we move on. So discuss, because about 1/3 of you got this one wrong. So discuss. So go ahead and put in your answers again. Good. All right, another five seconds. OK, and the answer is. Good, get that complex conjugation right. Little things like that can cause you an infinite amount of trouble. Especially as you'll notice on the problem set when you're computing time dependence of expectation values, getting that complex conjugation right is essential, because you need to see interference terms. And without the complex conjugation you will not get the right interference terms. Oh yeah, a question from the previous problem. Thank you. AUDIENCE: So since the summation with the constants, doesn't that technically give you the probability of each individual eigenstate? PROFESSOR: The norm squared gives you the probabilities. But if you know the coefficients--- so this is a good question. So let me rephrase the question. The question is, in the previous question we said, our wave function phi of x is some function which we could draw. So I could represent this in two ways. I could either draw it, or I could represent it as equal the sum over n of cn phi n where phi n are the known energy functions. And the question is, did the cn contain the same information? So that was the question. And the point that's being asked about here is a very good question. The question is, look, we know what these cn's mean. What the cn's mean is the probability that you measure the energy-- given that the state of psi, the probability [INAUDIBLE] energy En is equal to norm cn squared. And so the cn's tell you about probabilities. They don't tell you what the state is. They just tell you what a probability is that you'll measure a particular energy. Is that right? So how can it contain the same information? And there are two important things to say about this. First off, this is of course true. And this does give you the probability. And the probability comes from the norm squared. So the phase of cn doesn't particularly matter for that. However the claim that I can expand this function in a basis of the energy eigenfunctions, in a basis of other functions, is a statement that if know these cn's, then I take c1 multiplied by phi 1, c2 multiplied by phi 2, add them all back up, and I get nothing other than this original function. So if I know the cn's, I can just construct that original function explicitly. So this can't contain any more or any less information than the list of cn's, as long as I know what basis I'm talking about. So here's the disturbing thing about that. When we say the wave function is the state of the system-- it contains all knowledge you could possibly have access to. And not just knowledge that you could have access to, it contains all of the information of the state. And then we look at this and say, well look, it doesn't give us something deterministic. It gives us something probabilistic. It tells us the probability distribution for measuring the various different energies. What does that immediately tell you since they're equivalent information? This guy tells you nothing more than the probabilities. It doesn't tell you what energies you'll measure, it tells you the probabilities with which you will measure energies. So they're the same thing. It's a very good question. And I think the way to take the sort of discomfort-- I don't know about you-- but that causes me is just to read that as saying, look, the wave function is giving you probabilistic information about measurements. AUDIENCE: I get that. It's just that if you have a wave function written as that or a wave function written as the [INAUDIBLE],, I just see-- since it's written that way, then you know what all the cn's are and what all the possible eigenstates are. And just [INAUDIBLE]. I understand they both contain the same information, but just written as each. PROFESSOR: Excellent, excellent. So let me turn this around. Let me phrase this slightly different. Tell me if this is getting at the same point. If I gave you a list of numbers, 1, 14, 12, does that tell you what function I'm talking about? No. I have to tell you that they're the expansion coefficients in, say, sine waves of a particular wavelength. Or energy eigenfunctions of some particular energy operator. If I just give you a list of numbers, it doesn't tell you what function it is. I have to give you a list of numbers, and I have to tell you what basis I'm talking about. If you have the wave function, you don't need to know what basis you're talking about. The wave function is basis independent. It's just the function. In a few minutes, I'm going to come back and subvert what I just said. So the wave function is just what the wave function is. And if I give you a list of numbers, they're only equivalent when you also know the wave function. So in order for these two things to be equivalent, it's important that you know what bases are talking about. In particular, it's important that you know the energy eigenfunctions. Is that getting at the distinction you're making? OK, think more about it. And when the question becomes more sharp, ask. Other questions? AUDIENCE: I think you kind of covered what my question was. But for the future, if the question is phrased such that if you know the cn's, you know all the information, are we supposed to just infer that if we know the cn, we also know [INAUDIBLE]? PROFESSOR: Yeah, usually. I try on problem sets in exams to make that distinction clear, but that will generally be assumed. Other questions? I'm going to come back to this. I'm going to come back to this when we finish the clicker questions. Next question. In the simple harmonic oscillator, the eigenvalues-- as we've described twice-- our En is h bar omega times n plus 1/2 for an integer n, and a measurement of energy will always observe one of these values. That's from postulate four. Three? What can we say about the expectation value of the energy in some arbitrary state? This one isn't tricky, but it's not trivial. So think carefully about it. You have five seconds. Enter your best guess. And you are at complete chance. I think we've just put you through a hardness box. So discuss. OK go ahead and start putting in your modified response. You have another five seconds or so. OK, click. OK, there's still a lot of disagreement here. So let's see. So who would like to argue for C? Who would like to argue for D? Go ahead. AUDIENCE: OK, so I'm going to argue for D. And against C. So C suggests that we would measure the value of the En only for eigenstate that it addressed. But even if that changed, the stationary state consists of one and one eigenstate. So if we were to find the value, it can still be a combination of 1 and En. And it doesn't have to fall to a certain En value. PROFESSOR: Excellent, excellent. So let me restate that. So the statement-- if I'm following your logic. Thank you. So at the statement if I'm following your logic goes something like this. Look, if we had the wave function which was phi sub n times some phase, some ridiculous phase. Let's say the phase was 1. If we had the wave function as phi sub n, what will the expectation value of the energy operator be in this state? En, right? Because it's the energy. It's a sum of all possible n of En times the probability that we measure En, by definition, but this is equal to sum over n of En times the expansion coefficient of our wave function in the energy eigenstate basis cn squared. Now, what's the expansion coefficient in our energy eigenbasis here? It's 1 for a particular value of n, and 0 for everything else. So what will our answer be? It will be 0 for everything except 1 for a particular value En corresponding to little m. I should really call this m, so that dummy variables that we're summing over don't matter. So for this particular case the expectation value of e is equal to En. Everyone cool with that? So it certainly can be En. But is this the only way to get En? No. So for example, suppose we have alpha phi 1 plus beta phi 3. What will the answer be? Now first off, in order to make this normalizable, what does beta have to be? It needs to be square of 1 minus alpha norm squared. Just so that they sum to 1 when norm squared. So what I want to plot now the expectation value of the energy as a function of alpha. When alpha is 0, what is the expectation value of the energy? E3. When alpha is 1-- which it can't be anything greater than 1-- when alpha is 1, what is the expectation value of the energy? E1. And how does the expectation value of the energy vary between these two as a function of alpha? Well, it's smooth. In particular, it's quadratic because it goes like the square of these coefficients. So it's something smooth that does this and has slope 0 at the end. And somewhere in between, it will hit E2. But not only is this not equal to E2, it doesn't even contain E2 as an element of the superposition. So this is a counter-example to C. This is an important point. Now if I know that at some moment in time, the energy expectation value is this value, corresponding let's say to some particular alpha, and I let the system evolve in time, how does that expectation value evolve in time? How does the expectation value of the energy change over time? It doesn't. The expectation value of energy is always time independent. Does the expectation value of position change over time? In general, yes. When does it not change over time? In a stationary state. So working through this is what's on your previous problem set, as well as on your current problem set. AUDIENCE: So is this anything like conservation of energy, where the expectation value of energy needs to stay the same if there's nothing going on? PROFESSOR: Excellent, it sort of sounds like that. We'll make a sharp version of that later. So you should have that tinkling in the back of your head as like, oh look, we have a time independent potential, energy is conserved. That sounds kind of like the expectation value is constant. But we need to make that more precise. So we'll find a precise version of that statement later in the semester. But it's a very good intuition to have. Other questions? OK, so the answer is D. And you guys were about 4/5 on that at the end. Oops, that was not right. OK, next one. This is a three-step question. We have observable A and B with eigenstates psi 1 and psi 2, and phi 1 and phi 2 of A and B, and eigenvalues of a1, a2, and b1, b2. And the eigenstate's related in this linear fashion. I measure observable A and get the value of a1. What's the state immediately after that measurement? What is the wave function? Five seconds. 2-1. Click away. So there's pretty strong agreement among you all that the answer is A, and that is indeed the answer. Questions? So a slightly less obvious one. Immediately after this measurement of A, observable B is measured. What's the probability the b1 is found? Wow, there's some amazing initial transience there. I could use the first couple of seconds as a random number generator. OK, that's it. And there's actually a third of you disagree with the correct answer. So I'm going to invite you to chat with each other. Go ahead and update your answers. OK, five more seconds. One second. OK, nice. Excellent. You guys went from about 65% to 90%, so that was great. The answer is E. It's the norm squared of the coefficient of eigenfunction corresponding to b1 in the original state, which was psi 1. Now imagine, on the other hand, that the grad student doing the measurement failed to in fact measure the observable B-- it was a bad day-- but instead measured A again. What's the probability that the second measurement will yield a1? Trust me on this one, there's so many ways. They read the package wrong, they did the wrong measurement. AUDIENCE: [INAUDIBLE]. PROFESSOR: Sorry, thank you, thank you, thank you. Very good question. He accidentally measured B instead of A. AUDIENCE: [INAUDIBLE]. PROFESSOR: I mean A instead of B. A instead of B. See how easy it is to make that mistake? He accidentally measured A instead of B. I wish that was intentional. OK, and so and we got almost total unanimity. The answer is B. Yes, question. AUDIENCE: I was just going to ask how quickly after. PROFESSOR: Great question. Let's just assume it's instantaneous. Maybe sloppy, but he's very quick. OK, next question. A system is in a state which is a linear superposition of n equals 1 and n equals 2 energy eigenstates, blah. What's the probability that measurement of the energy will yield the first energy eigenvalue? Another five seconds, 2-3-4-5. Great, you guys totally agree that the answer is C. Great. That's it for the clicker questions for today. So here's the thing I want to emphasize. The first is that these were totally conceptual. They did not involve any computations. And yet, they're not trivial. Some of them got 50% consistently across the class. So it's not just the calculations that are hard. Just thinking through the basic premises, the basic postulates, is really essential at this point. It's more important than the computation. Yeah. AUDIENCE: Are you going to post this? PROFESSOR: I will post these. I will post these. Any questions remaining after this-- oh, shoot! I missed out on a whole opportunity. I'll do it next time. Poo. Any other questions before we move on to lecture part? AUDIENCE: [INAUDIBLE]. PROFESSOR: Oh, you totally have to wait. Yeah, you totally have to wait. Any other questions? Yeah. AUDIENCE: Do you ever have a [INAUDIBLE] function that's odd? Or just continues down [INAUDIBLE].. PROFESSOR: Well, that's a very good question. So the question is, can you ever have a potential function that is odd? So for example-- you're actually asking two questions, so let me disentangle them. So one question is, can I have an a potential function which is odd and linear? So it just ramps all the way down. So does that seem physical? Is the energy bounded from below? Well, that's a not a terribly good criterion because it's not bounded from below for the Coulomb potential either. But in the case of the Coulomb potential, it's not bounded from below in a little tiny region. Not in a huge swath. So we're going to need a better definition than this, but the short answer to this one is no. This is not good. On the other hand, it's not the oddness that's bad. It's that the energy is continuing to be unbounded from below. So let me be more precise than saying it's not happy. What important for an observable? What's important for the operator corresponding to an observable in quantum mechanics? What property must that operator have? It has to be Hermitian. And [INAUDIBLE] that implies real eigenvalues. It implies that there's a basis of its eigenfunctions, all these nice things. What we will find is that the Coulomb potential-- the hydrogen potential-- gives us a nice Hermitian energy operator. We'll prove that the energy operator self-adjoint on a problem set. However, this guy will not. So we're going to have one that is not bounded from below. And so while there's a sense in which it's Hermitian, there's not a good sense. It's got problems. But that's a more technical detail. However, there's a second part of your question, which is you could have been asking, look, can you have a potential that's odd but let's leave aside the diverging at infinity. So for example, could you have a potential that does this so that it's odd, but it's bounded? And that's fine. Nothing wrong with that. So odd isn't a problem. The problem is certain ways of diverging. AUDIENCE: So what would happen in the case where you did have some kind of linear potential? You had a uniform electric field and something was moving. Is the problem that that potential technically isn't a physical potential because it doesn't go away? PROFESSOR: How would you ever build a linear potential, right? So the question here is, look, I know how to build a linear potential. Turn a uniform electric field, and an electric field is the gradient to the electrostatic potential. So that means that the electrostatic potential is linear. And the potential energy of a charged particle with charge Q in an electrostatic potential is q times the electrostatic potential. So the potential energy for that charged particle is a linear function. So how do I build such a thing? Well, I take two capacitor plates, and I dump some charge on this one, and the opposite charge on this guy, and I build up a linear potential. Well yeah, exactly. So that's the problem. So can I make this linear over an arbitrarily large domain? No. I need an arbitrarily large amount of charge, and I need to push them apart from each other arbitrarily far, which takes an arbitrarily large amount of work. That's not terribly physical, right? Your arms are only so big. So at the end and the day, we're always going to discover that the pathologies that show up in a potential like this, like the Hamiltonian's not self-adjoint and compactly supported [INAUDIBLE] continuous functions, then OK. That's going to always be some mathematical version of, your arms are only so big. You cannot build an infinitely large apparatus like this. So yes, you're exactly right. It is not physical. Is it a mathematical problem one could analyze? Yes. One could write a dissertation on the singularity structure of these differential equations. But that dissertation would never be read by a physicist. Other questions? Yeah. AUDIENCE: Talking about the rising and lowering number. Can you find the raising and lowering for any [INAUDIBLE] potential. PROFESSOR: Excellent question. No. So here's what you can do. Suppose I have a bunch of energy eigenstates. I have some potential. This is the potential. It's a crazy potential. And it has some definite energy eigenvalues. As we've shown in 1D, or as I've mentioned and proved in office hours, we can't find degenerate eigenfunctions in a one-dimensional potential. So they're not degenerate. They're spaced a finite distance apart for wave functions that are convergent to 0 and infinity for bound states. So we can always construct an operator, which I will simply define in this way. A, and just for flourish I will put a dagger on it which does the following. A dagger is the defined as the operator that maps phi n to phi m, n plus 1. So what's the rule? It takes this state to this state, this state to this state, this state to this state. And since we can extend-- since these are a basis, then that tells us how this acts on an arbitrary function, because it acts on the superposition as acting on each term. So we can define it in this way, but here's the question. What does it mean for the raising operator to be the raising operator? It wasn't that it lifted to raise the operator. That's not what started out the whole machinery. Where the raising operator got its juice was from this computational relation, a dagger with E was equal to h bar omega a dagger. So can we build an operator a dagger that raises states and commutes with the energy operator in this fashion? No, that depends on the energy operator. This is not a general property. This is not always something we can do. If we could, then that energy operator would have evenly spaced energy eigenfunctions. On the other hand, it's not like it's never useful. So consider the following example. And I think this was on your problem set but I don't remember. So let me just say it. If we write a dagger a is equal to n, then the commutation relation for this-- so the energy operator for the harmonic oscillator was equal to h bar omega n plus 1/2. That's what we derived last time. If we define a dagger a as n, then the commutator of n with a dagger is a dagger. And the commutator of n with a is equal to minus a. Suppose the energy operator is equal to h bar omega n plus n cubed. Does the raising operator commute with the energy operator to give you a dagger again? No, because it commutes to give you an a dagger and it commutes with this guy to give you, well, a constant times n squared a dagger, which is a slightly funny [INAUDIBLE].. So this doesn't have the same computational relations. On the other hand, we know how to build the functions of the n operator because they live in a tower by this algebra. By this commutator relation, they live in a tower. So I can build the eigenfunctions of n. And then I can take those eigenfunctions of n and act on them with e and discover that they're also eigenfunctions of e. Because e acting on them is just n, and n cubed acting on the eigenfunctions of n. So the states with definite n are still eigenfunctions of the energy operator. But they're not evenly spaced. Now can every potential be written in terms of this n operator. So I leave that you as an exercise. The answer is no. Try it. AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah, exactly. So there's always something you could formally define in this fashion. It's not always useful to you, because it doesn't always commute with the energy operator in a nice way. So then you can ask, are there properties that aren't nice about this? Well, there's something called the supersymmetric method in 1D quantum mechanics. But that's sort of beyond the scope. Come to my office hours and ask me this question again. This is a miracle of the harmonic oscillator. Other questions? AUDIENCE: So to construct the [INAUDIBLE] we use a condition that a phi 0 is 0? PROFESSOR: Yes. AUDIENCE: So how did we come up with that? Did we demand that on physical grounds that it had to be 0? PROFESSOR: Yes, we did. So excellent question. So the question is this. When we constructed using the operator method for the harmonic oscillator, when we constructed the ground state, we said the ground state, phi 0, is that state which is annihilated by the learning operator. A phi 0 is equal to 0. So where did that come from? Were we forcing that for physical reasons? For mathematical reasons? What was the reason? So this was very important, so let's go through it slowly. So remember the step that came immediately before this step. Immediately before this step we said, look the expectation value of energy can always be written as a strictly positive thing. It's equal to the integral and you should derive this for yourself. I didn't derive it, but this is a good thing to derive for yourself. It takes only a couple of steps. Integral overall momentum of psi tilde of p-- the Fourier transformer norm squared. This is the probability that you have given the state psi. This is the probability density that you have momentum p, p squared upon 2 m. Plus the integral dx psi of x norm squared m omega squared upon 2 x squared. This is the probability that you're at position x, given the wave function x. And that's the value of the potential. Now this is a strictly positive quantity, as is this. Strictly positive, as is this. This is a plus. This quantity must always be greater than or equal to 0. Correct? However, we derive from the operator relations that if we take away function phi sub e and we act on it with the lowering operator a, this defines a new state, which up to some normalization is an eigenfunction of the energy operator with energy e minus h bar omega. And we show this by showing that the energy eigenfunction acting on this gives exactly this coefficient times this back. It's an eigenfunction. We just showed explicitly that it's an eigenfunction. So this suggest that if you have a state with energy e, you could also build a state with energy e minus h bar omega. And thus we can repeat e minus 2 h bar omega, and this turtles all the way down. The problem is, for any finite value of e, eventually this tower will get negative. Here's 0. But the energy expectation value cannot be less than 0. It's got to be strictly greater than 0. But if we were in this energy eigenstate and we measured the expectation value of energy, this is the value we would get, which would be negative. Whoops, that's a 2. So something is amiss. How can this possibly make sense? And the answer up with was, well look, it's inescapable that acting with A n phi gives us something that's an eigenfunction of the energy. However, it's possible that this function happens to be the 0 function. Not 0 energy, just 0, not a function. Is zero normalizable? No, the integral of 0 squared is 0. No coefficient times that is a finite number, is 1. So what must happen is it must be true that at some point we can't lower anymore. But we don't always act with a, so we always try to lower. The only out is if when we lower one particular guy, which I will call phi 0, the only out is if we try to lower phi 0, we don't get another state-- which we see we did here. We don't get another state. Instead, we just get 0. We get no state. Not energy 0. 0, the function is 0, identically vanishing. This does not describe the configuration of [INAUDIBLE].. The probability of it being anywhere is 0. That's bad. So in order for the tower to end, in order for the energy to be bounded from below as the potential is, we need that there's a lowest state in the tower. Does that answer your question? AUDIENCE: Yes. PROFESSOR: Great. Question? AUDIENCE: Are the energy eigenfunctions always strictly real? PROFESSOR: Very good question. You're going to show this on a problem set if you haven't already. I thought you did, but maybe not. The energy eigenfunctions for potentials with only bound states can always be expressed as purely real or purely imaginary. You can always decompose them in terms of purely real and purely imaginary. Proving that will be a problem on one of your problem sets. But for the moment, let me just say, yes, you can always show that the energy eigenfunctions for 1 D potential can always be expressed as purely real. I need that bound state condition, as we'll see later in the course. Yes. AUDIENCE: So we lowered the energy [INAUDIBLE].. And the next energy is 0, right? PROFESSOR: Yes. AUDIENCE: Isn't that supposed to mean that the [INAUDIBLE] is h omega, not h omega over 2? PROFESSOR: Sorry? No, good. So what this says is given some state with energy e, when we lower it we get h bar omega less. That doesn't tell you what e is. Other questions? AUDIENCE: Just to expand on that. So if you get the 0 function, that doesn't actually correspond to any energy eigenvalue, let alone 0, right? PROFESSOR: Correct. The function 0 is an eigenfunction of every operator. But it's a stupid eigenfunction of every operator. In particuar, it has nothing to do with the states we're interested in. We are interested in normalizable states, and that is not a normalizable state. You can't multiply it by any finite coefficient and get 1 when you square integrate it. One last question. AUDIENCE: Maybe a stupid question. But does that mean that the only particle that can have 0 energy is the particle that doesn't exist? PROFESSOR: Well, we need to add more work to that. So the question is, does that mean that the only particle that could really have 0 energy is some imaginary particle that doesn't exist. First off, imaginary particles may exist. We just haven't seen them. AUDIENCE: [INAUDIBLE] real particle [INAUDIBLE] 0 energy. PROFESSOR: Right. OK, so I need to say two things about this. So there's a technical complaint I need to make about your question, which is a very fair question. But I need to make it all the same. And the second is a more physical answer. So the technical complaint is, what do you mean by 0 energy? Do you ever measure energy directly? You measure energy differences. So relative to what? We have to decide, relative to what? So for example, in the harmonic oscillator what we mean when we say the [INAUDIBLE] energy is non-zero is if we draw the potential, and we call the minimum of the potential e is 0, classical e is 0, then what we discover is that the energy of the quantum mechanical ground state is 1/2 h bar omega above 0. So is there any state of the harmonic oscillator with energy 0? No. If we have a free particle, is there a configuration with energy 0? Well, that's an interesting question. So what are the eigenstates, e to the ikx. When the energy is 0-- so the energy is-- because it's just p squared upon 2m, which is h bar squared k squared upon 2m, the energy is h bar squared k squared up on 2m for a free particle, no potential. And by 0 we mean the asymptotic energy equals 0. Can you have a state with energy 0? This one's a little subtle, because if we take the energy 0, what must k be? 0, OK great. So what's the wave function? Constant. Is that normalizable? Not so much. So can you put a single particle in a state with energy 0? No, that would not seem to be possible, because that would be a non-normalzable state. Well you say, we deal with these exponentials all the time. We know how to deal with that. We don't use a single wave-- plane wave-- we use a wave packet with some width. So I can build a wave packet with some average width, but will that average width ever have an expectation value of k equals 0? No, because it will always have contributions from k not equal 0 so that it's specially localized. So can you build a normalizable state with energy 0, which is an eigenfunction of the energy 0? No, not in this case either. So there's the technical complaint that you need to talk about what you mean relative to what. That was about your question. And the second of, can you ever have an energy eigenstate which is at the minimum of the potential? And that's also 0. That's also no. You'll actually prove that on a later problem set. So with all that said, we have a mere-- that's awesome-- 15 minutes for a 14-page lecture. So can I do this fast? No, I'm kidding. I won't do that to you. So I want to just-- instead of going through it. So this was not directly relevant for the exam. And this can come a little bit later. So let me show you a couple of quick things. So the notes are posted online. Look at them. I'll go over them again in the future. What do I want to do? So I want to introduce to you two ideas. One is Dirac notation. And this is hearkening back to basics of vector spaces. And the second is, I want to tell you something about what the commutator means. So first off, let me ask you guys a question. I have one more clicker question for you. All right, are you all ready? Everyone up with the clickers? Here's is the question. [MUSIC PLAYING - JEOPARDY THEME] OK, go ahead and start clicking now. Sigma's the uncertainty. Sorry, sigma is the uncertainty. This is awesome. I swear the clicks are going in beat with the music. So let me give you three more seconds. 1-2-3. This is awesome. You guys are totally at chance. You have even probability distribution across the three. So this is good, because we haven't actually introduced this idea yet in 804. So let me quickly talk you through this. So I'll leave it off. So let me quickly talk you through this. And this is important. This is going to give you some intuition for what uncertainty means. And it's also going to give you some intuition for what the commutator means. So this is a little more experience, a little more practice with operators. And we'll pick up on the Dirac notation and everything next time. Because I just want to get through this physics. So consider two operators, A and B. So A and B are my two operators. They have hats. He has a top hot. So we have-- you've got to make this stuff a little more light hearted. So we have two operators, A and B. And I want to ask, is it possible for there to be a function phi little a little b which is simultaneously an eigenfunction of a and an eigenfunction of b? So now this is a pure math question. Given two operators, a and b, can you build a function which is simultaneously an eigenfunction of a and an eigenfunction of b? I will call this phi sub ab. Why not? OK, phi sub ab has this property. And phi ab is equal to little a. Why not? And b on phi ab is equal to little b on phi ab. You can't stop me, I have now created this object. From the existence of this state, what can you deduce about the operators a and b? And if you've already seen this before, that's cheating. So don't raise your hand. But if you haven't seen it before, just think through it. What does this tell you about the operators a and b? AUDIENCE: [INAUDIBLE]. PROFESSOR: Great. That's a nice guess. I like that guess. So let's check. So what would it mean for the commutator of a and b to be 0? So this is equal to ab minus ba. And what the commutator does is it takes two operators and it gives you a new operator called bracket ab. And sometimes it's useful to call it c, for commutator. So it gives you a new operator. Given two operators you build a new operator. And suppose now that we have a common eigenfunction of a and b. What does that tell you? Well, let's take the commutator-- a, b-- and let's take it and act on phi of sub ab on this common eigenstate. So what is this equal to? This is equal to ab minus ba phi b, which I can write as ab phi ab minus ba phi ab by linearity-- by addition, really. But b acting on phi ab gives me, by hypothesis, little b. And then a acting on a constant times phi ab gives me a little a times that constant minus-- now here a acting on phi ab gives me little a phi ab. And then b acting on constant times phi ab gives me little b. And now I can join these together. This is equal to little ab minus ba phi ab. But little a and little b are numbers. If you take 7 times 5 and 5 times 7 and you subtract them, what do you get? 0. So in order for a and b to share a single eigenfunction phi ab, what must be true of their commutator? It must have an a non-zero kernel, exactly. The commutator must annihilate that particular common eigenfunction. Does it tell you that it kills every function? No. But it must at the very least annihilate the shared eigenfunction. True? Consider two operators of the following form. Consider two operators who commutator ab is equal to the identity times a constant. Do these operators share any common eigenfunctions? They can't, because the commutator-- the identity-- doesn't annihilate any wave functions. Any functions. Nothing is 0 when acted upon by the identity. That's the definition of the identity. The identity takes any function, gives it that function back. Cool? Do a and b share any common eigenfunctions? Now let's think about what that means in terms of observables. Observables are represented by operators. What is the meaning of the eigenfunctions of those operators? AUDIENCE: It corresponds to the [INAUDIBLE].. PROFESSOR: It corresponds to the state with a definite value of that observable. And the value is the corresponding eigenvalue. So the eigenfunction of an observable operator is a possible state with a definite value of that operator. If I tell you that I have a state which is an energy eigenfunction, then that means it is a state with a definite value of the energy. If I measure that energy, I know exactly what I will get. Cool? If a and b are two observables and their commutator is proportional to the identity, is it possible that there is a state with a definite value of a and a definite value of b simultaneously? No, because a state with a definite value of a would be an eigenfunction of a. And a state with a definite value of b would be an eigenfunction of b. And there are no common states, no common eigenfunctions of a and b because the commutator never kills a state. It has no 0's. What is the commutator of x and p? Commit this to memory, i h bar. Now when we write i h bar, this is not really an operator. That's a number. What's the operator? The identity. Are there any states which are simultaneously eigenfunctions of x-- the operator x-- and eigenfunctions of p? Are there any states that have a definite position and a definite momentum simultaneously? Is that because we're ignorant? Are we ignorant? Yes, OK good. So it is because the kinds of things that position [INAUDIBLE] r forbid the existence of a state with a definite value of x and with a definite value of p simultaneously. It is neither here, nor there, nor both, nor neither. So this tells us something very lovely. This tells us that if we're in a state with a definite value of x, what must be true of our uncertainty in the value of p? It can't be 0, because if our uncertainty in the knowledge p were 0, that would mean we were in a p eigenstate. We would have definite value of p. So if a delta x is 0, delta p cannot be 0 for sure. Now how big does it have to be? Could it be arbitrarily small? So here's a commutation relation. You will prove the following relation later on in the course. But I want to tell you now, because I want to give you some intuition for what the commutator is. What it means. What the physics of it is. If I take two operators, a and b, which have a commutator-- which is a constant times the identity-- then the uncertainty in any state psi of a times the uncertainty in the same state psi of b must be greater than or equal to 1/2 the absolute value of the expectation value of the commutator-- that's an amazing set of symbols-- of a with b. Square root. AUDIENCE: You've got yo take the square root. PROFESSOR: Thank you. AUDIENCE: Oh, I was kidding. PROFESSOR: So the easier way to-- let's see a,b. Let's just make sure I'm getting this right. I'm the commutator of a and b. Oh no, no. There's no square root. Sorry. Why are you messing with me? Expectation value of a and b. Delta a. Delta p. H bar. Yes, good. So in particular, these expectation values should be taken with the state psi. Sorry, it's late and I'm tired. And this should be taken into state psi. So we compute the commutator. We take the expectation value in our state. We take the norm, multiply it by 1/2, and this is the bound This is the claim. So let's check this in the case of x and p. And do I have this using just math? This will come from just linear algebra. So let's check this in the case of x and p. What's the commutator? 1. So what's the expectation value in a state, psi, of i h bar times operator 1 psi? It's i h bar. Great. So what happens over here? What do we get for delta x in the state psi times delta p in the state psi? This must be greater than or equal to 1/2 times the absolute value of the expectation value of the commutator. But the expectation value of the commutator is i h bar. And the norm of that is just h bar. This is the uncertainty relation. So given some uncertainty in delta x, this tells us how much delta p is. Now, how does this right hand side depend on the wave function psi? It turns out it doesn't. It's explicitly independent of the particular wave function we're interested in. For any wave function whatsoever, the uncertainty in x times the uncertainty in p must be greater than or equal to 1/2 h bar. And this is a consequence of the commutation, and more importantly, the failure of x and p to commute, because the ability to commute and get 0 is necessary for there to be common eigenstates. Now imagine in this case a and b are that a and b that in fact do share a common eigenstate. So let's compute the uncertainty of a in the state phi sub ab, phi sub ab, phi sub ab, phi sub ab. And what is this equal to? Well, what's the commutator acting on the state phi sub ab? 0. So this is greater than or equal to 0. And in fact, you can just check by going through the proof of this that it's just equal to 0. So what's the uncertainty of a? Well, we know it's in the state with definite value of a. 0. Uncertainty in b in this state, what's the uncertainty? 0. We know that it's in an eigenstate of b. So 0 is indeed greater than or equal to 0, so it satisfies the uncertainty relation. Commuting is telling you about the possibility of the state having definite values of both operators simultaneously. And this is going to turn out to be enormously valuable when we talk about angular momentum, which is coming in a couple of weeks. OK, see you guys on Thursday here for the midterm.
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_2_Experimental_Facts_of_Life.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right. So today's task is going to be to outline some of the basic experimental facts that we will both have to deal with and that our aim should be to understand and model through the rest of the course. Physics doesn't tell you some abstract truth about why the universe is the way it is. Physics gives you models to understand how things work and predict what will happen next. And what we will be aiming to do is develop models that give us an intuition for the phenomena and allow us to make predictions. And these are going to be the experimental facts I would like to both explain, develop an intuition for, and be able to predict consequences of. So we'll start off with-- so let me just outline them. So, first fact, atoms exist. I'll go over some of the arguments for that. Randomness, definitely present in the world. Atomic spectre are discrete and structured. We have a photoelectric effect, which I'll describe in some detail. Electrons do some funny things. In particular electron diffraction. And sixth and finally, Bell's Inequality. Something that we will come back to at the very end of the class, which I like to think of as a sort of a frame for the entirety of 8.04. So... we'll stick with this for the moment. So everyone in here knows that atoms are made of electrons and nuclei. In particular, you know that electrons exist because you've seen a cathode ray tube. I used to be able to say you've seen a TV, but you all have flat panel TVs, so this is useless. So a cathode ray tube is a gun that shoots electrons at a phosphorescent screen. And every time the electron hits the screen it induces a little phosphorescence, a little glow. And that's how you see on a CRT. And so as was pithily stated long ago by a very famous physicist, if you can spray them, they exist. Pretty good argument. There's a better argument for the existence of electrons, which is that we can actually see them individually. And this is one of the most famous images in high-energy physics. It's from an experiment called Gargamelle, which was a 30-cubic meter tank of liquid freon pulsing just at its vapor pressure 60 times a second. And what this image is is, apart from all the schmut, you're watching a trail of bubbles in this de-pressurizing freon that wants to create bubbles but you have to nucleate bubbles. What you're seeing there in that central line that goes up and then curls around is a single electron that was nailed by a neutrino incident from a beam at CERN where currently the LHC is running. And this experiment revealed two things. First, to us it will reveal that you can see individual electrons and by studying the images of them moving through fluids and leaving a disturbing wake of bubbles behind them. We can study their properties in some considerable detail. The second thing it taught us is something new-- we're not going to talk about it in detail-- is that it's possible for a neutrino to hit an electron. And that process is called a weak neutral current for sort of stupid historical reasons. It's actually a really good name. And that was awesome and surprising and so this picture is both a monument to the technology of the experiment, but also to the physics of weak neutral currents and electrons. They exist if you can discover neutrinos by watching them. OK. Secondly, nuclei. We know that nuclei exist because you can shoot alpha particles, which come from radioactive decay, at atoms. And you have your atom which is some sort of vague thing, and I'm gonna make the-- I'm gonna find the atom by making a sheet of atoms. Maybe a foil. A very thin foil of stuff. And then I'm gonna shoot very high-energy alpha particles incident of this. Probably everyone has heard of this experiment, it was done by Rutherford and Geiger and Marsden, in particular his students at the time or post-docs. I don't recall-- and you shoot these alpha particles in. And if you think of these guys as some sort of jelly-ish lump then maybe they'll deflect a little bit, but if you shoot a bullet through Jello it just sort of maybe gets deflected a little bit. But Jello, I mean, come on. And I think what was shocking is that you should these alpha particles in and every once in a while, they bounce back at, you know, 180, 160 degrees. Rutherford likened this to rolling a bowling ball against a piece of paper and having it bounce back. Kind of surprising. And the explanation here that people eventually came up upon is that atoms are mostly zero density. Except they have very, very high density cores, which are many times smaller than the size of the atom but where most of the mass is concentrated. And as a consequence, most of the inertia. And so we know that atoms have substructure, and the picture we have is that well if you scrape this pile of metal, you can pull off the electrons, leaving behind nuclei which have positive charge because you've scraped off the electrons that have negative charge. So we have a picture from these experiments that there are electrons and there are nuclei-- which, I'll just write N and plus-- which are the constituents of atoms. Now this leads to a very natural picture of what an atom is. If you're a 19th-century physicist, or even an early 20th-century physicist, it's very natural to say, aha, well if I know if I have a positive charge and I have a negative charge, then they attract each other with a 1 over r minus q1 q2-- sorry, q1 q2 over r potential. This is just like gravity, right. The earth and the sun are attracted with an inverse-r potential. This leads to Keplerian orbits. And so maybe an atom is just some sort of orbiting classical combination of an electron and a nucleus, positively charged nucleus. The problem with this picture, as you explore in detail in your first problem on the problem set, is that it doesn't work. What happens when you accelerate a charge? It radiates. Exactly. So if it's radiating, it's gotta lose energy. It's dumping energy into this-- out of the system. So it's gotta fall lower into the potential. Well it falls lower, it speeds up. It radiates more. Because it's accelerating more to stay in a circular orbit. All right, it radiates more, it has to fall further down. So on the problem set you're going to calculate how long that takes. And it's not very long. And so the fact that we persist for more than a few picoseconds tells you that it's not that-- this is not a correct picture of an atom. OK. So in classical mechanics, atoms could not exist. And yet, atoms exist. So we have to explain that. That's gonna be our first challenge. Now interestingly Geiger who is this collaborator of Rutherford, a young junior collaborator of Rutherford, went on to develop a really neat instrument. So suppose you want to see radiation. We do this all the time. I'm looking at you and I'm seeing radiation, seeing light. But I'm not seeing ultra high energy radiation, I'm seeing energy radiation in the electromagnetic waves in the optical spectrum. Meanwhile I'm also not seeing alpha particles. So what Geiger wanted was a way to detect without using your eyes radiation that's hard to see. So the way he did this is he took a capacitor and he filled the-- surrounded the capacitor with some noble gas. It doesn't interact. There's no-- it's very hard to ionize. And if you crank up the potential across this capacitor plate high enough, what do you get? A spark. You all know this, if you crank up a capacitor it eventually breaks down because the dielectric in between breaks down, you get a spontaneous sparking. So what do you figure it would look if I take a capacitor plate and I charge it up, but not quite to breakdown. Just a good potential. And another charged particle comes flying through, like an alpha particle, which carries a charge of plus 2, that positive charge will disturb things and will add extra field effectively. And lead to the nucleation of a spark. So the presence of a spark when this potential is not strong enough to induce a spark spontaneously indicates the passage of a charged particle. Geiger worked later with-- Marsden? Muller. Heck. I don't even remember. And developed this into a device now known as the Geiger counter. And so you've probably all seen or heard Geiger counters going off in movies, right. They go ping ping ping ping ping ping ping ping ping, right. They bounce off randomly. This is an extremely important lesson, which is tantamount to the lesson of our second experiment yesterday. The 50-50, when we didn't expect it. The white electrons into the harness box then into a color box again, would come out 50-50, not 100 percent. And they come out in a way that's unpredictable. We have no ability to our knowledge-- and more than our knowledge, we'll come back to that with Bell's Inequality-- but we have no ability to predict which electron will come out of that third box, white or black, right. Similarly with a Geiger counter you hear that atoms decay, but they decay randomly. The radiation comes out of a pile of radioactive material totally at random. We know the probabilistic description of that. We're going to develop that, but we don't know exactly when. And that's a really powerful example-- both of those experiments are powerful examples of randomness. And so we're going to have to incorporate that into our laws of physics into our model of quantum phenomena as well. Questions? I usually have a Geiger counter at this point, which is totally awesome, so I'll try to produce the Geiger counter demo later. But the person with the Geiger counter turns out to have left the continent, so made it a little challenging. OK. Just sort of since we're at MIT, an interesting side note. This strategy of so-called hard scattering, of taking some object and sending it at very high velocity at some other object and looking for the rare events when they bounce off at some large angle, so-called hard scattering. Which is used to detect dense cores of objects. It didn't stop with Rutherford. People didn't just give up at that point. Similar experience in the '60s and '70s which are conducted at Slack, were involved not alpha particles incident on atoms but individual electrons incident on protons. So not shooting into the nucleus, but shooting and looking for the effect of hitting individual protons inside the nucleus. And through this process it was discovered that in fact-- so this was done in the '60s and '70s, that in fact the proton itself is also not a fundamental particle. The proton is itself composite. And in particular, it's made out of-- eventually people understood that it's made out of, morally speaking, and I'm gonna put this in quotation marks-- ask me about it in office hours-- three quarks, which are some particles. And the reason we-- all this tells you is that it's some object and we've given it the name quark. But indeed there are three point-like particles that in some sense make up a proton. It's actually much more complicated than that, but these quarks, among other things, have very strange properties. Like they have fractional charge. And this was discovered by a large group of people, in particular led by Kendall and Friedman and also Richard Taylor. Kendall and Friedman were at MIT, Richard Taylor was at Stanford. And in 1990 they shared the Nobel Prize for the discovery of the partonic structure out of the nucleons. So these sorts of techniques that people have been using for a very long time continue to be useful and awesome. And in particular the experiment, the experimental version of this that's currently going on, that I particularly love is something called the relativistic heavy ion collider, which is going on at Brookhaven. So here what you're doing is you take two protons and you blow them into each other at ultra high energy. Two protons, collide them and see what happens. And that's what happens. You get massive shrapnel coming flying out. So instead of having a simple thing where one of the protons just bounces because there's some hard quark, instead what happens is just shrapnel everywhere, right. So you might think, well, how do we interpret that at all. How do you make sense out of 14,000 particles coming out of two protons bouncing into each other. How does that make any sense? And the answer turns out to be kind of awesome. And so this touches on my research. So I want to make a quick comment on it just for color. The answer turns out to be really interesting. First off, the interior constituents of protons interact very strongly with each other. But at the brief moment when protons collide with each other, what you actually form is not a point-like quirk and another point-like quark. In fact, protons aren't made out of point-like quarks at all. Protons are big bags with quarks and gluons and all sorts of particles fluctuating in and out of existence in a complicated fashion. And what you actually get is, amazingly, a liquid. For a brief, brief moment of time the parts of those protons that overlap-- think of them as two spheres and they overlap in some sort of almond-shaped region. The parts of those protons that overlap form a liquid at ultra high temperature and at ultra high density. It's called the RHIC fireball or the quark-gluon plasma, although it's not actually a plasma. But it's a liquid like water. And what I mean by saying it's a liquid like water, if you push it, it spreads in waves. And like water, it's dissipative. Those waves dissipate. But it's a really funny bit of liquid. Imagine you take your cup of coffee. You drink it, you're drinking your coffee as I am wont to do, and it cools down over time. This is very frustrating. So you pour in a little bit of hot coffee and when you pour in that hot coffee, the system is out of equilibrium. It hasn't thermalized. So what you want is you want to wait for all of the system to wait until it's come to equilibrium so you don't get a swig of hot or swig of cold. You want some sort of Goldilocks-ean in between. So you can ask how long does it take for this coffee to come to thermal equilibrium. Well it takes a while. You know, a few seconds, a few minutes, depending on exactly how you mess with it. But let me ask you a quick question. How does that time scale compare to the time it takes for light to cross your mug? Much, much, much slower, right? By orders of magnitude. For this liquid that's formed in the ultra high energy collision of two protons, the time it takes for the system-- which starts out crazy out of equilibrium with all sorts of quarks here and gluons there and stuff flying about-- the time it takes for it to come to thermal equilibrium is of order the time it takes for light to cross the little puddle of liquid. This is a crazy liquid, it's called a quantum liquid. And it has all sorts of wonderful properties. And the best thing about it to my mind is that it's very well modeled by black holes. Which is totally separate issue, but it's a fun example. So from these sorts of collisions, we know a great deal about the existence of atoms and randomness, as you can see. That's a fairly random sorting. OK so moving on to more 8.04 things. Back to atoms. So let's look at specifics of that. I'm not kidding, they really are related to black holes. I get paid for this. So here's a nice fact, so let's get to atomic spectra. So to study atomic spectra, here's the experiment I want to run. The experiment I want to run starts out with some sort of power plant. And out of the power plant come two wires. And I'm going to run these wires across a spark gap, you know, a piece of metal here, a piece of metal here, and put them inside a container, which has some gas. Like H2 or neon or whatever you want. But some simple gas inside here. So we've got an electric potential established across it. Again, we don't want so much potential that it sparks, but we do want to excite the H2. So we can even make it spark, it doesn't really matter too much. The important thing is that we're going to excite the hydrogen, and in exciting the hydrogen the excited hydrogen is going to send out light. And then I'm going to take this light-- we take the light, and I'm gonna shine this on a prism, something I was taught to do by Newton. And-- metaphorically speaking-- and look at the image of this light having passed through the prism. And what you find is you find a very distinct set of patterns. You do not get a continuous band. In fact what you get-- I'm going to have a hard time drawing this so let me draw down here. I'm now going to draw the intensity of the light incident on the screen on this piece of paper-- people really used to use pieces of paper for this, which is kind of awesome-- as a function of the wavelength, and I'll measure it in angstroms. And what you discover is-- here's around 1,000 angstroms-- you get a bunch of lines. Get these spikes. And they start to spread out, and then there aren't so many. And then at around 3,000, you get another set. And then at around 10,000, you get another set. This is around 10,000. And here's the interesting thing about these. So the discovery of these lines-- these are named after a guy named Lyman, these are-- these are named after a guy named-- Ballmer. Thank you. Steve Ballmer. And these are passion, like passion fruit. So. Everyone needs a mnemonic, OK. And so these people identified these lines and explained various things about them. But here's an interesting fact. If you replace this nuclear power plant with a coal plant, it makes no difference. If you replace this prism by a different prism, it makes no difference to where the lines are. If you change this mechanism of exciting the hydrogen, it makes no difference. As long as it's hydrogen-- as long as it's hydrogen in here you get the same lines, mainly with different intensities depending upon how exactly you do the experiment. But you get the same position of the lines. And that's a really striking thing. Now if you use a different chemical, a different gas in here, like neon, you get a very different set of lines. And a very different effective color now when you eyeball this thing. So Ballmer, incidentally-- and I think this is actually why he got blamed for that particular series, although I don't know the history-- Ballmer noticed by being-- depending on which biography you read-- very clever or very obsessed that these guys, this particular set, could be-- they're wavelengths. If you wrote their wavelengths and labeled them by an integer n, where n ran from 3 to any positive integer above 3, could be written as 36. So this is pure numerology. 36, 46 angstroms times the function n squared over n squared minus 4, where N is equal to 3, 4, dot dot dot-- an integer. And it turns out if you just plug in these integers, you get a pretty good approximation to this series of lines. This is a hallowed tradition, a phenomenological fit to some data. Where did it come from? It came from his creative or obsessed mind. So this was Ballmer. And this is specifically for hydrogen gas, H2. So Rydberg and Ritz, R and R, said, well actually we can do one better. Now that they realized that this is true, they looked at the whole sequence. And they found a really neat little expression, which is that 1 over the wavelength is equal to a single constant parameter. Not just for all these, but for all of them. One single numerical coefficient times 1 over m squared minus 1 over n squared-- n is an integer greater than zero and greater in particular than m. And if you plug in any value of n and any value of m, for sufficiently reasonable-- I mean, if you put in 10 million integers you're not going to see it because it's way out there, but if you put in or-- rather, in here-- if you put any value of n and m, you will get one of these lines. So again, why? You know, as it's said, who ordered that. So this is experimental result three that we're going to have to deal with. When you look at atoms and you look at the specter of light coming off of them, their spectra are discrete. But they're not just stupidly discrete, they're discrete with real structure. Something that begs for an explanation. This is obviously more than numerology, because it explains with one tunable coefficient a tremendous number of spectral lines. And there's a difference-- and crucially, these both work specifically for hydrogen. For different atoms you need a totally different formula. But again, there's always some formula that nails those spectral lines. Why? Questions? OK. So speaking of atomic spectra-- whoops, I went one too far-- here's a different experiment. So people notice the following thing. People notice that if you take a piece of metal and you shine a light at it, by taking the sun or better yet, you know, these days we'd use a laser, but you shine light on this piece of metal. Something that is done all the time in condensed matter labs, it's a very useful technique. We really do use lasers not the sun, but still it continues to be useful in fact to this day. You shine light on a piece of metal and every once in a while what happens is electrons come flying off. And the more light and the stronger the light you shine, you see changes in the way that electrons bounce off. So we'd like to measure that. I'd like to make that precise. And this was done in a really lovely experiment. Here's the experiment. The basic idea of the experiment is I want to check to see, as I change the features of the light, the intensity, the frequency, whatever, I want to see how that changes the properties of the electrons that bounce off. Now one obvious way-- one obvious feature of an electron that flew off a piece of metal is how fast is it going, how much energy does it have. What's its kinetic energy. So I'd like to build an experiment that measures the kinetic energy of an electron that's been excited through this photoelectric effect. Through emission after shining light on a piece of metal. Cool? So I want to build that experiment. So here's how that experiment goes. Well if this electron comes flying off with some kinetic energy and I want to measure that kinetic energy, imagine the following circuit. OK first off imagine I just take a second piece of metal over here, and I'm going to put a little current meter here, an ammeter. And here's what this circuit does. When you shine light on this piece of metal-- we'll put a screen to protect the other piece of metal-- the electrons come flying off, they get over here. And now I've got a bunch of extra electrons over here and I'm missing electrons over here. So this is negative, this is positive. And the electrons will not flow along this wire back here to neutralize the system. The more light I shine, the more electrons will go through this circuit. And as a consequence, there will be a current running through this current meter. That cool with everyone? OK. So we haven't yet measured the kinetic energy, though. How do we measure the kinetic energy? I want to know how much energy, with how much energy, were these electrons ejected. Well I can do that by the following clever trick. I'm going to put now a voltage source here, which I can tune the voltage of, with the voltage V. And what that's going to do is set up a potential difference across these and the energy in that is the charge times the potential difference. So I know that the potential difference it takes, so the amount of energy it takes to overcome this potential difference, is q times V. That cool? So now imagine I send in an electron-- I send in light and it leads an electron to jump across, and it has kinetic energy, kE. Well if the kinetic energy is less than this, will it get across? Not so much. It'll just fall back. But if the kinetic energy is greater than the energy it takes to cross, it'll cross and induce a current. So the upshot is that, as a function of the voltage, what I should see is that there is some critical minimum voltage. And depending on how you set up the sign, the sign could be the other way, but there's some critical minimal voltage where, for less voltage, the electron doesn't get across. And for any greater voltage-- or, sorry, for any closer to zero voltage, the electron has enough kinetic energy to get across. And so the current should increase. So there's a critical voltage, V-critical, where the current running through the system runs to zero. You make it harder for the electrons by making the voltage in magnitude even larger. You make it harder for the electrons to get across. None will get across. Make it a little easier, more and more will get across. And the current will go up. So what you want to do to measure this kinetic energy is you want to measure the critical voltage at which the current goes to zero. So now the question is what do we expect to see. And remember that things we can tune in this experiment are the intensity of the light, which is like e squared plus b squared. And we can tune the frequency of the light. We can vary that. Now does the total energy, does that frequency show up in the total energy of a classical electromagnetic wave? No. If it's an electromagnetic wave, it cancels out. You just get the total intensity, which is a square of the fields. So this is just like a harmonic oscillator. The energy is in the amplitude. The frequency of the oscillator doesn't matter. You push the swing harder, it gets more kinetic energy. It's got more energy. OK. So what do we expect to see as we vary, for example, the intensity? So here's a natural gas. If you take-- so you can think about the light here as getting a person literally, like get the person next to you to take a bat and hit a piece of metal. If they hit it really lightly they're probably not going to excite electrons with a lot of energy. If they just whack the heck out of it, then it wouldn't be too surprising if you get much more energy in the particles that come flying off. Hit it hard enough, things are just gonna shrapnel and disintegrate. The expectation here is the following. That if you have a more intense beam, then you should get more-- the electrons coming off should be more energetic. Because you're hitting them harder. And remember that the potential, which I will call V0, the stopping voltage. So therefore V0 should be greater in magnitude. So this anticipates that the way this curve should look as we vary the current as a function of v, if we have a low voltage-- sorry, if we have a low-intensity beam-- it shouldn't take too much potential just to impede the motion. But if we have a-- so this is a low intensity. But if we have a high-intensity beam, it should take a really large voltage to impede the electric flow, the electric current, because high-intensity beam you're just whacking those electrons really hard and they're coming off with a lot of kinetic energy. So this is high intensity. Everyone down with that intuition? This is what you get from Maxwell's electrodynamics. This is what you'd expect. And in particular, as we vary-- so this is our predictions-- in particular as we vary-- so this is 1, 2, with greater intensity. And the second prediction is that V-naught should be independent of frequency. Because the energy density and electromagnetic wave is independent of the frequency. It just depends on the amplitude. And I will use nu to denote the frequency. So those are the predictions that come from 8.02 and 8.03. But this is 8.04. And here's what the experimental results actually look like. So here's the intensity, here's the potential. And if we look at high potential, it turns out that-- if we look, sorry, if we look at intermediate potentials, it's true that the high intensity leads to a larger current and the low intensity leads to a lower current. But here's the funny thing that happens. As you go down to the critical voltage, their critical voltages are the same. What that tells you is that the kinetic energy kicked out-- or the kinetic energy of an electron kicked out of this piece of metal by the light is independent of how intense that beam is. No matter how intense that beam is, no matter how strong the light you shine on the material, the electrons all come out with the same energy. This would be like taking a baseball and hitting it with a really powerful swing or a really weak swing and seeing that the electron dribbles away with the same amount of energy. This is very counter-intuitive. But more surprisingly, V-naught is actually independent of intensity. But here's the real shocker. V-naught varies linearly in the frequency. What does change V-naught is changing the frequency of the light in this incident. That means that if you take an incredibly diffuse light-- incredibly diffuse light, you can barely see it-- of a very high frequency, then it takes a lot of energy to impede the electrons that come popping off. The electrons that come popping off have a large energy. But if you take a low-frequency light with extremely high intensity, then those electrons are really easy to stop. Powerful beam but low frequency, it's easy to stop those electrons. Weak little tiny beam at high frequency, very hard to stop the electrons that do come off. So this is very counter-intuitive and it doesn't fit at all with the Maxwellian picture. Questions about that? So this led Einstein to make a prediction. This was his 1905 result. One of his many totally breathtaking papers of that year. And he didn't really propose a model or a detailed theoretical understanding of this, but he proposed a very simple idea. And he said, look, if you want to fit this-- if you want to fit this experiment with some simple equations, here's the way to explain it. I claim-- I here means Einstein, not me-- I claim that light comes in packets or chunks with definite energy. And the energy is linearly proportional to the frequency. And our energy is equal to something times nu, and we'll call the coefficient h. The intensity of light, or the amplitude squared, the intensity is like the number of packets. So if you have a more intense beam at the same frequency, the energy of each individual chunk of light is the same. There are just a lot more chunks flying around. And so to explain the photoelectric effect, Einstein observed the following. Look, he said, the electrons are stuck under the metal. And it takes some work to pull them off. So now what's the kinetic energy of an electron that comes flying off-- whoops, k3. Bart might have a laugh about that one. Kinetic, kE, not 3. So the kinetic energy of electron that comes flying off, well, it's the energy deposited by the photon, the chunk of light, h-nu well we have to subtract off the work it took. Minus the work to extract the electron from the material. And you can think of this as how much energy does it take to suck it off the surface. And the consequence of this is that the kinetic energy of an electron should be-- look, if h-nu is too small, if the frequency is too low, then the kinetic energy would be negative. But that doesn't make any sense. You can't have negative kinetic energy. It's a strictly positive quantity. So it just doesn't work until you have a critical value where the frequency times h-- this coefficient-- is equal to the work it takes to extract. And after that, the kinetic energy rises with the frequency with a slope equal to h. And that fits the data like a champ. So no matter-- let's think about what this is saying again. No matter what you do, if your light is very low-frequency and you pick some definite piece of metal that has a very definite work function, very definite amount of energy it takes to extract electrons from the surface. No matter how intense your beam, if the frequency is insufficiently high, no electrons come off. None. So it turns out none is maybe a little overstatement because what you can have is two photon processes, where two chunks hit one electron at the right, just at the same time. Roughly speaking the same time. And they have twice the energy, but you can imagine that the probability of two photon hitting one electron at the same time of pretty low. So the intensity has to be preposterously high. And you see those sorts of multi-photon effects. But as long as we're not talking about insanely high intensities, this is an absolutely fantastic probe of the physics. Now there's a whole long subsequent story in the development of quantum mechanics about this particular effect. And it turns out that the photoelectric effect is a little more complicated than this. But the story line is a very useful one for organizing your understanding of the photoelectric effect. And in particular, this relation that Einstein proposed out of the blue, with no other basis. No one else had ever seen this sort of statement that the electrons, or that the energy of a beam of light should be made up of some number of chunks, each of which has a definite minimum amount of energy. So you can take what you've learned from 8.02 and 8.03 and extract a little bit more information out of this. So here's something you learned from 8.02. In 8.02 you learned that the energy of an electromagnetic wave is equal to c times the momentum carried by that wave-- whoops, over two. And in 8.03 you should have learned that the wavelength of an electromagnetic wave times the frequency is equal to the speed of light, C. And we just had Einstein tell us-- or declare, without further evidence, just saying, look this fits-- that the energy of a chunk of light should be h times the frequency. So if you combine these together, you get another nice relation that's similar to this one, which says that the momentum of a chunk of light is equal to h over lambda. So these are two enormously influential expressions which come out of this argument from the photoelectric effect from Einstein. And they're going to be-- their legacy will be with us throughout the rest of the semester. Now this coefficient has a name, and it was named after Planck. It's called Planck's Constant. And the reason that it's called Planck's Constant has nothing to do with the photoelectric effect. It was first this idea that an electromagnetic wave, that light, has an energy which is linearly proportional not to its intensity squared, none of that, but just linearly proportional to the frequency. First came up an analysis of black body radiation by Planck. And you'll understand, you'll go through this in some detail in 8.044 later in the semester. So I'm not going to dwell on it now, but I do want to give you a little bit of perspective on it. So Planck ran across this idea that E is equal to h/nu. Through the process of trying to fit an experimental curve. There was a theory of how much energy should be emitted by an object that's hot and glowing as a function of frequency. And that theory turned out to be in total disagreement with experiment. Spectacular disagreement. The curve for the theory went up, the curve for the experiment went down. They were totally different. So Planck set about writing down a function that described the data. Literally curve-fitting, that's all he was doing. And this is the depths of desperation to which he was led, was curve-fitting. He's an adult. He shouldn't be doing this, but he was curve-fitting. And so he fits the curve, and in order to get it to fit the only thing that he can get to work even vaguely well is if he puts in this calculation of h/nu. He says, well, maybe when I sum over all the possible energies I should restrict the energies which were proportional to the frequency. And it was forced on him because it fit from the function. Just functional analysis. Hated it. Hated it, he completely hated it. He was really frustrated by this. It fit perfectly, he became very famous. He was already famous, but he became ridiculously famous. Just totally loathed this idea. OK. So it's now become a cornerstone of quantum mechanics. But he wasn't so happy about it. And to give you a sense for how bold and punchy this paper by Einstein was that said, look, seriously. Seriously guys. e equals h/nu. Here's what Planck had to say when he wrote a letter of recommendation to get Einstein into the Prussian Academy of Sciences in 1917, or 1913. So he said, there is hardly one among the great problems in physics to which Einstein has not made an important contribution. That he may sometimes have missed the target in his speculations as in his hypothesis of photons cannot really be held too much against him. It's not possible to introduce new ideas without occasionally taking a risk. Einstein who subsequently went on to develop special relativity and general relativity and prove the existence of atoms and the best measurement of Avogadro's Constant, subsequently got the Nobel Prize. Not for Avogadro's Constant, not for proving the existence of atoms, not for relativity, but for photons. Because of guys like Planck, right. This is crazy. So this was a pretty bold idea. And here, to get a sense for why-- we're gonna leave that up because it's just sort of fun to see these guys scowling and smiling-- there is, incidentally there's a great book about Einstein's years in Berlin by Tom Levenson, who's a professor here. A great writer and a sort of historian of science. You should take a class from him, which is really great. But I encourage you to read this book. It talks about why Planck is not looking so pleased right there, among many other things. It's a great story. So let's step back for a second. Why was Planck so upset by this, and why was in fact everyone so flustered by this idea that it led to the best prize you can give a physicist. Apart from a happy home and, you know. I've got that one. That's the one that matters to me. So why is this so surprising? And the answer is really simple. We know that it's false. We know empirically, we've known for two hundred and some years that light is a wave. Empirically. This isn't like people are like, oh I think it'd be nice if it was a wave. It's a wave. So how do we know that? So this goes back to the double-slit experiment from Young. Young's performance of this was in 1803. Intimations of it come much earlier. But this is really where it hits nails to the wall. And here's the experiment. So how many people in here have not seen a double-slit experiment described? Yeah, exactly. OK. So I'm just going to quickly remind you of how this goes. So we have a source for waves. We let the waves get big until they're basically plane waves. And then we take a barrier. And we poke two slits in it. And these plane waves induce-- they act like sources at the slits and we get nu. And you get crests and troughs. And you look at some distant screen and you look at the pattern, and the pattern you get has a maximum. But then it falls off, and it has these wiggles, these interference fringes. These interference fringes are, of course, extremely important. And what's going on here is that the waves sometimes add in-- so the amplitude of the wave, the height of the wave, sometimes adds constructively and sometimes destructively. So that sometimes you get twice the height and sometimes you get nothing. So just because it's fun to see this, here's Young's actual diagram from his original note on the double-slit experiment. So a and b are the slits, and c, d and f are the [INAUDIBLE] on the screen, the distant screen. He drew it by hand. It's pretty good. So we've known for a very long time that light, because of the double-slit experiment, light is clearly wavy, it behaves like a wave. And what are the senses in which it behaves like a wave? There are two important senses here. The first is answered by the question, where did the wave hit the screen? So when we send in a wave, you know, I drop a stone, one big pulsive wave comes out. It splits into-- it leads to new waves being instigated here and over here. Where did that wave hit the screen? Anyone? AUDIENCE: Everywhere. PROFESSOR: Yeah, exactly. It didn't hit this wave-- the screen in any one spot. But some amplitude shows up everywhere. The wave is a distributed object, it does not exist at one spot, and it's by virtue of the fact that it is not a localized object-- it is not a point-like object-- that it can interfere with itself. The wave is a big large phenomena in a liquid, in some thing. So it's sort of essential that it's not a localized object. So not localized. The answer is not localized. And let's contrast this with what happens if you take this double-slit experiment and you do it with, you know, I don't know, take-- who. Hmm. Tim Wakefield. Let's give some love to that guy. So, baseball player. And have him throw baseballs at a screen with two slits in it. OK? Now he's got pretty good-- well, he's got terrible accuracy, actually. So every once in a while he'll make it through the slits. So let's imagine first blocking off-- what, he's a knuckle-baller, right-- so every once in a while it goes, the baseball will go through the slit. And let's think about what happens, so let's cover one slit. And what we expect to happen is, well, it'll go through more or less straight, but sometimes it'll scrape the edge, it'll go off to the side, and sometimes it'll come over here. But if you take a whole bunch of baseballs, and-- so any one baseball, where does it hit? Some spot. Right? One spot. Not distributed. One spot. And as a consequence, you know, one goes here, one goes there, one goes there. And now, there's nothing like interference effects, but what happens is as it sort of doesn't-- you get some distribution if you look at where they all hit. Yeah? Everyone cool with that? And if we had covered over this slot, or slit, and let the baseballs go through this one, same thing would have happened. Now if we leave them both open, what happens is sometimes it goes here, sometimes it goes here. So now it's pretty useful that we've got a knuckle-baller. And what you actually get is the total distribution looks like this. It's the sum of the two. But at any given time, any one baseball, you say, aha, the baseball either went through the top slit, and more or less goes up here. Or it went through the bottom slit and more or less goes down here. So for chunks-- so this is for waves-- for chunks or localized particles, they are localized. And as a consequence, we get no interference. So for waves, they are not localized, and we do get interference. Yes, interference. OK. So on your problem set, you're going to deal with some calculations involving these interference effects. And I'm going to brush over them. Anyway the point of the double-slit experiment is that whatever else you want to say about baseballs or anything else, light, as we've learned since 1803 in Young's double-slit experiment, light behaves like a wave. It is not localized, it hits the screen over its entire extent. And as a consequence, we get interference. The amplitudes add. The intensity is the square of the amplitude. If the intensities add-- so sorry, if the amplitudes add-- amplitude total is equal to a1 plus a2, the intensity, which is the square of a1 plus a2 squared, has interference terms, the cross terms, from this square. So light, from this point of view, is an electromagnetic wave. It interferes with itself. It's made of chunks. And I can't help but think about it this way, this is literally the metaphor I use in my head-- light is creamy and smooth like a wave. Chunks are very different. But here's the funny thing. Light is both smooth like a wave, it is also chunky. It is super chunky, as we have learned from the photoelectric effect. So light is both at once. So it's the best of both worlds. Everyone will be satisfied, unless you're not from the US, in which case this is deeply disturbing. So of course the original Superchunk is a band. So we've learned now from Young that light is a wave. We've learned from the photoelectric effect that light is a bunch of chunks. OK. Most experimental results are true. So how does that work? Well, we're gonna have to deal with that. But enough about light. If this is true of light, if light, depending on what experiment you do and how you do the experiment, sometimes it seems like it's a wave, sometimes it seems like it's a chunk or particle, which is true? Which is the better description? So it's actually worthwhile to not think about light all the time. Let's think about something more general. Let's stick to electrons. So as we saw from yesterday's lecture, you probably want to be a little bit wary when thinking about individual electrons. Things could be a little bit different than your classical intuition. But here's a crucial thing. Before doing anything else, we can just think, which one of these two is more likely to describe electrons well. Well electrons are localized. When you throw an electron at a CRT, it does not hit the whole CRT with a wavy distribution. When you take a single electron and you throw it at a CRT, it goes ping and there's a little glowing spot. Electrons are localized. And we know that localized things don't lead to interference. Some guys at Hitachi, really good scientists and engineers, developed some really awesome technology a couple of decades ago. They were trying to figure out a good way to demonstrate their technology. And they decided that you know what would be really awesome, this thought experiment that people have always talked about that's never been done really well, of sending an electron through a two-slitted experiment. In this case it was like ten slits effectively, it was a grading. Send an electron, a bunch of electrons, one at a time, throw the electron, wait. Throw the electron, wait. Like our French guy with the boat. So do this experiment with our technology and let's see what happens. And this really is one of my favorite-- let's see, how we close these screens-- aha. OK. This is going to take a little bit of-- and it's broken. No, no. Oh that's so sad. AUDIENCE: [LAUGHTER] PROFESSOR: Come on. I'm just gonna let-- let's see if we can, we'll get part of the way. I don't want to destroy it. So what they actually did is they said, look, let's-- we want to see what happens. We want to actually do this experiment because we're so awesome at Hitachi Research Labs, so let's do it. So here's what they did. And I'm going to turn off the light. And I set this to some music because I like it. OK here's what's happening. One at a time, individual photons. [MUSIC PLAYING] PROFESSOR: So they look pretty localized. There's not a whole lot of structure. Now they're going to start speeding it up. It's 100 times the actual speed. [MUSIC PLAYING] PROFESSOR: Eh? Yeah. AUDIENCE: [APPLAUSE] PROFESSOR: So those guys know what they're doing. Let's-- there were go. So I think I don't know of a more vivid example of electron interference than that one. It's totally obvious. You see individual electrons. They run through the apparatus. You wait, they run through the apparatus. You wait. One at a time, single electron, like a baseball being pitched through two slits, and what you see is an interference effect. But you don't see the interference effect like you do from light, from waves on the sea. You see the interference effect by looking at the cumulative stacking up of all the electrons as they hit. Look at where all the electrons hit one at a time. So is an electron behaving like a wave in a pond? No. Does a wave in a pond at a spot? No. It's a distributed beast. OK yes, it interferes, but it's not localized. Well is it behaving like a baseball? Well it's localized. But on-- when I look at a whole bunch of electrons, they do that. They seem to interfere, but there's only one electron going through at a time. So in some sense it's interfering with itself. How does that work? Is an electron a wave? AUDIENCE: Yes. PROFESSOR: Does an electron hit at many spots at once? AUDIENCE: No. PROFESSOR: No. So is an electron a wave. No. Is an electron a baseball? No. It's an electron. So this is something you're going to have to deal with, that every once in awhile we have these wonderful moments where it's useful to think about an electron as behaving in a wave-like sense. Sometimes it's useful to think about it as behaving in a particle-like sense. But it is not a particle like you normally conceive of a baseball. And it is not a wave like you normally conceive of a wave on the surface of a pond. It's an electron. I like to think about this like an elephant. If you're closing your eyes and you walk up to an elephant, you might think like I've got a snake and I've got a tree trunk and, you know, there's a fan over here. And you wouldn't know, like, maybe it's a wave, maybe it's a particle, I can't really tell. But if you could just see the thing the way it is, not through the preconceived sort of notions you have, you'd see it's an elephant. Yes, that is the Stata Center. So-- look, everything has to happen sometime, right? AUDIENCE: [LAUGHTER] PROFESSOR: So Heisenberg-- it's often, people often give the false impression in popular books on physics, so I want to subvert this, that in the early days of quantum mechanics, the early people like Born and Oppenheimer and Heisenberg who invented quantum mechanics, they were really tortured about, you know, is it an electron, is it a wave. It's a wave-particle duality. It's both. And this is one of the best subversions of that sort of silliness that I know of. And so what Heisenberg says, the two mental pictures which experiments lead us to form, the one of particles the other waves, are both incomplete and have the validity of analogies, which are accurate only in limited cases. The apparent duality rises in the limitation of our language. And then he goes on to say, look, you developed your intuition by throwing rocks and, you know, swimming. And, duh, that's not going to be very good for atoms. So this will be posted, it's really wonderful. His whole lecture is really-- the lectures are really quite lovely. And by the way, that's him in the middle there, Pauley all the way on the right. I guess they were pleased. OK so that's the Hitachi thing. So now let's pick up on this, though. Let's pick up on this and think about what happens. I want to think in a little more detail about this Hitachi experiment. And I want to think about it in the context of a simple two-slit experiment. So here's our source of electrons. It's literally a gun, an electron gun. And it's firing off electrons. And here's our barrier, and it has two slits in it. And we know that any individual electron hits its own spot. But when we take many of them, we get an interference effect. We get interference fringes. And so the number that hit a given spot fill up, construct this distribution. So then here's the question I want to ask. When I take a single electron, I shoot one electron at a time through this experiment, one electron. It could go through the top slit, it could go through the bottom slit. While it's inside the apparatus, which path does it take? AUDIENCE: Superposition. PROFESSOR: Good. So did it take the top path? AUDIENCE: No. PROFESSOR: How do you know? [INTERPOSING VOICES] PROFESSOR: Good, let's block the bottom, OK, to force it to go through the top slit. So we'll block the bottom slit. Now the only electrons that make it through go through the top slit. Half of them don't make it through. But those that do make it through give you this distribution. No interference. But I didn't tell you these are hundreds of thousands of kilometers apart, the person who threw in the electron didn't know whether there was a barrier here. The electron, how could it possibly know whether there was a barrier here if you went through the top. This is exactly like our boxes. It's exactly like our box. Did it go through-- an electron, when the slits are both open and we know that ensemble average it will give us an interference effect, did the electron inside the apparatus go through the top path? No. Did it go through the bottom path? Did it go through both? Because we only see one electron. Did it go through neither? It is in a-- AUDIENCE: Superposition. PROFESSOR: --of having gone through the top and the bottom. Of being along the top half and being along the bottom path. This is a classic example of the two-box experiment. OK. So you want to tie that together. So let's nuance this just a little bit, though, because it's going to have an interesting implication for gravity. So here's the nuance I want to pull on this one. Let's cheat. OK. Suppose I want to measure which slit the electron actually did go through. How might I do that? Well I could do the course thing I've been doing which is I could block it and just catch the-- catch electrons that go through in that spot. But that's a little heavy-handed. Probably I can do something a little more delicate. And so here's the more delicate thing I'm going to do. I want to build a detector that uses very, very, very weak light, extremely weak light, to detect whether the particle went through here or here. And the way I can do that is I can sort of shine light through and-- I'm gonna, you know, bounce-- so here's my source of light. And I'll be able to tell whether the electron went through this slit or it went through this slit. Cool? So imagine I did that. So obviously I don't want to use some giant, huge, ultra high-energy laser because it would just blast the thing out of the way. It would destroy the experiment. So I wanna something very diffuse, very low energy, very low intensity electromagnetic wave. And the idea here is that, OK, it's true that when I bounce this light off an electron, let's say it bounces off an electron here, it's true it's going impart some momentum and the electron's gonna change its course. But if it's really, really weak, low energy light, then it's-- it's gonna deflect only a little tiny bit. So it will change the pattern I get over here. But it will change it in some relatively minor way because I've just thrown in very, very low energy light. Yeah? That make sense? So this is the experiment I want to do. This experiment doesn't work. Why. AUDIENCE: You know which slit it went through. PROFESSOR: No. It's true that it turns out that those are correlated facts, but here's the problem. I can run this experiment without anyone actually knowing what happens until long afterwards. So knowing doesn't seem to play any role in it. It's very tempting often to say, no, but it turns out that it's really not about what you know. It's really just about the experiment you're doing. So what principle that we've already run into today makes it impossible to make this work? If I want to shine really low-energy, really diffuse light through, and have it scatter weakly. Yeah. AUDIENCE: Um, light is chunky. PROFESSOR: Yeah, exactly. That's exactly right. So when I say really low-energy light, I don't-- I really can't mean, because we've already done this experiment, I cannot possibly mean low intensity. Because intensity doesn't control the energy imparted by the light. The thing that controls the energy imparted by a collision of the light with the electron is the frequency. The energy in a chunk of light is proportional to the frequency. So now if I want to make the effect the energy or the momentum, similarly-- the momentum, where did it go-- remember the momentum goes like h over lambda. If I want to make the energy really low, I need to make the frequency really low. Or if I want to make the momentum really low, I need to make the wavelength what? Really big. Right? So in order to make the momentum imparted by this photon really low, I need to make the wavelength really long. But now here's the problem. If I make the wavelength really long, so if I use a really long-wavelengthed wave, like this long of a wavelength, are you ever going to be able to tell which slit it went through? No, because the particle could have been anywhere. It could have scattered this light if it was here, if it was here, if it was here, right? In order to measure where the electron is to some reasonable precision-- so, for example, to this sort of wavelength, I need to be able to send in light with a wavelength that's comparable to the scale that I want to measure. And it turns out that if you run through and just do the calculation, suppose I send in-- and this is done in the books, in I think all four, but this is done in the books on the reading list-- if you send in a wave with a short enough wavelength to be able to distinguish between these two slits, which slit did it go through, the momentum that it imparts precisely watches-- washes out is just enough to wash out the interference effect, and break up these fringes so you don't see interference effects. It's not about what you know. It's about the particulate nature of light and the fact that the momentum of a chunk of light goes like h over lambda. OK? But this tells you something really interesting. Did I have to use light to do this measurement? I could have sent in anything, right? I didn't have to bounce light off these things. I could have bounced off gravitational waves. So if I had a gravitational wave detector, so-- Matt works on gravitational wave detectors, and so, I didn't tell you this but Matt gave me a pretty killer gravitational wave detector. It's, you know, here it is. There's my awesome gravitational wave detector. And I'm now going to build supernova. OK. And they are creeping under this black hole, and it's going to create giant gravitational waves. And we're gonna use those gravitational waves and detect them with the super advanced LIGO. And I'm gonna detect which slit it went through. But gravitational waves, those aren't photons. So I really can make a low-intensity gravitational wave, and then I can tell which slit it went through without destroying the interference effect. That would be awesome. What does that tell you about gravitational waves? They must come in chunks. In order for this all to fit together logically, you need all the interactions that you could scatter off this to satisfy these quantization properties. But the energy is proportional to the frequency. The line I just gave you is a heuristic. And making it precise is one of the great challenges of modern contemporary high-energy physics, of dealing with the quantum mechanics and gravity together. But this gives you a strong picture of why we need to treat all forces in all interactions quantum-mechanically in order for the world to be consistent. OK. Good. OK, questions at this point? OK. So-- oh, I forgot about this one-- so there are actually two more. So I want to just quickly show you-- well, OK. So, this is a gorgeous experiment. So remember I told you the story of the guy with the boat and the opaque wall and it turns out that's a cheat. It turns out that this opaque screen doesn't actually give you quantum mechanically isolated photons. They're still, in a very important way, classical. So this experiment was done truly with a source that gives you quantum mechanically isolated single photons, one at a time. So this is the analogue of the Hitachi experiment. And it was done by this pretty awesome Japanese group some number of years ago. And I just want to emphasize that it gives you exactly the same effects. We see that photons-- this should look essentially identical to what we saw at the end of the Hitachi video. And that's because it's exactly the same physics. It's a grating with something like 10 slits and individual particles going through one at a time and hitting the screen and going, bing. So what you see is the light going, bing, on a CCD. It's a pretty spectacular experience. So let's get back to electrons. I want another probe of whether electrons are really waves or not. So this other experiment-- again, you're going to study this on your problem set-- this other experiment was done by a couple of characters named Davisson and Germer. And in this experiment, what they did is they took a crystal, and a crystal is just a lattice of regularly-located ions, like diamond or something. Yeah? AUDIENCE: Before you go on I guess, I wanted to ask if the probability of a photon or an electron going through the 10 slits is about the same? PROFESSOR: Is what, sorry? AUDIENCE: Is exactly the same. PROFESSOR: You mean for different electrons? AUDIENCE: Yeah. PROFESSOR: Well they can be different if the initial conditions are different. But they could be-- if the initial conditions are the same, then the probabilities are identical. So every electron behaves identically to every other electron in that sense. Is that what you were asking? AUDIENCE: It is actually like through any [INAUDIBLE] the probability of it going like [INAUDIBLE]? PROFESSOR: Sure, absolutely. So the issue there is just a technological one of trying to build a beam that's perfectly columnated. And that's just not doable. So there's always some dispersion in your beam. So in practice it's very hard to make them identical, but in principle they could be if you were infinitely powerful as an experimentalist, which-- again, I was banned from the lab, so not me. So here's our crystal. You could think of this as diamond or nickel or whatever. I think they actually use nickel but I don't remember exactly. And they sent in a beam of electrons. So they send in a beam of electrons, and what they discover is that if you send in these electrons and watch how they scatter at various different angles-- I'm going to call the angle here of scattering theta-- what they discover is that the intensity of the reflected beam, as a function of theta, shows interference effects. And in particular they gave a whole calculation for this, which I'm not going to go through right now because it's not terribly germane for us-- you're going to go through it on your problem set, so that'll be good and it's a perfect thing for your recitation instructors to go through. But the important thing is the upshot. So if the distance between these crystal planes is L-- or, sorry, d-- let me call it d. If the distance between the crystal planes is d, what they discover is that the interference effects that they observed, these maxima and minima, are consistent with the wavelength of light. Or, sorry, with the electrons behaving as if they were waves with a definite wavelength, with a wavelength lambda being equal to some integer, n, over 2d sine theta. So this is the data-- these are the data they actually saw, data are plural. And these are the data they actually saw. And they infer from this that the electrons are behaving as if they were wave-like with this wavelength. And what they actually see are individual electrons hitting one by one. Although in their experiment, they couldn't resolve individual electrons. But that is what they see. And so in particular, plugging all of this back into the experiment, you send in the electrons with some energy, which corresponds to some definite momentum. This leads us back to the same expression as before, that the momentum is equal to h over lambda, with this lambda associated. So it turns out that this is correct. So the electrons diffract off the crystal as if they have a momentum which comes with a definite wavelength corresponding to its momentum. So that's experimental result-- oh, I forgot to check off four-- that's experimental result five, that electrons diffract. We already saw the electron diffraction. So something to emphasize is that-- so these experiments as we've described them were done with photons and with electrons, but you can imagine doing the experiments with soccer balls. This is of course hard. Quantum effects for macroscopic objects are usually insignificantly small. However, this experiment was done with Buckyballs, which are the same shape as soccer balls in some sense. But they're huge, they're gigantic objects. So here's the experiment in which this was actually done. So these guys are just totally amazing. So this is Zellinger's lab. And it doesn't look like all-- I mean it looks kind of, you know. It's hideous, right? I mean to a theorist it's like, come on, you've got to be kidding that that's-- But here's what a theorist is happy about. You know, because it looks simple. We really love lying to ourselves about that. So here's an over. We're going to cook up some Buckyballs and emit them with some definite known thermal energy. Known to some accuracy. We're going to columnate them by sending them through a single slit, and then we're going to send them through a diffraction grating which, again, is just a whole bunch of slits. And then we're going to image them using photo ionization and see where they pop through. So here is the horizontal position of this wave along the grating, and this is the number that come through. This is literally one by one counts because they're going bing, bing, bing, as a c60 molecule goes through. So without the grating, you just get a peek. But with the grating, you get the side bands. You get interference fringes. So these guys, again, they're going through one by one. A single Buckyball, 60 carbons, going through one by one is interfering with itself. This is a gigantic object by any sort of comparison to single electrons. And we're seeing these interference fringes. So this is a pretty tour de force experiment, but I just want to emphasize that if you could do this with your neighbor, it would work. You'd just have to isolate the system well enough. And that's a technological challenge but not an in-principle one. OK. So we have one last experimental facts to deal with. And this is Bell's Inequality, and this is my favorite one. So Bell's Inequality for many years languished in obscurity until someone realized that it could actually be done beautifully in an experiment that led to a very concrete experiment that they could actually do and that they wanted to do. And we now think of it as an enormously influential idea which nails the coffin closed for classical mechanics. And it starts with a very simple question. I claim that the following inequality is true: the number of undergraduate-- of the number of people in the room who are undergraduates, which I'll denote as U-- and not blonde, which I will denote as bar B-- so undergraduates who are not blonde-- actually let me write this out in English. It's gonna be easier. Number who are undergrads and not blonde plus the number of people in the room who are blonde but not from Massachusetts is strictly greater than or equal to the number of people in the room who are undergraduates and not from Massachusetts. I claim that this is true. I haven't checked in this room. But I claim that this is true. So let's check. How many people are undergraduates who are not blonde? OK this is going to-- jeez. OK that's-- so, lots. OK. How many people are blonde but not from Massachusetts? OK. A smattering. Oh God, this is actually going to be terrible. AUDIENCE: [LAUGHTER] PROFESSOR: Shoot. This is a really large class. OK. Small. And how many people are undergraduates who are not from Massachusetts? Yeah, this-- oh God. This counting is going to be-- so let's-- I'm going to do this just so I can do the counting with the first two rows here. OK. My life is going to be easier this way. So how many people in the first two rows, in the center section, are undergraduates but not blonde? One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen. We could dispute some of those, but we'll take it for the moment. So, fourteen. You're probably all undergraduates. So blonde and not from Massachusetts. One. Awesome. Undergraduates not from Massachusetts. One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen. Equality. AUDIENCE: [LAUGHTER] PROFESSOR: OK. So that-- you might say well, look, you should have been nervous there. You know, and admittedly sometimes there's experimental error. But I want to convince you that I should never, ever ever be nervous about this moment in 8.04. And the reason is the following. I want to prove this for you. And the way I'm gonna prove it is slightly more general, in more generality. And I want to prove to you that the number-- if I have a set, or, sorry, if the number of people who are undergraduates and not blonde which, all right, is b bar plus the number who are blonde but not from Massachusetts is greater than or equal to the number that are undergraduates and not from Massachusetts. So how do I prove this? Well if you're an undergraduate and not blonde, you may or you may not be from Massachusetts. So this is equal to the number of undergraduates who are not blonde and are from Massachusetts plus the number of undergraduates who are not blonde and are not from Massachusetts. It could hardly be otherwise. You either are or you are not from Massachusetts. Not the sort of thing that you normally see in physics. So this is the number of people who are blonde and not from Massachusetts, number of people who are blonde, who are-- so if you're blonde and not from Massachusetts, you may or may not be an undergraduate. So this is the number of people who are undergraduates, blonde, and not from Massachusetts plus the number of people who are not undergraduates, are blonde and are not from Massachusetts. And on the right hand side-- so, adding these two together gives us plus and plus. On the right hand side, the number of people that are undergraduates and not from Massachusetts, well each one could be either blonde or not blonde. So this is equal to the number that are undergraduates, blonde, and not from Massachusetts, plus-- remember that our undergraduates not blonde and not from Massachusetts. Agreed? I am now going to use the awesome power of-- and so this is what we want to prove, and I'm going to use the awesome power of subtraction. And note that U, B, M bar, these guys cancel. And U, B bar, M bar, these guys cancel. And we're left with the following proposition: the number of undergraduates who are not blonde but are from Massachusetts plus the number of undergrad-- of non-undergraduates who are blonde but not from Massachusetts must be greater than or equal to zero. Can you have a number of people in a room satisfying some condition be less than zero? Can minus 3 of you be blonde undergraduates not from Massachusetts? Not so much. This is a strictly positive number, because it's a numerative. It's a counting problem. How many are undergraduates not blonde and from Massachusetts. Yeah? Everyone cool with that? So it could hardly have been otherwise. It had to work out like this. And here's the more general statement. The more general statement is that the number of people, or the number of elements of any set where each element in that set has binary properties a b and c-- a or not a, b or not b, c or not c. Satisfies the following inequality. The number who are a but not b plus the number who are b but not c is greater than or equal to the number who are a but not c. And this is exactly the same argument. And this inequality which is a tautology, really, is called Bell's Inequality. And it's obviously true. What did I use to derive this? Logic and integers, right? I mean, that's bedrock stuff. Here's the problem. I didn't mention this last time, but in fact electrons have a third property in addition to-- electrons have a third property in addition to hardness and color. The third property is called whimsy, and you can either be whimsical or not whimsical. And every electron, when measured, is either whimsical or not whimsical. You never have a boring electron. You never have an ambiguous electron. Always whimsical or not whimsical. So we have hardness, we have color, we have whimsy. OK. And I can perform the following experiment. From a set of electrons, I can measure the number that are hard and not black, plus the number that are black but not whimsical. And I can measure the number that are hard and not whimsical. OK? And I want to just open up the case a little bit and tell you that the hardness here really is the angular momentum of the electron along the x-axis. Color is the angular momentum of the electron along the y-axis. And whimsy is the angular momentum of the electron along the z-axis. These are things I can measure because I can measure angular momentum. So I can perform this experiment with electrons and it needn't be satisfied. In particular, we will show that the number of electrons, just to be very precise, the number of electrons in a given set, which have positive angular momentum along the x-axis and down along the y-axis, plus up along the y-axis and down along the z-axis, is less than the number that are up. Actually let me do this in a very particular way. Up... zero down at theta. Up at theta, down at-- two theta is greater than the number that are up at zero and down at theta. Now here's the thing-- two theta. You can't at the moment understand what this equation means. But if I just tell you that these are three binary properties of the electron, OK, and that it violates this inequality, there is something deeply troubling about this result. Bell's Inequality, which we proved-- trivially, using integers, using logic-- is false in quantum mechanics. And it's not just false in quantum mechanics. We will at the end of the course derive the quantum mechanical prediction for this result and show that at least to a predicted violation of Bell's Inequality. This experiment has been done, and the real world violates Bell's Inequality. Logic and integers and adding probabilities, as we have done, is misguided. And our job, which we will begin with the next lecture, is to find a better way to add probabilities than classically. And that will be quantum mechanics See you on Tuesday. AUDIENCE: [APPLAUSE]
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_16_Eigenstates_of_the_Angular_Momentum_Part_2.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right. Welcome, everyone. Hi. So today we're going to pick up where we left off last time in our study of angular momentum and rotations in quantum mechanics. Before I get started, let me open up for questions, pragmatic and physics related. Yeah? AUDIENCE: When we were solving for the 3D harmonic oscillator we solved for the energy eigenfunction that was a product of phi x, phi y, and phi z. We made an assumption that phi e was equal to phi sub n of x plus that sum. How did you get from [INAUDIBLE]?? PROFESSOR: Good. So what we had was that we had that the energy-- we wanted to find the energy of the 3D harmonic oscillator. And we wanted to find the energy eigenfunctions and eigenvalues. And they way we did this was by saying, look, the energy of the 3D harmonic oscillator, which I can think of as a function of x and px and y and py and z and pz, has this nice form. We could write it as the energy operator purely in terms of x, p squared x upon 2m plus m omega squared upon 2x squared. Plus-- so this is a single 1D harmonic oscillator energy operator in the x direction. Plus E 1D in the y direction, plus a harmonic oscillator energy 1D in the z direction. So that was the first observation. And then we said that given that this splits in this fashion, I'm going to write my energy eigenfunction, phi of x, y, and z in separated form as a product. Phi x of x, phi y of y, and phi z of z. And we used this and deduced that in order for this to be an eigenfunction of the 3D harmonic oscillator it must be true that 5x of x was itself an eigenfunction of the x harmonic oscillator equals with some energy epsilon, which I will call epsilon sub x, phi sub x. And ditto for y and z. And if this is true, if phi sub x is an eigenfunction of the 1D harmonic oscillator, then this is an eigenfunction of the 3D harmonic oscillator. But we know what the eigenfunctions are of the harmonic oscillator. The eigenfunctions of the 1D harmonic oscillator we gave a name. We call them pi sub n. So what we then said was, look, if this is an eigenfunction of the 1D harmonic oscillator in the x direction, then it's labeled by an n. And if this one is in the y direction, it's labeled by an l. And if this one is in the z direction it's labeled by a z. But on top of that we know more. We know what these energy eigenvalues are. The energy eigenvalues corresponding to these guys are if this is phi n, this is En. I can simply write this as En for the 1D harmonic oscillator. And taking this, using the fact that they're 1D eigenfunctions and plugging it into the energy eigenvalue equation for the 3D harmonic oscillator tells us that the energy eigenvalues for the 3D case are of the form E1d x plus E1d in the y plus E1d in the z, which is equal to, since these were the same frequency, h bar omega times n plus l plus m, from each of them a 1/2, so plus 3/2. Cool? That answer your question? Excellent. Other questions? Yeah? AUDIENCE: Energy and angular momentum have to be related somehow, right? PROFESSOR: Yeah. AUDIENCE: I mean, of course. Because it's both. PROFESSOR: Indeed. AUDIENCE: [INAUDIBLE]. The thing is, we have a ladder. Is there limits on them? PROFESSOR: Very good question. So this is where we're going to pick up. Let me rephrase this. This is really two questions. Question number one. Look, there should be a relationship between angular momentum and energy. But we're just talking about angular momentum. Why? Second question, look, we've got a ladder. But is the ladder infinite? So let me come back to the second question. That's going to be the beginning of the lecture. On the first question, yes angular momentum is going to play a role when we calculate the energy. But two quick things to note. First off, consider a system which is spherically symmetric, rotationally invariant. That means that the energy doesn't depend on a rotation. If I rotate the system I haven't changed the energy. So if the system is rotationally invariant, that's going to imply some constraints on the energy eigenvalues and how they depend on the angular momentum, as we discussed last time. Let me say that slightly differently. When we talk about the free particle, 1D free particle-- we've talked about this one to death. Take the 1D free particle. We can write the energy eigenfunctions as momentum eigenfunctions, because the momentum commutes with the energy. And so the way the eigenfunctions of the energy operate are indeed e to iKX there are plane waves, they're eigenfunctions of the momentum operator as well. Similarly, when we talk about a 3D system it's going to be useful in talking about the energy eigenvalues to know a basis of eigenfunctions of the angular momentum operator. Knowing the angular momentum operator is going to allow us to write energy eigenfunctions in a natural way and a simply way, in the same way that knowing the momentum operator allowed us to write energy eigenfunctions in a simple way in the 1D case. That make sense? We're going to have a glorified version of the Fourier theorum where instead of something over e to iKX, we're going to have something over angular momentum eigenstates. And those are called the spherical harmonics. And they are the analog of Fourier expansion for this year. But you're right. We're going to have to understand how that interacts with the energy. And that'll be the topic of the next lecture. We're going to finish up angular momentum today. Other questions? OK, so from last time, these are the commutation relations which we partially derived in lecture and which you will be driving on your problem set. It's a really good exercise. Commit these to memory. They're your friends. Key thing here to keep in mind. h bar has units of angular momentum, so this makes sense. Angular momentum, angular momentum, angular momentum. So when you see an h bar in this setting, its job, in some sense, is to make everything dimensionally sensible. So the important things here are that lx and ly do not commute. They commute to lz. Can you have a state with definite angular momentum in the x direction and definite angular momentum in the y direction simultaneously? No, because of this commutation relation. It would have to vanish. This is, say, the x component of the angular momentum. Can you have a state with definite angular momentum in the x direction and total angular momentum all squared? Yes. OK, great. That's going to be important for us. So in order to construct the eigenfunctions it turns out to be useful to construct these so-called raising and lowering operators, which are Lx plus iLy. They have a couple of nice properties. The first is, since these are built up out of Lx and Ly, both of which commute with L squared, the L plus minuses commute with L squared. So these guys commute. So if we have an eigenfunction of L squared, acting with L plus does not change its eigenvalue. Similarly, L plus commuting with Lz gives us h bar L plus or minus, with a plus or minus out front. This is just like the raising and lowering operators for the harmonic oscillator. But instead of the energy we have the angular momentum. So this is going to tell us that the angular momentum eigenvalues, the eigenvalues of Lz, are shifted by plus or minus h bar when we raise or lower with L plus or L minus, just like the energy was shifted-- for the 1d harmonic oscillator the energy was shifted, plus h bar omega a dagger, was shifted by h bar omega when we acted with a plus on an energy eigenstate. Same thing. Questions on the commutators before we get going? In some sense, we're going to just to just take advantage of these commutation relations and explore their consequences today. So our goal is going to be to build the eigenfunctions and eigenvalues of the angular momentum operators, and in particular of the most angular momentum operators we [INAUDIBLE] complete set of commuting observables, L squared and Lz. You might complain, look, why Lz? Whoops. I don't mean a commutator. I mean the set. You might say, why Lz? Why not Lx? And if you call this the z direction then I will simply choose a new basis where this is called the x direction. So it makes no difference whatsoever. It's just a name. The reason we're going to choose Lz is because that coordinate system plays nicely with spherical coordinates, just the conventional choice of spherical coordinates where theta equals 0 is the up axis. But there's nothing deep about that. We could have taken any of these. OK. So this is our goal. So let's get started. So first, because of these commutation relations and in particular this one, we know that we can find common eigenfunctions of L squared and Lz. Let us call those common eigenfunctions by a name, Y sub lm such that-- so let these guys be the common eigenfunctions of L squared and Lz. I.e., L squared Ylm is equal to-- well first off, units. This has units of angular momentum squared, h bar squared. So that got rid of the units. And we want our eigenvalue, lm. And because I know the answer, I'm going to give-- instead of calling this a random dimensionless number, which would be the eigenvalue, I'm going to call it a very specific thing, l, l plus 1. This is a slightly grotesque thing to do, but it will make the algebra much easier. So similarly, so that's what the little l is. Little l is labeling the eigenvalue of L squared and the actual value of that eigenvalue. I'm just calling h bar squared ll plus 1. That doesn't tell you anything interesting. This was already a real positive number. So this could have been any real positive number as well, by tuning L. Similarly, Lz Ylm-- I want this to be an eigenfunction-- this has units of angular momentum. So I'll put an h bar. Now we have a dimensionless coefficient Ylm. And I'll simply call that m. So if you will, these are the definitions of the symbols m and little l. And Ylm are just the names I'm giving to the angular momentum eigenfunctions. Cool? So I haven't actually done anything. I've just told you that these are the eigenfunctions. What we want to know is what properties do they have, and what are the actual allowed values of the eigenvalues. So the two key things to note are first that L plus and minus leave the eigenvalue of L squared alone. So they leave l alone. So this is the statement that L squared on L plus Ylm is equal to L plus on L squared Ylm is equal to h bar squared ll plus 1, the eigenvalue of L squared acting on Ylm, L plus Ylm. And so L plus Ylm is just as much an eigenfunction of L squared as Ylm was itself, with the same eigenvalue. Two. So that came from-- where are we--- this commutation relation. OK, so similarly, L plus minus raise or lower m by one. And the way to see that is to do exactly the same computation, Lz on L plus, for example, Ylm is equal to L plus-- now we can write the commutator-- Lz, L plus, plus L plus Lz Ylm. But the commutator of Lz with L plus we already have, is plus h bar L plus. So this is equal to h bar L plus. And from this term, L plus Lz of Ylm, Lz acting on Ylm gives us h bar m, plus h bar m L plus Ylm is equal to, pulling this out, this is h bar times m plus 1 times L plus. h bar m plus 1 L plus Ylm. So L plus has raised the eigenvalue m by one. This state, what we get by acting on Ylm with the raising operator is a thing with m greater by one. And that came from the commutation relation. And if we had done the same thing with minus, if you go through the minus signs, it just gives us this. What does that tell us? What this tells us is we get a ladder of states. Let's look at what they look like. Each ladder, for a given value of L, if you raise with L plus and lower with L minus, you don't change L but you do change m. So we get ladders that are labeled by L. So for example, if I have some value L1, this is going to give me some state labeled by m. Let me put the m to the side. I can raise it to get m plus 1 by L plus. And I can lower it to get m minus 1 by L minus. So I got a tower. And if we have another value, a different value of L, I'll call it L2, we got another tower. You get m, m plus 1, m plus 2, dot, dot, dot, minus 1, dot, dot, dot. So we have separated towers with different values of L squared. And within each tower we can raise and lower by L plus, skipping by one. OK, questions? So that was basically the end of the last lecture, said slightly differently. Now here's the question. So this is the question that a student asked right at the beginning. Is this tower infinite? Or does it end? So I pose to you the question. Is the tower infinite, or does it end? And why? AUDIENCE: [INAUDIBLE] direction. PROFESSOR: OK, so it's tempting to say it's infinite in one direction, because--? AUDIENCE: There are no bounds to the angular momentum. PROFESSOR: OK, so it's because there are no bounds to the angular momentum one can have. That's tempting. AUDIENCE: But at the same time, when you act a raising or lowering operator on L, the eigenvalue of L squared remains the same. So then you can't raise the z [INAUDIBLE] of the angular momentum above the actual momentum. PROFESSOR: Thank you. Exactly. So here's the statement. Let me restate that. That's exactly right. So look, L squared is the eigenvalue of the total angular momentum. Roughly speaking, it's giving you precisely in the state Ylm it tells you the expected value of the total angular momentum, L squared. That's some number. Now, if you act with L plus you keep increasing the expected value of Lz. But if you keep increasing it and keep increasing it and keep increasing it, Lz will eventually get much larger than the square root of L squared. That probably isn't true. That sounds wrong. That was the statement. Excellent. Exactly right. So let's make that precise. So is the tower infinite? No. It's probably not, for precisely that reason. So let's make that precise. So here's the way we're going to do it. This is a useful trick in general. This will outlive angular momentum and be a useful trick throughout quantum mechanics for you. I used it in a paper once. So here's the nice observation. Suppose it's true that the tower ends. Just like for the raising and lowering operators for the harmonic oscillator in one dimension, that tells us that in order for the power to end that state must be 0 once we raise it. The last state, so Yl, and I'll call this m plus, must be 0. There must be a max. Oh, sorry. Let me actually, before I walk through exactly this statement-- So let me make, first, this, the no, slightly more obvious and precise. So let's turn that argument into a precise statement. L squared is equal to, just from the definition, Lx squared plus Ly squared plus Lz squared. Now let's take the expectation value in this state Ylm of both sides of this equation. So on the left-hand side we get h bar squared ll plus 1. And on the right-hand side we get the expectation value of Lx squared plus the expectation value of Ly squared plus the expectation value of Lz squared. But we know that the expectation value of Lz squared is h bar squared, m squared. But the expectation of Lx squared and Ly squared are strictly positive. Because this can be written as a sum over all possible eigenvalues of Lx squared, which is the square of the possible eigenvalues of Lx times the probability distribution. That's a sum of positive, strictly positive [INAUDIBLE].. These are positive [INAUDIBLE]. So ll plus 1 is equal to positive plus positive plus h bar squared m squared. In particular that tells you that it's greater than or equal to h bar squared m squared. Maybe these are 0. So the least it can be is-- so the most m squared can possibly be is the square root of L plus 1. Or the most m can be. So m is bounded. There must be a maximum m, and there must be a minimum m. Because this is squared. The sign doesn't matter. Everyone cool with that? OK, so let's turn this into, now, a precise argument. What are the values of m plus and m minus? What is the top of the value of each tower? Probably it's going to depend on total L, right? So it's going to depend on each value of L. Let's check that. So here's the nice trick. Suppose we really do have a maximum m plus. That means that if I try to raise the state Ylm plus, I should get the state 0, which ends the tower. So suppose this is true. In that case, in particular here's the nice trick, L plus Ylm, if we take its magnitude, if we take the magnitude of this state, the state is 0. It's the zero function. So what's its magnitude? Zero. You might not think that's all that impressive an observation. OK, but note what this is. We know how to work with this. And in particular this is equal to-- I should have done this, dot, dot, dot-- I'm now going to use the Hermitian adjoint and pull this over to the right-hand side. The adjoint of L plus is L minus. So this gives us Ylm, L minus, L plus Ylm. This also doesn't look like much of an improvement, until you notice from the definition of L plus and L minus that what's L minus L plus? Well, L minus L plus, we're going to get an Lx squared. So Ylm, we get an Lx squared plus an Ly squared. I'm going to get my sign right, if I'm not careful. Plus-- well, let's just do it. So we have L minus L plus. So we're going to get a Lx iLy minus i, iLyLx. So plus i commutator of Lx with Ly. Good. So that's progress, Ylm. But it's still not progress, because the natural operators with which to act on the Ylm's are L squared and Lz. So can we put this in the form L squared and Lz? Sure. This is L squared minus Lz squared. Ylm L squared minus Lz squared. And this is i h bar Lz times i is minus h bar Lz. So minus h bar Lz, Ylm. And this must be equal to 0. But this is equal to-- from L squared we get h bar squared, ll plus 1. From the Lz squared we get minus m squared h bar squared. And from here we get minus h bar, h bar m, so minus m each bar squared. Notice that the units worked out. And all of this was multiplying Ylm, Ylm. But Ylm, Ylm, if it's a properly normalized eigenstate, which is what we were assuming at the beginning, is just 1. So we get this times 1 is 0. Aha. And notice that in all of this, this was m plus. We were assuming, we were working here with the assumption that L plus annihilated this top state. And so grouping this together, this says therefore h bar squared times-- pulling out a common h bar squared from of all this-- times l, l plus 1 minus m plus, m plus, plus 1 is equal to 0. And this tells us that m plus is equal to l. And if you want to be strict, put a plus sign. Everyone cool with that? This is a very useful trick. If you know something is 0 as function, you know its norm is 0. And now you can use things like Hermitian adjoints. Very, very useful. OK questions about that? Yeah. AUDIENCE: [INAUDIBLE] PROFESSOR: Excellent. Where this came from is I was just literally taking L plus and L minus, taking the definitions, and plugging them in. So if I have L minus L plus, L minus is, just to write it out explicitly, L minus is equal to Lx. Because the [INAUDIBLE] are observables, and so they're Hermitians, so they're self-adjoint, the minus iLy. So L minus L plus gives me an LxLx. It gives me an LxiLy. It gives you a minus iLyLx. And it gives me a minus ii, which is plus 1 Ly squared. Cool? Excellent. I'm in a state of serious desperation here. Good. Other questions? Yeah. AUDIENCE: [INAUDIBLE] How do you get from that to that? PROFESSOR: This to here? OK, good. Sorry, I jumped a step here. So here what I said is, look, I've got this nice set of operators acting on Ylm. I know how each of these operators acts on Ylm. Lz gives me an h bar m, minus some h bar squared. Lz squared, L squared. And then overall that was a just some number times Ylm, which I can pull out of the inner product. Cool? Yeah? AUDIENCE: [INAUDIBLE] PROFESSOR: Good. This subscript plus meant that, look, there was a maximum value of m. So m squared had to be less than or equal to this. There's a maximum value and a minimum value. Maybe they're different. I don't know. I'm just going to be open-minded about that. Others? Yeah? AUDIENCE: Will that mean, then, that you can't ever have all the angular momentum [INAUDIBLE]?? PROFESSOR: Yeah, awesome observation. Exactly. We were going to get there in a little bit. I like that. That's exactly right. Let me go through a couple more steps, and then I'll come back to that observation. And I promise I will say so. So if we go through exactly a similar argument, let's see what happens if we did L minus on Ylm minus, just to walk through the logic. So if this were-- you lower the lowest one, you get 0. Then we'd get the same story, L minus L minus. When we take the adjoint we'd get L plus, L minus. And L plus L minus, what changes? The only thing that's going to change is that you get this commutator the other direction. And there are minus signs in various places. The upshot of which is that m minus is equal to minus l. So this is quite parsimonious. It's symmetric. If you take z to minus z, if you switch the sign of the angular momentum you get the same thing back. That's satisfying, perhaps. But it's way more than that. This tells us a lot about the possible eigenvalues, in the following way. Look back at our towers. In our towers we have that the angular momentum is raised and lowered by L plus. So the Lz angular momentum is raised and lowered by L plus. l remains the same. But there's a maximum state now, which is plus l. And there's a minimum stay in here, l1. And here there's a minimum state, minus l1. Similarly for the other tower, there's a minimum state minus l2 and a maximum state, l2. So how big each tower is depends on the total angular momentum. That kind of makes sense, right? If you've got more angular momentum and you can only step Lz by one, you've got more room to move with a large value of l than with a small value of l. OK. So what does that tell us about the values of m? Notice that m spans the values from its minimal value, minus l to l in integer steps, in unit steps. So if you think about m is l, then m is l minus 1. And if I keep lowering I get down to m is minus l plus 1. And then m is equal to minus l. So the difference in the Lz eigenvalue between these guys, the difference is 2l. But the number of unit steps in here is one fewer, because one, dot, dot, dot 2l minus 1. So this is some integer, which is the number of states minus 1. So if these are n states, and I'll call this N sub l states, then this difference twice l is N sub l minus 1, because it's unit steps. Cool? So for example, if there were two states, so N is 2, 2l is equal to-- well, that's 1, which is 2 minus 1, the number of states minus 1, also known as 1. So l is 1/2. And more generally we find that l must be, in order for this process to make sense, in order for the m plus and the m minus to match up, we need that l is of the form an integer N sub l minus 1 upon 2. So this tells us that where Nl is an integer-- Nl is just the number of states in this tower, and it's a strictly positive integer. If you have zero states in the tower, then that's not very interesting. So this tells us that l is an integer a half integer, but nothing else. Cool? In particular, if it's a half integer, that means 1/2, 3/2, 5/2, if it's an integer or it can be 0. There's nothing wrong with it being 0. But then 1 on. So that means we can plot our system in the following way. If we have l equals 0, how many states do we have? If little l is 0, how many states are? What's the largest value of Lz? What's the largest allowed value of Lz or of m if little l is equal to 0? AUDIENCE: 0. PROFESSOR: 0. Because it goes from plus 0 to minus 0. So that's pretty much 0. So we have a single state with m equals 0 and l equals 0. If l is equal to-- what's the next possible value?-- 1/2, then m can be either 1/2 or we lower it by 1, which is minus 1/2. So there are two states, m equals 1/2 and m equals minus 1/2. So there's one power. It's a very short tower. This is the shortest possible tower. Then we also have the state l equals 1, which has m equals 0. It has m equals 1. And it has m equals minus 1. And that's it. And so on and so forth. Four states for l equals 3/2, with states 3/2, 1/2, this is m equals 1/2, m equals minus 1/2, and minus 3/2. The values of m span from minus l to l with integer steps. And this is possible for all values of l which are of the form all integer or half integer l's. So for every different value of the total angular momentum, the total amount of angular momentum you have, you have a tower of states labeled by Lz's eigenvalue in this fashion. Questions? Does anyone notice anything troubling about these? Something physically a little discomfiting about any of these? What does it mean to say I'm in the state l equals 0, m equals 0? What is that telling you? Zero angular momentum. What's the expectation value of l squared? Zero. The expectation value of Lz? And by rotational invariance, the expected value of Lx or Ly? That thing is not rotating. There's no angular momentum. So the no angular momentum state is one with l equals 0, m equals 0. So this is not spinning. What about this guy? L equals 1, m equals 0. Does this thing carry angular momentum? Yeah, absolutely. So it just doesn't carry any angular momentum in the z direction. But its total angular momentum on average, its expected total angular momentum is ll plus 1, which is 2, times h bar squared. So if you measure Lx and Ly what do you expect to get? Well, if you measure Lx squared or Ly squared you probably expect to get something non-zero. We'll come back to that in just a second. And what about the state l equals 1, m equals 1? Your angular momentum-- you've got as much angular momentum in the z direction as you possibly can. So that definitely carries angular momentum. So there's a state that has no angular momentum in the z direction. And there's a state that has some, and there's a state that has less. That make sense. Yeah? AUDIENCE: If m equals 1 and l equals 1, that means that the angular momentum for the z direction is the total angular momentum? PROFESSOR: Is it? AUDIENCE: And then Lx and Ly was 0? PROFESSOR: OK, that's an excellent question. Let me answer that now, and then I'll come back to the point I wanted to make. Hold on a second. Let me just answer that question. I'll work here. So here's the crucial thing. Even in this state, so you were asking about the state l equals 1, m equals 1. And the question that was asked, a very good question, is look, does that mean all the angular momentum is in the z direction, and Lx and Ly are 0? But let me just ask this more broadly. Suppose we have a state with angular momentum l, and m is equal to l. The most angular momentum you can possibly have in the z direction, same question. Is all the angular momentum in the z direction? And what I want to emphasize you is no, that's absolutely not the case. So two arguments for that. The first is, suppose it's true that Lx and Ly are identically 0. Can that satisfy the uncertainty relation due to those commutators? No. There must be uncertainty in Lx and Ly, because Lz has a non-zero expectation value. So it can't be that Lx squared and Ly squared have zero expectation value. But let's be more precise about this. The expectation value of L squared is easy to calculate. It's h bar squared l, plus 1. Because we're in the state Ylm. This is the state Y l sub l, or ll. The expectation of Lz squared is equal to h bar squared, m squared. And m squared now, m is equal to l. So h bar squared, l squared. Aha. So the expected value of l square is not the same as Lx squared, but this is equal to the expected value of Lx squared plus Ly squared plus Lz squared. Therefore the expected value of Lx squared plus the expected value of Ly squared is equal to the difference between this and this. We just subtract this off h bar squared ll plus 1 minus h bar squared, l squared, h bar squared l. And by symmetry you don't expect the symmetry to be broken between Lx and Ly. You can actually do the calculation and not just be glib about it. But both arguments give you the correct answer. The expectation value of Lx squared is equal to 1/2 h bar squared l. And ditto for y, in this state. So notice two things. First off is we make the total angular momentum little l large. The amount by which we fail to have all the angular momentum in the z direction is getting larger and larger. We're increasing the crappiness of putting all the angular momentum in the z direction. However, as a ratio of the total angular momentum divided by L squared, so this divided by L squared, and this is h bar squared, l, l plus 1, and in particular this is l squared plus l, so if we took the ratio, the rational mismatch is getting smaller and smaller. And that's good. Because as we go to very large angular momentums where things should start getting classical in some sense, we should get back the familiar intuition that you can put all the angular momentum in the z direction. Yeah? AUDIENCE: Why are we imposing the [INAUDIBLE] condition that the expectation values of Lx and Ly should be identical? PROFESSOR: Excellent. That's why I was saying this is a glib argument. You don't have to impose-- that needs to be done a little bit more delicately. But we can just directly compute this. And you do so on your problem set. Yeah? AUDIENCE: Why doesn't the existence of the l equals 0 eigenfunction where the angular momentum the L squared is definitely 0 violate the uncertainty principle? PROFESSOR: Awesome. On your problem set you're going to answer that, but let me give you a quick preview. This is such a great response. Just Let me give you a quick preview. So from that Lx-- OK, this is a really fun question. Let me go into it in some detail. Wow, I'm going fast today. Am I going way too fast? No? A little? Little too fast? OK, ask me more questions to slow me down. I'm excited. I didn't get much sleep last night. One of the great joys of being a physicist is working with other physicists. So yesterday one of my very good friends and a collaborator I really delight in talking with came to visit. And we had a late night dinner. And this led to me late at night, not doing my work, but reading papers about what our conversation was about. And only then at the very end, when I was just about to die did I write the response to the reviewer on the paper that I was supposed to be doing by last night's deadline. So I'm kind of tired, but I'm in a really good mood. It's a really good job. Now I've totally lost my train of thought. What was the question again? AUDIENCE: The existence of the l equals 0 state where the total angular momentum is definitely-- PROFESSOR: Excellent, and the uncertainty. So the question is, why don't we violate the uncertainty when we know that L equals-- where am I; I just covered it-- when we know that L equals 0 and m equals 0, doesn't that destroy our uncertainty? Because we know that the angular momentum Lz is 0. Angular momentum for Lx is 0. Angular momentum for Ly is 0. All of them vanish. Doesn't that violate the uncertainty principle? So let's remind ourselves what the form of that uncertainty relation is. The form of the uncertainty relation following from Lx Ly is i h bar Lz. Recall the general statement. The uncertainty in A times the uncertainty in B, squared, squared, is equal to-- let me just write it as h bar upon 2-- sorry, 1/2. The absolute value of the expectation value of the commutator, A with B. Good lord. Dimensionally, does this work? Yes. OK, good. Because units of A, units of B. Unites of A, units of B. Triumph. And these are going to be quantum-mechanically small, because commutators have h bars. And commutators have h bars why? Not because God hates us. Why do commutators have h bars? What happens classically? In classical mechanics, do things commute? Yes. Why are the h bars and commutators physical observables? Because we exist. Because there's a classical limit. OK so this is going to make quantum-mechanically small, so we expect the uncertainty relation to also be quantum-mechanically small. Just important intuition. So let's look at the specific example of Lx, Ly, and Lz. So the uncertainty in Lx, in any particular-- remember that this is defined as the uncertainty in a particular state psi, in a particular state psi. And this expectation value is taken in a particular state psi, that same state. So the uncertainty of Lx, in some state Ylm, times the uncertainty of Ly in that same state Ylm should be greater than or equal to 1/2 the absolute value of the expectation value of the commutator of Lx and Ly. But the commutator of Lx and Ly is i h bar Lz. And i, when we pull it through this absolute value, is going to give me just 1. h bar is going to give me h bar. So h bar upon 2, expectation value of Lz, absolute value. Yeah? Can Lx and Ly have zero uncertainty? When? Expectation value of Lz is 0. So that sounds good. It sounds like if Lz has expectation value of 0, then we can have Lx and Ly, definite. But that's bad. Really? Really, can we do that? Why not? AUDIENCE: [INAUDIBLE] [LAUGHTER] PROFESSOR: Bless you, my son. Can we have Lx and Ly both take definite values, just because Lz? Why? What else do we have to satisfy? What other uncertainty relations must we satisfy? There are two more. And I invite you to go look at what those two more are and deduce that this is only possible if Lx squared, Ly squared, and Lz squared all have zero expectation value. In fact, I think it's just a great question that I think it's on your problem set. So thank you for that question. It's a really good question. There was another question in here. Yeah? AUDIENCE: Something that you said earlier [INAUDIBLE] half integer. So were you deriving this by counting the number of equations and somehow asserting-- Why does--? PROFESSOR: Great. So the question is, wait, really? Why is L and integer a half integer. That was a little too quick. Is that roughly the right statement? AUDIENCE: [INAUDIBLE]. PROFESSOR: Good. Excellent. Let me go through the logic. So the logic goes like this. I know that the Ylm's-- if the Ylm's are eigenfunctions of L squared and Lz, and I've constructed this tower of them using the raising and lowering operators, we've already shown that the largest possible value of m is l and the least possible value is minus l. And these states must be separated by integer steps in m. OK, good. So pick a value of l, a particular tower. And let the number of states in that tower be N sub l. So there's N sub l of them. Great. And what's the distance between these guys? We haven't assumed l is an integer yet. We haven't assumed that. So this N sub l is an integer. Because it's the number of states. And the number of states can't be a pi. Now let's count that-- that so how many states are there? There are Nl. But if I count one, two, three, four, the total angular momentum down here is 2l. I had some pithy way of giving this a fancy name. But I can't remember what it was. So the length of the tower in units of h bar, the height of this tower is 2l. But the number of steps I took in here was Nl minus 1. And that number of steps is times 1. So we get that 2l is N minus 1. There's nothing fancy here. I'm just saying if L is 0 we go from here to here. There's just one element. So number states is 1, L is 0. So it's that same logic just repeated for every value of L. Other questions? Coming back to this, something on this board should cause you some serious physical discomfort. We've talked about the l equals 0 m equals 0 state. This is a state which has no angular momentum whatsoever, in any direction at all. We've talked about the l equals 1 m equals 0 state. It carries no angular momentum in the z direction, but it presumably has non-zero expectation value for L squared x and a Ly squared. These guys are also fine. What about these guys? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yes! That's disconcerting. Do I have to have angular momentum in the z direction? AUDIENCE: [INAUDIBLE] [LAUGHTER] PROFESSOR: That should go on a shirt somewhere. Let me ask the question more precisely, or in a way that's a little less threatening to me. Do you have to have angular momentum in the z direction? I'm sorry, what's your name? Does David need to have angular momentum in the z direction? AUDIENCE: Yes. PROFESSOR: Does this chalk need to have angular dimension in the-- well, the chalk's-- OK, classically no. It can have some total angular momentum, which is 0, and it can be rotating not in the z direction. It can be rotating in the zx plane. If it's rotating in the zx plane, it's got total angular momentum L squared as non-zero. But its angular momentum in the z direction is 0. Its axis is exactly along the y direction. And so it's got no angular moment-- that's perfectly possible classically. And that's perfectly possible when L is an integer. Similarly when L is 2. This is the particular tower that I love the most. 2, 1, 0, minus 1, minus 2. The reason I love this the most is that it's related to gravity, which is pretty awesome. That's a whole other story. I really shouldn't have said that. That's only going to confuse you. So for any integer, it's possible to have no angular momentum in the z direction. That means it's possible to rotate around the x-axis or the y-axis, orthogonal to the z-axis. That make sense? But for these half integer guys, you are inescapably spinning. There is no such thing as a state of this guy that carries no angular momentum. Anything well described by these quantum states is perpetually rotating or spinning. It carries angular momentum, we say precisely. Perpetually carries angular momentum in the z direction. Any time you measure it, it carries an angular momentum, either plus a half integer or minus a half integer. But never, ever zero. AUDIENCE: But in the classical limit where you have very large L, the m equals one half state, or the m equals minus a half state is going to get arbitrarily small compared to the angular momentum. So isn't it just like where we said, well, OK, you can never have your angular momentum only in the z direction, but we don't care? Because in the classical limit it gets arbitrarily close to there. PROFESSOR: See, one of the nice things about writing lectures like this is that you get to leave little landlines. So this is exactly one of those. Thank you for asking this question. Let me rephrase that question. Look, we all took high school chemistry. We all know about spin. The nuclei have spin. They have some angular momentum. But if you build up a lot of them, you build up a piece of chalk, look, as we said before, while it's true that there's some mismatch in the angular momentum in the z direction for some large L state, it's not only angular momentum. Some is in Lz, Lx, and Ly as well. It's preposterously small for a macroscopic object where L is macroscopic. It's the angular momentum in Planck units, in units of the Planck constant. 10 to the 26th-- something huge. Why would we even notice? But here's the real statement. The statement isn't just that this is true of macroscopic objects. But imagine you take a small particle. Imagine you take a single atom. We're deep in the quantum mechanical regime. We're not in the classical regime. We take that single atom. And if it carries angular momentum and it's described by L equals 1/2 state, that atom will never, ever, ever be measured to have its angular momentum in the z direction, or indeed any direction, be 0. You will never, in any direction, measure its angular momentum to be 0. That atom perpetually carries angular momentum. And that is weird. OK, maybe you don't find it weird. This is good. You've grown up in an era when that's not a weird thing. But I find this deeply disconcerting. And you might say, look, we never actually measure an atom. But we do. We do all the time. Because we measure things like the spectre of light, as we'll study when we study atoms in a week, a couple weeks because of the exam-- sorry guys, there has to be an exam-- as we will find when we study atoms in more detail, or indeed, at all, we'll be sensitive to the angular momentum of the constituents of the atom. And we'll see different spectra. So it's an observable property when you shine light on gases. This is something we can really observe. So what this suggests is one of two things. Either these are just crazy and ridiculous and we should ignore them, or there's something interesting and intrinsically quantum-mechanical about them that's not so familiar. And the answer is going to turn out to be the second, the latter of those. The ladder? That was not intentional. Maybe it was subconscious. OK, so I want to think about some more consequences of the structure of the Ylm's and the eigenvalues, in particular of this tower structure. I want to understand some more. What other physics can we extract from this story? First, a very useful thing is just to get a picture of these guys in your head. Let's draw the angular momentum eigenfunctions. So what does that mean? Well first, when we talked about the eigenfunctions of momentum, linear momentum in one dimension, we immediately went to the wave function. We talked about how the amplitude to beat a particular spot varied in space. And the amplitude was just e to the iKX. So the amplitude was an oscillating-- the phase rotated. And the absolute value of the probability density was completely constant. Everybody cool with that? That was the 1D plane wave. So the variables there, we had an angular momentum eigenstate. And that's a function of the position. Or, sorry, a linear momentum eigenstate. And that's a function of the position. Angular momentum eigenstates are going to be functions of angular position. So I want to know what these wave functions look like, not just the eigenvalues. But I want to know what is the wave function associated to eigenvalues little l and little m look like, y sub lm of the angles theta and phi. What do these guys look like? How do we get them? This is going to be your goal for the next two minutes. So the first thing to notice is that we know what the form of the eigenvalues and eigenfunctions are. If we act with Lz we get h bar m back. If we act with L squared we get h bar squared ll plus 1 back. But we also have other expressions for Lz and L squared. In particular-- I wrote them down last time in spherical coordinates. So I'm going to working in the spherical coordinates where the declination from the vertical is an angle theta, and the angle around the equator is an angle phi. And theta equals 0 is going to be up in the z direction. It's just a choice of coordinates. There's nothing deep. There's nothing even shallow. It's just definitions. So we're going to work in the spherical coordinates. And in spherical coordinates we observe that this angular momentum, just following the definition from r cross p, takes a particularly simple form. That's a typo. h bar upon i, dd phi. And instead of writing L squared I'm going to write L plus minus, because it's shorter and also because it's going to turn out to be more useful. So this takes the form h bar, e to the plus minus i phi, d theta, plus or minus cotangent of theta, d phi. So these are the expressions for Lz and L plus minus in spherical coordinates, in these spherical coordinates. I want to know how Ylm depends on theta and phi. And it's clearly going to be easier to ask about the Lz eigenequation. So let's look at that. Lz on Ylm gives me h-- Oh yeah? AUDIENCE: Is it h or h bar? PROFESSOR: Oh, Jesus. When I write letters by hand, which is basically when I write to my mom, all my h's are crossed. I can't help it. So this is like the inverse. It's been a long time since I made that mistake. It's usually the other. So Lz acting on Ylm gives me h bar m, acting on Ylm. But that's what I get when I act with Lz, so let's just write out the differential equation . So h bar m Ylm where m is an integer, or a half integer, depending on whether l is an integer or half integer. h bar m Ylm is equal to h bar upin i, d phi of Ylm. Using the awesome power of division and multiplication I will divide both sides by h bar. And I will multiply both sides by i. And we now have the equation for the eigenfunctions of Lz, which we actually worked on last time. And we can solve this very simply. This says that, remember, Ylm is a function of theta and phi. Here we're only looking at the phi dependence, because that's all that showed up in this equation. So this tells us that the eigenfunctions Ylm are of the form of theta and phi, are of the form e to the im phi times some remaining dependence on theta, which I'll write as p of theta. And that p could depend on l and m. Cool? Already, before we even ask about that dependence on theta, the p dependence, we learn something pretty awesome. Look at this wave function. Whatever else we know, its dependence on phi is e to the im phi. Now, remember what phi is. Phi is the angle around the equator. So it goes from 0 to 2 pi. And when it comes back to 2 pi it's the same point. Phi equals 0 and phi equals pi are two names for the same point. Yes? But that should worry you. Because note that as a consequence of this, Ylm of theta 0 is equal to, well, whatever it is. Sorry, theta of 2 pi is equal to e to the im 2 pi times Plm to theta. But this is equal to-- oh, now we're in trouble. If m is an integer, this is equal to e to the i integer 2 pi 1. Yes, OK. You're supposed to cheer at that point. It's like the coolest identity in the world. So e to the i 2 pi, that's one. So if m is an integer then this is just Plm of theta, which is also what we get by putting in phi equals 0. So this is equal to Ylm of theta, comma, 0. But if m is a half integer then e to the i half integer times 2 pi is minus 1. And so that gives us minus Ylm of theta and 0, if m is a half integer. So let me say that again, in the same words, actually. But let me just say it again with different emphasis. What this tells us is that Ylm at 0, as function of the coordinates theta and phi, Ylm-- so let's take m as an integer. Ylm at theta and 0 is equal to Ylm at theta and 0. That's good. But if Y is a half integer, then Ylm at theta and 2 pi, which is the same point as Ylm at theta and 0, is equal to minus Ylm at theta and 0. That's less good. What must be true of Ylm, of theta and 0? 0. And was there anything special about the point 0? No. I could have just taken any point and rotated it around by pi. So this tells us that Ylm of theta and phi is identically equal to 0 if m is a half integer. Huh. That's bad. Because what's the probability density of being found at any particular angular position? 0. Can you normalize that state? No. That is not a state that describes a particle. That is a state that describes the absence of a particle. That is not what we want. So these states cannot describe these values of l and m, which seem like perfectly reasonable values of l and m, perfectly reasonable eigenvalues of L squared and lm. They cannot be used to label wave functions of physical states corresponding to wave functions on a sphere. You can't do it. Because if you try, you find that those wave functions identically vanish. OK? So these cannot be used. These do not describe wave functions of a particle in quantum mechanics. They cannot be used. Those values, those towers cannot be used to describe particles moving in three dimensions. Questions about that? This is a slightly subtle argument. Yeah? AUDIENCE: You said something earlier about how atoms would never have any zero angular momentum. And so the ones that have no zero angular momentum, we just said they're not possible. So [INAUDIBLE]? PROFESSOR: Excellent question. I said it slightly diff-- so the question is, look, earlier you were saying, yeah, yeah, yeah. There are atoms in the world and they have half integer angular momentum. And you can shine a light on them and you can tell and stuff. But you just said these can't exist. So how can those two things both be true? Thank you for this question. It's a very good question. I actually said a slightly different thing. What I said was, states where angular momentum lm are half integers cannot be described by a wave function of the coordinates. We're going to need some different description. And in particular, we're going to need a different description that does what? Well, as we take phi from 0 to 2 pi, as we rotate the system, we're going to pick up a minus sign. So in order to describe an object with lm being a half integer, we can't use the wave function. We need something that is allowed to be doubly valued. And in particular, we need something that behaves nicely. When you rotate around by 2 pi, we need to come back to a minus sign, not itself. So at some object that's not a function, it's called a spinner. So we'll talk about it later. We need some object that does that. So there's this classic demonstration at this point, which is supposed to be done in a quantum mechanics class. So at this point the lecturist is obliged to do the following thing. You say, blah, blah, blah, if things rotate by 2 pi they have to come back to themselves. And then you do this. I'm going to rotate my hand like a record by 2 pi. And it will not come back to itself. OK? And it quite uncomfortably has not come back to itself. But I can do a further rotation to show you that it's a minus sign. I can do a further the rotation by 2 pi and have it come back to itself. And I kept the axis vertical, yeah? OK, so at this point you're all supposed to go like, oh, yes, uh-huh, mmm. So now that we've got that out of the way, I have an arm. So the story is a little more complicated than that. This is actually a fair demonstration, but it's a slightly subtle story. If you want to understand it, ask your recitation instructor or come to my office hours. AUDIENCE: What about USB sticks? PROFESSOR: USB sticks? AUDIENCE: You insert them here and they don't go and insert the other way. [LAUGHTER] PROFESSOR: What about USB sticks? You insert them this way, they don't work. You insert them this way, they don't work. But if you do it again, then they do. [LAUGHTER] [APPLAUSE] PROFESSOR: OK. That's pretty good. So for the moment, as long as we want to describe our system with a wave function of position, which means we're thinking about where will we find it as a function of angle, we cannot use the half integer l or m. So if we can't use the half interger l or m, fine. We'll just throw them out for the moment and we'll use the integer l and m. And let's keep going. What we need to determine now is the P. We've determined the phi dependence, but we need the theta dependence. We can get the theta dependence in a sneaky fashion. Remember the harmonic oscillator in 1D. When we wanted to find the ground state wave function, we could either solve the energy eigenvalue equation, which is a second order differential equation and kind of horrible, or we could solve the ground state equation that said that the ground state is annihilated by the annihilation operator, which is a first order difference equation and much easier to solve. Yeah? Let's do the same thing. We have an annihilation condition. If we have the top state, L plus on Y ll is equal to 0. But L plus is a first order differential operator. This is going to be easier. So we need to find a solution to this equation. And do I want to go through this? Yeah, why not. OK, so I want this to be equal to 0. But L plus Yll is equal to-- well, it's h bar, e to the plus i phi, d theta, plus cotangent theta, d phi on Yll. And Yll is e to the i l phi times Plm of theta. Cool? So dd phi on e to the il phi, there's no phi dependence here. It's just going to give us a vector of il. Did I get the-- yeah. So that's going to give us a plus il and no dd phi. And this e to the il phi we can pull out. But this has to be equal to 0. So this says that 0 is equal to d theta plus il cotangent theta P lm of theta. And this is actually much better than it seems for the following reason. dd theta-- find the dd theta. Cotangent of theta is cosine over sine. That's what you get if you take the derivative of lots of sine functions-- sine to the l, say. Take a derivative. You lose a power of sine and you pick up a power of cosine. So multiplying by cotangent gets rid of a power of sign and gives you a power of cosine, which is a derivative of sine. So Plm, noticing the l-- and I screwed up an i somewhere. I think I wanted an i up here. Let's see. i. Sorry. Yes, I want an i cotangent. OK, that's much better. So i cotangent-- good, good. And then the i squareds give me a minus l. And this tells us that Plm, so if this is sine to the l, then d theta gives us an l sine to the l minus 1 cosine, which is what I get by taking sine to the l and multiplying by cosine, dividing by sine, and multplying by l. So this gives me Pll. This is for the particular state ll. We're looking at the top state and we're annhilating it. So Pll is equal to some coefficient, so I'll just say proportional to sine to the l of theta. So this tells us that Yll is equal to some normalization, sub ll, just some number, times from the phi dependence e to the il phi, and from the theta dependence, sine to the l of theta. So this is the form. Sorry, go ahead. AUDIENCE: What's the symbol there? PROFESSOR: Oh, this twisted horrible thing? It's proportional to. But it was a long night. So now we have the wave function explicitly as a function phi and as a function of theta completely understood for the top state in any tower. This is for any L. The top state in any tower is e to the il phi. Does that make sense? Well Lz is h bar upon id phi. So that gives us h bar as m as l. So that's good. That's the top state. And from the sine theta we just checked. We constructed that this indeed has the-- well, if you then check you will find that the L squared eigenvalue, which you'll do on the problem set, the l squared eigenvalue is h bar squared ll plus 1. AUDIENCE: Quick question. The expression we have for the L plus minus operators, how did we construct the expressions for Lx and Ly? PROFESSOR: Good. It's much easier than you think. So Lx is equal to-- L is r cross b, right? So this is going to be yPz minus zPy. And this is equal to h bar upon i, ydz minus zdy. But you know what y is in spherical coordinates. And you know what derivitive with respect to z is in spherical coordinates. Because you know what z is, and you know the chain rule. So taking this and just plugging in explicit expression for the change of variables to spherical coordinates takes care of it. Yeah? AUDIENCE: What does the superscript of the sine indicate? Is that sine to the power of l? PROFESSOR: Sorry. This is bad notation. It's not bad notation, it's just not familiar notation. It's notation that is used throughout theoretical physics. It means this, sine theta to the l. The lth power of sine. For typesetting reasons we often put the power before the argument. Yeah, no, it's a very good question. Thank you for asking, because it was unclear. I appreciate that. Other questions? AUDIENCE: [INAUDIBLE] L hat [INAUDIBLE]?? PROFESSOR: How did we come up with the L hat plus minus? That was from this. So we know what the components of the angular momentum are in Cartesian coordinates. And you know how, because it's coordinates, to change variables from Cartesian to spherical. So you just plug this in for Lx. But L plus is Lx plus ioy, and so you just take these guys in spherical form and add them together with the relative i. And that gives you that expression. AUDIENCE: Maybe I missed this, but can you just explain the distinction between Y sub ll and Y sub lm? PROFESSOR: Yeah, absolutely. So Y sub ll, it means Y sub lm where m is equal to l. AUDIENCE: Oh, OK. It was just the generic [INAUDIBLE].. PROFESSOR: It's a generic eigenfunction of the angular momentum, with the angular momentum in the z direction being equal to the angular momentum in the total angular momentum. At least for these numbers. OK? Cool. Good, so now, if we know this, how do we just as a side note-- suppose we know-- well, suppose we know this? We know this. We know what the top state in the tower looks like. How do I get the next state down in the tower? how do I get Yl l minus 1? AUDIENCE: Lower it. PROFESSOR: Lower it. Exactly. So this is easy, L minus on Yll. And we have to be careful about normalization. So again, it's proportional to. But this is easy. We don't have to solve any difference equations. We just have to take derivatives. So it's just like the raising operator for a harmonic oscillator. We can raise and lower along the tower and get the right wave function. To give you some examples-- yeah, let's do that here. Let me just quickly give you a few examples of the first few spherical harmonics. Sorry, I should give these guys a name. These functions, Ylm of theta and phi, they're called the spherical harmonics. They're called these because they solve the Laplacian equation on the sphere, which is just the eigenvalue equation. L squared on them is equal to a constant times those things back. Just to tabulate a couple of examples for you concretely, consider the l equals 0 states. What are the allowed values of m for little l equals 0? 0. So Y0,0 is the only state. And if you properly normalize it, it's 1 over root 4 pi. OK, good. what about l equals 1? Then we have Y1, 0 and we have Y1 minus 1. And we have Y1,1. So these guys take a particularly simple form. Root 3-- I'm not even going to worry about the coefficient. They're in the notes. You can look them up anywhere. So first off let's look at Y1, 1. So non-linear today. So Y1, 1, it's going to be some normalization. And what is the form? It's just e to the il phi sine theta to the l. l is 1. So this is some constant times e to the i phi, sine theta. Who 1, 0? Well, it's got angular momentum 0 in the z direction, in Lz. So that means how does it depend on phi? It doesn't. And you can easily see that, because when we lower we get an e to the i minus i phi. Anyway, so this gives us a constant times no e to the i phi, no phi dependents, and cosine of theta. And if you get a cosine of theta, the d theta and the cotangent d phi will give you the same thing. And Y1 minus 1 is again a constant, times e to the minus i phi. So it's got m equals minus 1 and sine theta again. Notice a pleasing parsimony here. The theta dependence is the same for plus m and minus m. So what about Y2, 2? Some constant e to the i 2 phi, sine squared theta, dot, dot, dot, Y0, 0. And here it's interesting. Here we just got one term from taking the derivative. They both give you cosine. But now there are two ways to act with the two derivatives. And this gives you a constant. Now what's the phi dependence? It's nothing, because it's got m equals 0. And so the only dependence is on theta. And we get a cos squared whoops, there's a 3-- 3 cos squared theta minus 1. AUDIENCE: Do you mean Y2, 0? PROFESSOR: Oh, shoot. Thank you. Yes, I mean Y2, 0. Thank you. And then if we continue lowering to Y2 minus 2, this is equal to, again, a constant. And the it's going to be the same dependence on theta, but a different dependence of phi. e to the minus 2i phi, sine squared theta. OK? Yeah. AUDIENCE: Aren't these not normalizable? PROFESSOR: Why? AUDIENCE: Oh, never mind. PROFESSOR: Good. So let me turn that into a question. The question is, are these normalizable? Yeah, so how would we normalize them? What's the check for if they're normalizable or not? AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah, we integrate them norm squared over a sphere. Not over a volume, just over a sphere. Because they're only wave functions on the sphere. We haven't dealt with the radial function. We'll deal with that later. That will come in the next lecture. Other questions? Yeah? AUDIENCE: Can you explain one more time why m equals 0 doesn't have [INAUDIBLE]? PROFESSOR: Yeah, why m equals 0 doesn't have any phi dependence? If m equals 0 had phi dependence, then we know that the eigenvalue of Llz is what we get when we take a derivative with respect to phi. But if the Lz eigenvalue is 0, that means that when we act with dd phi we get 0. That means it can't depend on phi. Cool? Other questions? What I'm going to do next is I'm going show you, walk you through some of these angular momentum eigenstates, graphically on the computer. Before we do that, any questions about the calculation so far? OK. So this mathematical package I'll post on the web page. And at the moment I think it's doing the real part. So what we're looking at now is the real part. Actually, let's look at the absolute value. Good. So here we are, looking at the absolute value of-- that's not what I wanted to do. So what we're looking at in this notation is some horrible parametric plot. You don't really need to see the mathemat-- oh, shoot. Sorry. You don't need to see the code, particularly. So I'm not going to worry about it, but it will be posted. So here we're looking at the absolute value, the norm squared of the spherical harmonic Y. And the lower eigenvalue here, lower coordinate is the l. And the upper is m. So when l is 0, what we get-- here's what this plot is indicating. The distance away from the origin in a particular angular direction is the absolute value of the wave function. So the further away from the origin the colored point you see is, the larger the absolute value. And the color here is just to indicate depth and position. It's not terribly meaningful. So here we see that we get a sphere. So the probability density or the norm squared of the wave function of the spherical harmonic is constant. So that makes sense. It's spherically symmetric. It has no angular momentum. As we start increasing the angular momentum, let's take the angular momentum l is 1, m is 0 state, now something interesting happens. The total angular momentum is 1. And we see that there are two spheres. Let me sort of rotate this. So there are two spheres, and there's the z-axis passing through them. And so the probability is much larger around this lobe on the top or the lobe on the bottom. And it's 0 on the plane. Now, that's kind of non-intuitive if you think, well, Lz is large, so why is it along the vertical? Why is that true? So here, this is the Lz equals 0 state. That means it carries no angular momentum along the z-axis. That means it's not rotating far out in the xy plane. So your probability of finding it in the xy plane is very small. Because if it was rotating in the xy plane it would carry a large angular momentum in Lz. But Lz is 0. So it can't be extended out in the xy plane. Cool? On the other hand, it can carry angular momentum in the x or the y direction. But if it carries angular momentum in the x direction for example, that means the system is rotating around the z-- sorry. If it carries angular momentum in the x direction, it's rotating in the zy plane. So there's some probability to find it out of the plane in y and z. But it can't be in the xy plane. Hence it's got to be in the lobes up above. That cool? So it's very useful to develop an intuition for this stuff if you're going to do chemistry or crystallography or any condensed manner of physics. It's just very useful. So I encourage you to play with these little applets. And I'll post this mathematics package. But let's looked at what happens now if we crank up the angular momentum. So as we crank up the l angular momentum, now we're getting this sort of lobe-y thing, which looks like some sort of '50s sci-fi apparatus. So what's going on there? This is the 2, 0 state. And the 2, 0 state has a 3 cos squared theta minus 1. But cos squared theta, that means it's got two periods as it goes from vertical to negative. And if you take that and you square it, you get exactly this. OK? So they're using the cos squared as a function of the angle of declination from vertical. And it's m equals 0, so you're saying no dependence on phi. Of course, that's a little cheap. Because the angular dependence on phi is just an overall phase. So we're not going to see it in the absolute value. Everyone agree on that? We're not going to see the absolute-- OK. So let's check that. Let's take l2 and m2. So now when l is 2 and m is 2, we just get this donut. So what that's saying is, we've got some angular momentum. l is 2. But all the angular momentum, almost all of it, anyway, is in the z direction. And is that what we're seeing here? Well, yeah. It seems like it's most likely to find the particle, the probability is greatest, out in this donut around the plane. Now, if it were Lz is equal to l, it would be flat. It would be a strictly 0 thickness pancake. But we have some uncertainty in what Lx and Ly are, which is why we got this fattened donut. Everyone cool with that? And if we crank up l and we make-- yeah, right? So you can see that you've got some complicated shapes. But as we crank up l and crank up m, we just get a thinner and thinner donut. And the fact that the donut's getting thinner and thinner is that l over L squared that we did earlier. Cool? It's still a donut, but it's getting relatively thinner and thinner, by virtue of getting wider and wider. And a last thing to show you is let's take a look now at-- in fact, let's go to the l equals 0 state. Let's take a look at the real part. So now we're looking at the real part. And nothing much changed for the Y0, 0. But for Y2, 0, well, still not much changed. For Y2, 0 let's now-- this is sort of disheartening. Nothing really has changed. Why? Because it's real, exactly. So for Y2, 0, as long as m is equal to 0, this is real. There's no phase. The phase information contains information about the Lz eigenvalue. So we can correct this by changing the angular momentum. Let's-- oh shoot, how do I do that? I can't turn it off at the moment. OK, whatever. So here we have a large M. And now we've got this very funny lobe-y structure. So this is the Y2, 2, which a minute ago looked rotationally symmetric. And now it's not rotationally symmetric. It's this lobe-y structure, where the lobes are-- remember, previously we had a donut. Now we have these lobes when we look at the real part. How does that make sense? Well, we've got an e to the i 2 phi. And if we look at the real part, that's cosine of 2 pi. And so we're getting a cosine function modulating the donut. And if we look at the real part, let's do the same thing. Let's look at the imaginary part. And the imaginary part of Y2, 0, we get nothing. That's good, because Y2, 0 was real. That would have been bad. But if we look at the imaginary part of Y2, 2, we get the corresponding lobes, the other lobes, so that cos squared plus sine squared is 1. Play with these. Develop some intuition. They're going to be very useful for us when we talk about hydrogen and the structure of solids. And I will see you next week. [APPLAUSE]
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_12_The_Dirac_Well_and_Scattering_off_the_Finite_Step.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare ocw.mit.edu. PROFESSOR: All right. Hello. So today, as people slowly trickle in, so today we're going to talk about scattering states. So we've spent a lot of time talking about bound states, states corresponding to particles that are localized in a region and that can't get off to infinity. And in particular, when we distinguish bound states from scattering states, we're usually talking about energy eigenstates. They're eigenstates that are strictly localized in some region. For example in the infinite well, all of the states are strictly localized to be inside the well. In the finite well, as we saw last time, there are a finite number of states, of energy eigenstates which are bound inside the well. But there are also states that aren't bound inside the well, as hopefully you'll see in recitation, and as we've discussed from qualitative structure of wave functions, but under the eigenfunction. So why would we particularly care? We also did the harmonic oscillator where everything was nice and bound. So why would we particularly care about states that aren't bound for scattering states? And the obvious answer is, I am not bound. Most things in the world are either bound or they are scattering states. By scattering state just mean things that can get away. Things that can go away. So it's easy to under emphasize, it's easy to miss the importance of scattering experiments. It's easy to say look, a scattering experiment is where you take some fixed target and you throw something at it and you look at how things bounce off and you try to do something about this object by helping it bounce off. For example, one of the great experiments of our day is the LEC, and it's a gigantic ring, it's huge, at the foot of the Jura Mountains outside of Geneva. And those are people and that's the detector-- that's one of the detectors, I think, at CMS. And what are you doing in this experiment? In this experiment you're taking two protons and you're accelerating them ridiculously fast, bounded only by the speed light, but just barely. Until they have a tremendous amount of kinetic energy, and you're colliding them into each other. And the idea is if you collide them into each other and you watch how the shrapnel comes flying off, you can deduce what must have been going on when they actually collided. You could do something about the structure. How are protons built up out of corks. And it's easy to think of a scattering experiment as some bizarre thing that happens in particle detectors under mountains. But I am currently engaged in a scattering experiment. The scattering experiments I'm currently engaged in is light is bouncing from light sources in the room, off of very bits of each of you. And some of it, and probably, scatters directly into my eye. That is shocking. And through the scattering process and deducing what I can from the statistics of the photons that bounce into my eye, I could deduce that you're sitting there and not over there, which is pretty awesome. So we actually get a tremendous amount of the information about the world around us from scattering processes. These are not esoteric, weird things that happen. Scattering is how you interact with the world. So when we do scattering problems, it's not because we're thinking about particle colliders. It's because at the end of the day I want to understand-- AUDIENCE: [SNEEZE] PROFESSOR: --how surfaces-- bless you-- how surfaces reflect. I want to understand why when you look in the mirror light bounces back. I want to understand how you see things. So this is a much broader-- scattering's a much broader idea than just the sort of legacy of how we study particle physics. Scattering is what we do constantly. And we're going to come back to the computer in a bit. I'm going to close this for now. Yes, good. So today we're going to begin a series of lectures in which we study scattering processes in one dimension in various different settings. So in particular that means we're going to be studying quantum particles being sent in from some distance, incident on some potential, incident on some object. And we're going to ask how likely are they to continue going through, how likely are they to bounce back. And so we're going to start with an easy potential, an easy object off which to scatter something, which is the absence of an object, the free particle. And I want to think about how the system evolves. So for the free particle where we just have something of mass, m, and its potential is constant, we know how to solve the-- Question? Yeah. AUDIENCE: This may be a little bit dumb, but given that these states, [INAUDIBLE]? PROFESSOR: Fantastic question. I'm going to come back in just a second. It's a very good question. So we're interested in energy eigenstates, and we know the energy eigenstates for the free particle-- we've written these down many times-- and I'm going to write it in the phonic form. Suppose we have a free particle in an energy eigenstate with energy e, then we know it's a superposition of clean waves, e to the i kx plus b e to minus ikx. And the normalization is encoded in a and b, but, of course, these are normalizable states. We typically normalize them with a 1 over root 2 pi for delta function normalizability. Now, in order for this to be a solution to the Schrodinger equation, we need the frequency or the energy e is equal to h-bar squared k squared upon 2m, and omega is this upon h-bar. And knowing that, we can immediately write down the solution. Because these are energy eigenstates, we can immediately write down the solution of the Schrodinger equation, which begins at time t 0 in this state. So there's the solution of the time dependent Schrodinger equation. So we've now completely solved this in full generality, which is not so shocking since it's a free particle. But let's think about what these two possible states are. What do these mean? So the first thing to say is this component of our superposition is a wave, and I'm going to call it a right moving wave. And why am I calling it a right moving wave? If you draw the real part of this, some moment in time, it's got some crests. And if you draw on it a short bit of time later, I want to ask how the wave has moved. And, for example, a peak in the wave will be when kx minus omega t is 0. If kx minus omega t is 0, then kx is equal to omega t, or x is equal to omega over kt. And assuming as we have that k is positive, then this says that as time increases the position increases. And this one, by exactly the same logic, x is equal to minus omega over kt. So we'll call this a right going or right moving, and we'll call this a left moving wave. So the general solution is a superposition, an arbitrary superposition, of a left moving and a right moving wave. This should be familiar from 803 in your study of waves. But as was pointed out just a moment ago, we've got a problem. Phi e or phi e is not normalizable. And this is obvious from if we-- let's have b equal 0 for simplicity. This is a pure phase, and the norm squared of a pure phase is 1. And so the probability density is constant. It's 1 from minus infinity to infinity. So the probability of finding it at any given point is equal. So assuming this is on the real line, which is what we mean by saying it's a free particle, there's no constant you can multiply that by to make it normalizable. This is not a normalizable function. So more precisely, what we usually do is we take our states phi sub k equal 1 over root 2 pi, e to the i kx. Such that phi k, phi k prime is equal to delta. Sorry-- that's not what I wanted to write. Such that-- well, fine. Phi k, phi k prime is equal to delta of k minus k prime. This is as good as we can do. This is good as we get to normalizing it. It's not normalized to 1. It's normalized 1k is equal to k prime, to the value of the delta function, which is ill-defined. So we've dealt with this before, we've talked about it. So how do we deal with it? Well, we've just learned, actually, a very important thing. This question, very good question, led to a very important observation which is that can a particle, can a quantum mechanical particle be placed, meaningfully said to be placed, in an energy eigenstate which is bound? Can you put a particle in the ground state of a harmonic oscillator? AUDIENCE: [INAUDIBLE]? PROFESSOR: Sure. That's fine. Can you put the particle in the k equals 7 state of a free particle? AUDIENCE: No. PROFESSOR: No. So these scattering states, you can never truly put your particle truly in a scattering state. It's always going to be some approximate plane wave. Scattering states are always going to be some plane wave asymptotically. So we can't put the particles directly in a plane wave. What can we do, however? AUDIENCE: [INAUDIBLE] PROFESSOR: Right. We can build a wave packet. We can use these as a basis for states, which are normalizable and reasonably localized. So we need to deal with wave packets. So today is going to be, the beginning of today is going to be dealing with the evolution of a wave packet. And in particular we'll start by looking at the evolution in time of a well localized wave packet in the potential corresponding to a free particle. And from that we're going to learn already some interesting things, and we'll use that intuition for the next several lectures as well. Questions before we get going? All right. So here's going to be my first example. So consider for the free particle or consider a free particle in a minimum uncertainty wave packet. So what is a minimum uncertainty wave packet. We've talked-- AUDIENCE: A Gaussian. PROFESSOR: A Gaussian, exactly. So the minimum uncertainly wave packet's a Gaussian. Suppose that I take my particle and I place it at time 0, x0 in a Gaussian. I'm going to properly normalize this Gaussian. I hate these factors of pi, but whatever. a root pi, e to the minus x squared over 2a squared. So this is the wave function at time 0, I declare. I will just prepare the system in this state. And what I'm interested in knowing is how does the system evolve in time? What is x psi of x and t? So how do we solve this problem? Well, we could just plug it into the Schrodinger equation directly and just by brute force try to crank it out. But the easier thing to do is to use the same technique we've used all the way along. We are going to take the wave function, known wave function, known dependence on x, expand it as a superposition of energy eigenstates. Each energy eigenstates evolved in a known fashion will evolve those energy eigenstates in that superposition, and do the sum again, re-sum it, to get the evolution as a function of position. Cool? Yeah. AUDIENCE: So given that the x-basis and the t-bases are uncountable, [INAUDIBLE]? PROFESSOR: Good. Here's the theorem. So the question is roughly how do you know you can do that? How do you know you can expand in the energy. And we have to go back to the spectral theorem which tells us for any observable, the corresponding operator has a basis of eigenfunctions. So this is a theorem that I haven't proven, but that I'm telling you. And that we will use over and over and over again, spectral theorem. But what it tells you is that any operator has an eigenbasis. So if we find any good operator corresponding to an observable has a good eigenbasis. That means that if we find the eigenfunctions of the energy operator, any state can be expanded as a superposition in that state. And the question of accountable versus uncountable is a slightly subtle one. For example, how many points are there on the circle? Well, there's an uncountable number. How many momentum modes are there on a circle? Well, that's countable. How does that work? How can you describe that both in position and in momentum space. Ask me afterwards and we'll talk about that in more detail. But don't get too hung up on countable versus uncountable. It's not that big a difference. You're just summing-- it's just whether you sum over a continuous or discrete thing. I'm being glib, but it's useful to be glib at that level. Other questions before we-- OK. Good. Yeah. AUDIENCE: So the reason we are going to use energy eigenstates is just because [INAUDIBLE]? PROFESSOR: Almost. So the question is are we using the energy eigenstates because at the end of the day it's going to boil down to doing a Fourier transform, because the energy eigenstates are just plain waves, e to the ix? And that will be true, but that's not the reason we do it. It's a good question. So let me disentangle these. Well, I guess I can use this. The point is that if I know that si-- for any wave function in any system, if I know that phi of x at 0 is some function, and I know the energy eigenstates of that system, then I can write this as sum over n, of cn, phi n of x for some set of coefficient cn. So that's the spectral theorem. Pick an operator here, the energy. Consider its eigenfunctions. They form a basis and any state can be expanded. But once we've expanded in the energy eigenbasis, specifically for the energy eigenbasis where e phi n is en phi n, then we know how these guys evolve in time. They evolve by an e to the minus i en t upon h-bar. And we also know that the Schrodinger equation is a linear equation, so we can superpose solutions and get a new solution. So now, this is the expression for the full solution at time t. That's why we're using energy eigenstates. In this simple case of the free particle, it is a felicitous fact that the energy eigenstates are also Fourier modes. But that won't be true in general. For example, in the harmonic oscillator system, the harmonic oscillator energy eigenstates are Gaussians times some special functions. And we know how to compute them. That's great. We use a raising, lowering operators. It's nice. But they're not plane waves. Nonetheless, despite not being Fourier modes, we can always expand an arbitrary function in the energy eigenbasis. And once we've done so, writing down the time evolution is straightforward. That answer your question? AUDIENCE: Yeah. PROFESSOR: Great. Anything else? OK. So let's do it for this guy. So what are the energy eigenstates. They're these guys, and I'm just going to write it as e to the i kx without assuming that k is positive. So k could be positive or negative, and energy is e to the h-bar squared k squared. And so we can write this as-- we can write the wave function in this fashion, but we could also write it in Fourier transformed form, which is integral dk upon root 2 pi for minus infinity to infinity. I'm mostly not going to write the bounds. They're always going to be minus infinity to infinity. Of dk 1 over root 2 pi e to the i kx. The plane wave times the Fourier transform, psi tilde of k. So here all I'm doing is defining for you the psi tilde. But you actually computed this, so this is the Fourier transfer. You actually computed this on, I think, the second problem set. So this is equal to the integral dk over root 2 pi e to the i kx. And I'm going to get the coefficients-- I'm going to be careful about the coefficients a, so root a over 4 [INAUDIBLE] pi. e to the minus k squared a squared over 2. So here, this is just to remind us if we have a Gaussian of width a, the Fourier transform is Gaussian of width 1 upon a. So that's momentum and position uncertainty in Fourier space. And you did this in problem set two. So this is an alternate way of writing this wave function through its Fourier transform. Everyone cool with that? But the nice thing about this is that we know how to time evolve this wave function. This tells us that that psi of xt is equal to integral-- and I'm going to get these coefficients all pulled out-- integral of root a over root pi. Integral dk upon root 2 pi. e to the i kx minus omega t. And remember, Omega it depends on k, because it's h-bar squared k squared upon 2m divided by h-bar. Times our Gaussian, e to the minus k squared a squared upon 2. So everyone cool with that? So all I've done is I've taken this line, I pulled out the constant, and I've added the time evolution of the Fourier mode. So now we have this integral to give us the full, OK, so now we just have to do the integral. And this is not so hard to do. But what makes it totally tractable is that if I just do this with some function of omega of k-- I mean that's complicated, know how to do that integral. But I happen to know that e is h-bar squared k squared upon 2m, which means that omega is equal to h-bar k squared upon 2m. So I can rewrite this as minus h-bar k squared upon 2m. Yes? AUDIENCE: Where is the fourth [INAUDIBLE]? PROFESSOR: That's a square root of a square root. AUDIENCE: Oh, I didn't see it. My bad. PROFESSOR: No, no. That's OK. It's a horrible, horrible factor. So what do we take away from this? Well, this can be written in a nice form. Note that here we have a k squared, here we have a k squared, here we have a k. This is an integral over k, so this is still Gaussian. It's still the exponential of a quadratic function of k. So we can use our formulas for Gaussian integrals to do this integral. So to make that a little more obvious, let's simplify the form of this. So the form of this is again going to be square root of a over square root pi. Integral dk upon root 2 pi. e to the i kx. And I'm going to take this term and this term and group them together because they both have a k squared in the exponential. e to the minus k squared upon 2 times-- now instead of just a squared, we have a squared plus i h-bar upon 2m t-- there's a typo in my notes. Crap. Good. So before we do anything else, let's just check dimensional analysis. This has units of one of length squared. So this had better have units of length squared. That's a length squared, good. This, is this a length squared? So that's momentum times length, which is mass times length over time. Divide by mass, multiplied by time, so that's length squared. Good. So our units make sense. AUDIENCE: [INAUDIBLE]. PROFESSOR: Sorry? AUDIENCE: [INAUDIBLE]. PROFESSOR: This, too, we need. AUDIENCE: In terms of 2m. PROFESSOR: Oh, this one. Good. Thank you. Yes. Thanks. So this is again a Gaussian. And look, before we knew we have a Gaussian, the Fourier transform is just this Gaussian. The only difference is it's now a Gaussian with width. Not a, but a complex number. But if you go through the analysis of the Gaussian integral, that's perfectly fine. It's not a problem at all. The effective width is this. So what does that tell you about psi of t? So this tells you that psi of x and t is equal to root a over root pi. And now the width. So we have the 1 over root a. But now the effective width is this guy, a squared plus-- so it's square root of a squared plus i h-bar upon mt. e to the minus x squared over 4a squared plus-- oops, where did my 4-- there's a 2 that-- there's a spurious 2 somewhere. I'm not sure where that 2 came from. There should be a 2 there. a squared plus i h-bar upon mt. So at this point it's not totally transparent to me what exactly this wave function is telling me, because there's still a phase downstairs in the width. So to get a little more intuition let's look at something that's purely real. Let's look at the probability to find the particle at position x at time t. That's just the norm squared of this guy. And if you go through a little bit of algebra, this is equal to 1 over root pi times-- I'm going to write out three factors-- times 1 over square root of-- yes, good-- a squared plus h-bar upon 2ma squared t squared times e to the minus x squared upon 2a squared plus h-bar squared over 2ma squared. Oops-- h-bar over 2ma squared t. So what do we get? The probability distribution is again a Gaussian. But at any given moment in time-- this should be t squared-- at any given moment in time the width is changing. The width of this Gaussian is changing in time. Notice this is purely real again. And meanwhile, the amplitude is also changing. And notice that it's changing in the following way. At time 0, this is the least it can possibly be. At time 0 this denominator is the least it can possibly be because that's gone. At any positive time, the denominator is larger so the probability has dropped off. So the probability at any given point is decreasing. Except for we also have this Gaussian whose width is increasing in time, quadratically. It's getting wider and wider and wider. And you can check-- so let's check dimensional analysis again. This should have units of length squared, and so this is momentum times length divided by length divided by mass-- that's just a velocity. So this is the velocity squared times squared. That's good. So this has the correct units. So it's spreading with the velocity v is equal to h-bar upon 2ma. And what was a? a was the width at times 0. a was the minimum width. So graphically, let's roll this out-- what is this telling us? And incidentally, a quick check, if this is supposed to be the wave function at time t, it had better reproduce the correct answer at time 0. So what is this wave function at time 0? We lose that, we lose that, and we recover the original wave function. So that's good. So graphically what's going on? So let's plot p of x and t. At time 0 we have a Gaussian and it's centered at 0, and it's got a width, which is just a. And this is at time t equals 0. At a subsequent time, it's again centered at x equals 0-- modulating my bad art-- but it's got a width, which is a squared square root of a squared plus h-bar upon 2ma squared, t squared. So it's become shorter and wider such that it's still properly normalized. And at much later time it will be a much lower and much broader Gaussian. Again, properly normalized, and again centered at x equals 0. Is that cool? So what this says is if you start with your well-localized wave packet and you let go, it disperses. Does that make sense? Is that reasonable? How about this. What happens if you ran the system backwards in time? If it's dispersing and getting wider, then intuitively you might expect that it'll get more sharp, if you integrated it back in time. But, in fact, what happens if we take t negative in this analysis? We didn't assume it was positive, anyway. AUDIENCE: [INAUDIBLE]. PROFESSOR: It disperses again. It comes from being a very dispersed wave, to whap, to being very well [INAUDIBLE], and whap, it disperses out again. And notice that the sharper the wave function was localized, the smaller a was in the first place, the faster it disperses. This has 1 upon a. It has an a downstairs. So the more sharp it was at time 0, the faster it disperses away. This is what our analysis predicts. So let me give you two questions to think about. I'm not going to answer them right now. I want you to think about them-- not right-- well, think about them now, but also think about them more broadly as you do the problem set and as you're reading through the reading for the next week. So the width in this example was least at time t equals 0. Why? How could you build a wave function, how could you take this result and build a wave function whose width was minimum at some other time, say, time t0. So that's one question. Second question. This wave function is sitting still. How do you make it move? And either the second or the third problem on the problem set is an introduction to that question. So here's a challenge to everyone. Construct a well-localized wave packet that's moving, in the sense that it has well-defined momentum expectation value. And again, hint, look at your second or third problem on the problem set. And then repeat this entire analysis and check how the probability distribution evolves in time. And in particular, what I'd like to do is verify that the probability distribution disperses, but simultaneously moves in the direction corresponding to the initial momentum with that momentum. The center of the wave packet moves according to the momentum that you gave it in the first place. It's a free particle so should the expectation value, the momentum change over time? No. And indeed it won't. So that's a challenge for you. OK, questions at this point? AUDIENCE: Yeah. So this wave or this Gaussian's not moving, but it's centered over 0 at time 0 or x0? PROFESSOR: Sorry, this is at x is equal to 0. AUDIENCE: OK. So is that-- so all those [INAUDIBLE] are at x equals 0 over here. Shouldn't the amplitude increase and not the-- PROFESSOR: Good. So what I'm plotting-- sorry. So I'm plotting p of x and t as a function of x for different times. So this is time 0. This was time t equals something not 0, equals 1. And this is some time t equal to large. Right. AUDIENCE: And so but is it if x is always 0, then shouldn't the [INAUDIBLE] not change because it would equal [INAUDIBLE]? PROFESSOR: Good. So if x is 0, then the Gaussian contribution is always giving me 1. But there's still this amplitude. And the amplitude is the denominator is getting larger and larger with time. So, indeed, the amplitude should be falling off. Is that-- AUDIENCE: Yeah, but will the width change? PROFESSOR: Yeah, absolutely. So what the width is, the width is saying, look, how rapidly as we increase x, how rapidly does this fall off as a Gaussian. And so if time is increasing, if time is very, very large, than this denominator's huge. So in order for this Gaussian to suppress the wave function, x has to be very, very large. AUDIENCE: So we're assuming [INAUDIBLE] x hasn't changed. PROFESSOR: Ah, good. Remember, so this x is not the position of the particle. x is the position at which we're evaluating the probability. So what we're plotting is we're plotting-- good. Yeah, this is an easy thing to get confused by. What is this quantity telling us? This quantity is telling us the probability, the probability [INAUDIBLE, that we find the particle at position x at time t. So the x is like, look, where do you want to look? You tell me the x and I tell you how likely it is to be found there. So what this is telling us is that the probability distribution to find the particles start out sharply peaked, but then it becomes more more and more dispersed. So at the initial time, how likely am I to find the particle, say, here? Not at all. But at a very late time, who knows, it could be there. I mean my confidence is very, very limited because the probability distribution's very wide. Did that answer your question. AUDIENCE: Yes. PROFESSOR: Great. Yup? AUDIENCE: [INAUDIBLE] what happens after [INAUDIBLE]. PROFESSOR: Excellent question. That's a fantastic question. So we've talked about measurement of position. We've talked about measure-- So after you measure an observable, the system is left in an eigenstate of the observable corresponding to the observed eigenvalue. So suppose I have some potential or I have some particle and it's moving around. It's in some complicated wave function, and some complicated state. And then you measure its position to be here. But my measurement isn't perfect. I know it's here with some reasonable confidence, with some reasonable width. What's going to be the state thereafter? Well, it's going to be in a state corresponding to being more or less here with some uncertainty. Oh look, there's that state. And what happened subsequently? What happened subsequently is my probability immediately decreases. And this is exactly what we saw before. Thank you for asking this question. It's a very good question. This is exactly what we said before. If we can make an arbitrarily precise measurement of the position, what would be the width? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah. It'd be delta, so with width would be 0. That little a would be 0. So how rapidly does it disperse? AUDIENCE: [INAUDIBLE]. PROFESSOR: Pretty fast. And you have to worry a little bit because that's getting a little relativistic. Yeah. AUDIENCE: If you make a measurement that's not perfect, how do you account for that [INAUDIBLE]? PROFESSOR: Yeah. So OK. That's a good question. So you make a measurement, it's not perfect, there was some uncertainty. And so let me rephrase this question slightly. So here's another version of this question. Suppose I make a measurement of the system. If there are two possible configurations, two possible states, and they measure an eigenvalue corresponding to one of them, then I know what state the system is in. And then again, if I'm measuring something like position where I'm inescapably going to be measuring that position with some uncertainty, what state is it left in afterwards? Do I know what state it's left in afterwards? No, I-- I know approximately what state. It's going to be a state corr-- OK, so then I have to have some model for the probability distribution of being in a state. But usually what we'll do is we'll say we'll approximate that. We'll model that unknown. We don't know exactly what state it is because we don't know exactly what position we measured. Well, model that by saying, look, by the law of large numbers it's going to be roughly a Gaussian centered around the position with the width, which is our expected uncertainty. Now, it depends on exactly what measurement you're doing. Sometimes that's not the right thing to do, but that's sort of a maximally naive thing. This is an interesting but complicated question, so come ask me in office hours. Yeah. AUDIENCE: So if this probability isn't moving, what is that velocity we [INAUDIBLE] over there? PROFESSOR: Yeah, exactly. So what is this velocity? So this velocity, what is it telling us? It's telling us how rapidly the wave spreads out. So does that indicate anything moving? No. What's the expected value of the position at any time? AUDIENCE: 0. PROFESSOR: 0, because this is an even function. What's the position? Well, it's just as likely to be here as here. It's just that the probability of being here gets greater and greater over time for a while and then it falls off because the Gaussian is spreading and falling. Yeah. One last question. Yup. AUDIENCE: Can you tell me a little bit more about negative time is the same as positive time? PROFESSOR: Yeah. So the question is can you talk a little bit more about why the behavior as time increases is the same as the behavior as we go backwards in time. And I'm not sure how best to answer that, but let me give it a go. So the first thing is to say look at our solution and look at the probability distribution. The probability distribution is completely even in time. It depends only on t squared. So if I ask what happens at time t, it's the same as what happened at time minus t. That's interesting in a slightly unusual situation. Let's look back at the wave function. Is that true of the wave function? AUDIENCE: [INAUDIBLE]. PROFESSOR: No. Because the phase and the complex parts depend on-- OK, so the amplitude doesn't depend on whether it's t or minus t. But the phase does. And that's a useful clue. Here's a better way. What was the weird thing that made this t reversal invariant? There's another weird property of this, which is that it's centered at 0. It's sitting still. If we had given it momentum that would also not have been true. So the best answer I can give you, though, requires knowing what happens when you have a finite momentum. So let me just sketch quickly. So when we have a momentum, how do we take this wave packet and give it momentum k naught? So this is problem three on your problem set, but I will answer it for you. You'd multiply by phase e to the i k naught x. And the expectation value will now shift to be h-bar k naught for momentum. So what's that going to do for us? Well, as we do the Fourier transform, that's going to shift k to k minus k naught. And so we can repeat the entire analysis and get roughly k and k naught here. The important thing is that we end up with phases. So the way to read off time 0 is going to be-- well, many answers to this. As time increases or decreases, we'll see from this phase that the entire wave packet is going to continue moving, it's going to continue walking across. And that breaks the t minus t invariance. Because at minus t it would have been over here. And then think about that first question. How would you make it become minimum wave packet at some time which is not time equals 0? So when you have answers to both of those you'll have the answer to your question. Yeah. AUDIENCE: What does it really mean for us to say that we have a probability distribution and measure something like negative time? Because it seems like we did something at time equals 0, and we say the probability that we would have found it [INAUDIBLE] is here. PROFESSOR: Right. AUDIENCE: It seems that you have to be aware [INAUDIBLE]. PROFESSOR: Good. So that's not exactly what the probability of distribution is telling us. So what the probability and distribution is telling you is not what happens when you make a measurement. It's given this state what would happen? And so when you ask about the probability distribution at time minus 7, what you're saying is given the wave function now, what must the wave function have been a time earlier, such that Schrodinger evolution, not measurement, but Shrodinger evolution gives you this. So what that's telling you is someone earlier-- had someone earlier been around to do that experiment, what value would they have got? AUDIENCE: But wouldn't that have changed the wave function of that? PROFESSOR: Absolutely, if they had done the measurement. There's a difference between knowing what the probability distribution would be if you were to do a measurement, and actually doing a measurement and changing the wave function. AUDIENCE: So for the negative time, is it the earlier statement about what the wave function-- PROFESSOR: It's merely saying about what the wave function was at an earlier time. AUDIENCE: It doesn't really have measurements. PROFESSOR: No one's done any measurements, exactly. Writing down the probability distribution does not do a measurement. Exactly. One last question. Sorry, I really-- Yup. AUDIENCE: Why define it negative 2, and then prepares the states and its width times 0? PROFESSOR: Well, you could ask the question slightly differently. If you had wanted to prepare the state at time minus 7 such that you got this state at time 0-- AUDIENCE: OK. PROFESSOR: This is the inverse of asking given that the state is prepared now, what is it going to be at time 7. So they're equally reasonable questions. What should you have done last week so that you got your problem set turned in today? There was no problem set due today. OK, so with that said, we've done an analytic analysis-- that was ridiculous-- we've done a quick computation that showed analytically what happens to a Gaussian wave packet. We've done an analysis that showed us how the wave function evolved over time. What I'd like to do now is use some of the PhET simulations to see this effect, and also to predict some more effects. So what I want you to think about this as, so this is the PhET quantum tunneling wave packet simulation. If you haven't played with these, you totally should. They're fantastic. They're a great way of developing some intuition. All props to the Colorado group. They're excellent. So what I'm going to do is I'm going to run through a series of experiments that we could, in principle, do on a table top, although it would be fabulously difficult. But the basic physics is simple. Instead I'm going to run them on the computer on the table top because it's easier and cheaper. But I want you to think of these as experiments that we're actually running. So here, what these diagram show, for those of you who haven't played with this, is this is the potential that I'm going to be working with. The green, this position, tells me the actual value of the energy of my wave packet. And the width tells me how broad that wave packet is. And roughly speaking, how green it is tells me how much support I have on that particular energy eigenstate. And I can change that by tuning the initial width of the wave packet. So if I make width very narrow, I need lots and lots of different energy eigenstates to make an narrow well-localized wave packet. And if I allow the wave packet to be quite wide, then I don't need as many energy eigenstates. So let me make that a little more obvious. So fewer energy eigenstates are needed, so we have a thinner band in energy, more energy eigenstates are needed, so I have more wide band. Is everyone cool with that? And what this program does is it just integrates the Schrodinger equation-- well, yeah, well it just integrates the Schrodinger equation. So what I want to do is I want to quickly-- oops. So this is the wave function, its absolute value. And this is the probability density, the norm squared. So let's see what happened. So what is this initial state? This initial state-- in fact, I'm going to re-load the configuration. So this initial state corresponds to being more or less here with some uncertainty. So that's our psi of x0. And let's see how this evolves. So first, before you actually have it evolved, what do you think is going to happen? And remember, it's got some initial momentum as well. OK. So there it is evolving and it's continuing to evolve, and it's really quite boring. This is a free particle. Wow, that was dull. OK, so let's try that again. Let's look also at the real part and look at what the phase of the wave function's doing. So here there's something really cool going on. It might not be obvious. But look back at this. Look at how rapidly the wave packet moves and how rapidly the phase moves. Which one's moving faster? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah, the wave packet. And the wave packet, it's kind of hard to tell exactly how much faster it's moving. But we can get that from this. What's the velocity of a single plane wave or the phase velocity? So the phase velocity from problem set one is h-bar k squared upon 2m-- let me write this as omega upon k, which is just equal to h-bar k upon 2m. But that's bad because that's the momentum. Divide the momentum by the mass, that's the classical velocity. So this says that the phase of any plane wave is moving with half the classical velocity. That's weird. On the other hand, the group velocity of a wave packet, of a normalizable wave packet is do mega dk. And do mega dk will pull it down an extra factor of 2. This Is equal to h-bar k upon m. This is the classical velocity. So this is the difference between the phase and the group velocity, and that's exactly what we're seeing here. So the first thing you notice is you see the difference between the phase and the group velocity. So that's good. That should be reassuring. But there's something else that's really quite nice about this. Let's make the wave packet really very narrow. So here it's much more narrow, and let's let the system evolve. And now you can see that the probability density is falling off quite rapidly. Let's see that again. Cool. The probability distribution is falling off very rapidly. And there's also kind of a cool thing that's happening is you're seeing an accumulation of phase up here and a diminution down here. We can make that a little more sharp. Oops, sorry. Yeah. So that is a combination of good things and the code being slightly badly behaved. So the more narrow I make that initial-- so let's-- so I made that really narrow, and how rapidly does it fall off? So that very narrow wave packet by time 10 is like at a quarter. But if we made it much wider, that very narrow wave packet started out much lower and it stays roughly the same height. It's just diminishing very, very slowly. And that's just the 1 over a effect. So this is nice. And I guess the way I'd like to think about this is this is confirming rather nicely, our predictions. So now I want to take a slightly different system. So this system is really silly because we all know what's going to happen. Here's a hill, and if this were the real world I would think of this as some potential hill that has some nice finite fall off. The sort of place you don't want to go skiing off. AUDIENCE: [INAUDIBLE]. PROFESSOR: Well, I don't. So imagine you take an object of mass m and you let it slide along in this potential, what will happen? It will move along at some velocity. It will get to this cliff, it will fall off the cliff and end up going much faster at the bottom. Yeah? So if you kick it here, it will get over there and it will end up much faster. What happens to this wave packet. So we're solving the Schrodinger equation. It's exactly the same thing we've done before. Let's solve it. So here's our initial wave packet, nice and well-localized. There's our wave function. And what happens? So at that point something interesting has happened. You see that for the most part, most of the probability is over here, but not all the probability is over here. There's still a finite probability that the particle stays near the wall for a while. Let's watch what happens to that probability as time goes by. And now you should notice that it's decayed into a wave packet that's over here, and then a superposition of a wave packet over here moving to the left. This wave has scattered off the barrier going downhill. It's mostly transmitted, but some of it reflected. Which is kind of spooky. So let's watch that-- I'm going to make this much more extreme. So here's a much more extreme version of this. I'm now going to make the energy very close to the height of the potential. The energy is very, very close to the height of the potential, and the potential is far away. We're still reasonably local-- I can make it a little less localized just to really crank up the suspense. So watch what happens now. So first off, what do you expect to happen? So we've made the energy of the wave packet just barely higher than the energy of the hill. So it's almost classically disallowed from being up here. What does that mean about its velocity? It's effective loss. AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah. The momentum, the expectation value for momentum should be very low because it just has a little bit of spare energy. So the expected value for the momentum should be very low. It should dribble, dribble, dribble, dribble, dribble, dribble, dribble till it gets to the cliff, and then-- [WHISTLE]-- go flying off. So let's see if that's what we see. So it's dribbling, but look at the probability density down here. Oh, indeed, we do see little waves, and those waves are moving off really fast. But the amplitude is exceedingly small. Instead, what we're seeing is a huge pile up of probability just before the wall, similar to what we saw a minute ago. Except the probability is much, much larger. And what's going to happen now? AUDIENCE: [INAUDIBLE]. PROFESSOR: It's leaking. It's definitely leaking. But an awful lot of the probability density is not going off the cliff. This would be Thelma and Louise cruising off the cliff and then just not falling. This is a very disconcerting-- you guys get to-- yeah. [LAUGHTER] So see what happens, see what happened here. We've got this nice big amplitude to go across or to reflect back, to scatter back. You really want to fall down classically, but quantum mechanically you can't. It's impossible. So, in fact, the probability they reflect back is 41, and we're going to see how to calculate that in the next few lectures. Yup. AUDIENCE: So the transmission was actually really high. PROFESSOR: Transition was reasonably high, yup. AUDIENCE: And it was really [INAUDIBLE]. PROFESSOR: It wasn't really, sorry? AUDIENCE: It wasn't really apparent graphically. PROFESSOR: Yeah, it wasn't apparent. The way it was apparent was just the fact that the amplitude that bounced back was relatively small. So what was going is any little bit of probability that fell down had a large effective momentum. Just ran off the screen very rapidly. So it's hard to see that in the simulation, it's true. We can make the-- we can squeeze-- ah, there we go. So here is-- that's about as good as I'm going to be able to get. And it's just going to be glacially slow. But this is just going to be preposterously slow. I'm not even sure there's much point in-- But you can see what's going to happen. They're going to build up, we're going to get a little leak off, but for the most part, the wave packet's going to go back to the left. Questions about this one before we move on to the next? Yeah. AUDIENCE: [INAUDIBLE]. PROFESSOR: Are the-- Yeah, exactly. So r and t here are going to be defined as the probability that this wave packet gets off to infinity in either direction. AUDIENCE: What happens if you make the energy lower than-- PROFESSOR: Then you can't have a wave packet on the left that's moving. Because remember that the wave packet in order to have an expected momentum needs to be oscillating. But if the energy is lower than the potential, its exponential. So the expectation value for the momentum is 0. So you can't have a particle moving in from the left if it's got energy less than the height of the barrier. Cool? Yeah. AUDIENCE: Sorry, what's this green thing? PROFESSOR: The green thing is the energy of the wave packet that I'm sending in. OK, so let's go to the next one. So this is something we'd like to understand. The reason I'm showing you this is I want to explain this phenomena. We already did the dispersion. I want to explain this phenomena that you reflect when going downhill, which is perhaps surprising. And I want to ask how efficiently do reflect off any given barrier. Yeah. Question. AUDIENCE: Can you describe the [INAUDIBLE] physical experiment that would show that done, though? PROFESSOR: Oh yeah, absolutely. So suppose I have a little capacitor plate and there's a potential difference across it and a hole so a particle could shoot threw it. I mean it doesn't have to be an insanely small hole, but a hole. So that if a particle goes from here to the other side of the capacitor plate, it will have accelerated across the potential difference. So if you think about the effective potential energy, the potential is decreasing linearly. It's a constant electric field. So over this short domain between the capacitor plates, we have effectively a linear potential energy. So I have a particle that I send with very little momentum, but it carries some charge. It gets on capacitor plates, and if it goes through the capacitor plate, it ends with a lot more energy relative to the potential energy. That cool? So a little capacitor plate with a hole in it is a beautiful example of this system. Good? Now, making it infinitely sharp, well, that would require-- you know. But making it very sharp is no problem. So here's the inverse of what we just did, which is sending a particle into a barrier. And so there you see it does some quite awesome things. And I want to pause it right here. This is the energy band. So the green represents-- it says that at any given energy inside this green band, there's some contribution from the corresponding energy eigenstate to our wave packet. There isn't any contribution from states up here where there's no green, which means that all the contributions come from energy eigenstates with energy below the height of the barrier. None of them have enough energy to cross the barrier. And what we see when we send in our wave packet is some complicated mucking around near the barrier, but in particular, right there, there's a finite non-zero probability that you're found in the classically disallowed region. In the region where you didn't have enough energy to get there. We see right there. And meanwhile, there's also a huge pile up here. This is a hard wall. Normally classically from a hard wall, you roll along, you hit the hard wall, and you bounce off instantaneously with the same velocity. But here what we're saying is no, the wave doesn't just exactly bounce off instantaneously. We get this complicated piling up and the interference effect. So what's going on there? Well, it's precisely an interference effect. So look at the wave function. The wave function is a much easier thing to look at. So the red is the amplitude. The red is the real part. The black is the phase. So the real part is actually behaving quite reasonably. It looks like it's just bouncing right off. But the physical thing is the probability distribution. The probability distribution has interference terms from the various different contributions to the wave function. So at the collision those interference effects are very important. And at late times notice what's happened. At late times the probability distribution all went right back off. So that's something else we'd like to understand. Penetration into the barrier from a classically disallowed wave packet. Yup. AUDIENCE: [INAUDIBLE] the wave packet is also expanding again? PROFESSOR: Yeah. The wave packet, exactly. So the wave packet is always going to expand. And the reason is we're taking our wave packet, which is a reasonably well-localized approximately Gaussian wave packet. So whatever else is going on, while it's in the part of the potential where the potential is constant, it's also just dispersing. And so we can see that here. Let's make that more obvious. Let's make the initial wave packet much more narrow. So if the wave packet's much more narrow, we're going to watch it disperse, and let's move the wall over here. So long before it hits the wall it's going to disperse, right? That's what we'd expect. And lo, watch it disperse. OK, now gets to the wall. And it's sort of a mess. Now, you see that there's contributions to the wave function over here, there's support for the wave function moving off to the right. But now notice that we have support on energy eigenstates with energy above the barrier. So, indeed, it was possible for some contribution to go off to the right. So one more of these guys. I guess two more. These are fun. So here's a hard wall. Let's make the-- So here's a hard wall. But it's a finite width barrier. What happens? Well, this is basically the same as what we just saw. We collide up against that first wall, and since there's an exponential fall off it doesn't really matter that there's this other wall where it falls down over here because the wave function is just exponentially suppressed in the classically disallowed region. So let's see that again. It's always going to be exponentially suppressed in the classically disallowed region. As you see, there's that big exponential suppression. So there's some finite particle here, but you're really unlikely to find it out here exponentially. On the other hand, if we make the barrier-- so let's think about this barrier. Classically, this is just as good as a thick barrier. You can't get passed this wall. But quantum mechanically what happens? You go right through. Just-- AUDIENCE: [INAUDIBLE]. PROFESSOR: --right on-- [LAUGHTER] Yeah, uh-huh. Right through. It barely even changes shape. It barely even changes shape. Just goes right on through. Now, there's some probability that you go off to the left, that you bounce off. [LAUGHTER] I share your pain. I totally do. So I would like to do another experiment which demonstrates the difference between classical and quantum mechanical physics. Right. [LAUGHTER] [APPLAUSE] So we have some explaining to do. And this is going to turn out to be-- surprisingly, this is not a hard thing to explain. It's going to be real easy. Just like the dispersion, this is going to be not a hard thing to explain. But the next one and this is-- oh yeah, question. AUDIENCE: Does the height of the spike matter? PROFESSOR: It does, absolutely. So the height of the spike and the width of the spike matter. So let's make it a little wider. And what we're going to see now is oh, it's a little less likely to go through. If we make it just a tiny bit wider we'll see that it's much less likely to go through. We'll see much stronger reflected wave. So both the height and width are going to turn out to matter. So here you can see that there's an appreciable reflection, which there wasn't previously. Let's make it just a little bit wider. And now it's an awful lot closer to half and half. See the bottom two lumps? Let's make it just ever so slightly wider. And again, looking down here we're going to have a transmitted bit, but we're also going to have a reflected-- now it looks like the reflected bit is even a little bit larger maybe. And we can actually compute the reflection transmission here. 72% gets reflected and 28% get transmitted. I really strongly encourage you to play with these simulations. They're both fun and very illuminating. Yeah. AUDIENCE: [INAUDIBLE] just the higher it got, the less got transmitted? PROFESSOR: Yes. AUDIENCE: So if this were a delta function would [INAUDIBLE]? PROFESSOR: Well, what's the width? AUDIENCE: 0. PROFESSOR: So what happens when we make the well less and less-- or more and more thin? More, yeah, through. So we've got a competition between the heights and the width. So one of the problems on your problem set, either this week or next week, I can't recall, will be for the delta function barrier, compute the transmission probability. And, in fact, we're going to ask you to compute the transmission probability through two delta functions. And, in fact, that's this week. And then next week we're going to show you a sneaky way of using something called the S matrix to construct bound states for those guys. Anyway, OK. So next, the last simulation today. So this is the inverse of what we just did. Instead of having a potential barrier, we have a potential well. So what do you expect to happen? Well, there's our wave packet, and it comes along, and all heck breaks loose inside. We get some excitations and it tunnels across. But we see there's all this sort of wiggling around inside the wave function. And we get some support going off to the right, some support going off to the left. But nothing sticks around inside, which it looks like there is, but it's going to slowly decay away as it goes off to the boundary. These are scattering states. So one way to think about what's going on here is it first scatters off, it scatters downhill, and then it scatters uphill. But there's a very funny thing that happens when you can scatter uphill and scatter downhill. As we saw, any time you have scattering uphill and there's some probability-- that you reflected some probability that you transmit. And we scatter uphill. There's some probability you transmit, and some probability that you reflect. But if you have both an uphill and a downhill when you have a well, something amazing happens. Consider this well. This didn't work very well. Let's take a much larger one. Uh-hmm! So at some point, I think it was last year, some student's cell phone went off, and I was like oh, come on, dude. You can't do that. And like five minutes later my cell phone went off. [LAUGHTER] So I feel your pain, but please don't let that happen again. So here we go. So this is a very similar system-- well, the same system, but with a different wave packet, and a slightly different width. And now what happens? Well, all this wave packet just sort of goes on, but how much probability is going back off to the left? Very little. In fact, we can really work this. Sorry, I need this, otherwise I'm very bad at this. There's some probability that reflects in the first well. There's some probability that reflects in the second well. But the probability that reflects from the combination of the two, 0. How can that be? So this is just like the boxes at the very beginning. There's some probability. So to go in this system-- so let's go back to the beginning. There's some amplitude to go from the left side to the right-hand side by going-- so how would you go from the left-hand side to the right-hand side? You transmit and then transmit, right? So it's the transmission amplitude times transmission amplitude gives you the product. But is that the only thing that could happen? What are other possible things that could happen? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah. You could transmit, reflect, reflect, transmit. You could also transmit, reflect, reflect, reflect, reflect, transmit. And you could do that an arbitrary number of times. And each of those contributions is a contribution to the amplitude. And when you compute the probability of transmit, you don't take some of the probabilities. You take the square of the sum of the amplitudes. And what we will find when we-- I guess we'll do this next time when we actually do this calculation-- is that that interference effect from taking the square of the sum of the amplitudes from each of the possible bouncings can interfere destructively and it can interfere constructively. We can find points when the reflection is extremely low. And we can find points where the transmission is perfect. Yeah. AUDIENCE: So two questions. One, the computer says that the reflection coefficient is 0. Why did it look like it there was a wave packet going off back to the left? PROFESSOR: Excellent question. And, indeed, there is a little bit here. So now I have to tell you a little bit about how the computer is casting this reflection amplitude. The way it's doing that is it's saying suppose I have a state which has a definite energy at the center of this distribution. Then the corresponding reflection amplitude would be 0. But, in fact, I have a superposition. I've taken a superposition of them. And so the contributions from slightly different energies are giving me some small reflection. AUDIENCE: And also, why do you have to fine-tune the width of that as well? PROFESSOR: Yeah. That's a really-- let me rephrase the question. The question is why do you have to fine-tune the width of that well? Let me ask the question slightly differently. How does the reflection probability depend on the width of the well? What's going on at these special values of the energy or special values of the width of the well when the transmission is perfect? Which I will call a resonance. Why is that happening? And we'll see that. So that's a question I want you guys to ask. So I'm done now with the experiments. You guys should play with these on your own time to get some intuition. Yeah. AUDIENCE: Isn't the [INAUDIBLE] you just showed actually a classical [INAUDIBLE] objects? PROFESSOR: So excellent. So here's what I'm going to tell. I'm going to tell you two things. First off, if I take a classical particle and I put it through these potentials, for example, this potential. If I take a classical particle and I put it in this potential, it's got an energy here, does it ever reflect back? AUDIENCE: No. PROFESSOR: No. Always, always it rolls and continues on. Always. So if I take a quantum particle and I do this, will it reflect back? AUDIENCE: Sometimes. PROFESSOR: Apparently sometimes. Now, how do I encode that? I encode that in studying the solutions to the wave equation dt minus-- sorry-- i h-bar dt psi is equal to minus h-bar squared upon 2m, dt squared psi plus v psi. So this property of the quantum particle that it reflects is a property of solutions to this equation. When I interpret the amplitude squared as a probability. But this equation doesn't just govern-- this isn't the only place that this equation's ever shown up. An equation at least very similar to this shows up in studying the propagation of waves on a string, for example. Where the potential is played by the role of the tension or the density of the string. So as you have a wave coming along the string, as I'm sure you did in 803, you have a wave coming along a string. And then the thickness or density of the string changes, becomes thicker or thinner, then sometimes you have reflections off that interference-- from impedance mismatch is one way to phrase it. And as you move across to the other side you get reflections again. And the thing that matters is not the amplitude of the-- one of the things that matters is not the amplitude, but it's the intensity. And the intensity's a square. And so you get interference effects. So, indeed, this phenomena of reflection and multiple reflection shows up in certain classical systems, but it shows up not in classical particle systems. It shows up in classical wave systems, like waves on a chain or waves in a rope. And they're governed by effectively the same equations, not exactly, but effectively the same equations, as the Schrodinger evolution governs the wave function. But the difference is that those classical fields, those classical continuous objects have waves on them, but those waves are actual waves. You see the rope. The rope is everywhere. In the case of the quantum mechanical description of a particle, the particle could be anywhere, but it is a particle. It is a chunk, it is a thing. And this wave is a probability wave in our knowledge or our command of the system. Other questions? OK. So let's do the first example of this kind of system. So let's do the very first example we talked about, apart from the free particle, which is the potential step. First example is the potential step. And what we want to do is we want to find the eigenfunctions of the following potential. Constant, barrier, constant. And I'm going to call the height here V naught, and the height here 0. And I'll call this position X equals 0. And I want to send in a wave-- think about the physics of sending in a particle that has an energy E naught. Now to think about the dynamics, before we think about time evolution of a localized wave packet, as we saw in the free particle, it behooves us to find the energy eigenstates, then we can deduce the evolution of the wave packet by doing a re-summation by using the superposition principle. So let's first find the energy eigenstates. And, in fact, this will turn out to encode all the information we need. So we want to find phi e of x. What is the energy eigenfunction with energy E, let's say. And we know the answer on this side and we know the answer on this side because it's just constant. That's a free particle and we know how to write the solution. So we can write this as this is equal to. On the left-hand side it's just a sum of plane waves. It's exactly of that form. a e to the i kx plus b e to the minus i kx on the left. And on the right we have c e to the minus i alpha x-- sorry. On this side it's disallowed, classically disallowed. So this is e to the minus-- sorry, do you-- plus alpha x plus d e to minus alpha x. Where h-bar squared k squared upon 2m is equal to e. And h-bar squared alpha squared upon 2m is equal to v naught minus e, the positive quantity. So we need to satisfy our various continuity and normalizability conditions. And in particular, what had the wave function better do out this way? AUDIENCE: [INAUDIBLE]. PROFESSOR: It had better not divert-- are we going to be able to build normalizable wave packets-- or are we going to be able to find normalizable wave functions of this form? No. Because there are always going to plane waves on the left. So the best we'll be able to do is find delta function normalizable energy eigenstates. That's not such a big deal because we can build wave packets, as we discussed before. But on the other hand, it's one thing to be delta function normalizable going to just a wave, it's another thing to diverge. So if we want to build something that's normalizable up to a delta function normalization condition, this had better vanish. Yeah? But on top of that, we need that the wave function is continuous, so at x equals we need that phi and phi prime are continuous. So that turns out to be an easy set of equations. So for phi, this says that a plus b is equal to d. And for phi prime, this says that ik a. And then the next one is going to give me minus ik a minus b is equal to, on the right-hand side, minus alpha c. Because the exponential's all value to 0. So we can invert these to get that d is equal to 2-- that's weird. AUDIENCE: You have a d in the bottom. There's [INAUDIBLE]. PROFESSOR: Oh, sorry. d. d. d. d. d. Thank you. To invert those you get 2k over k plus i alpha, I think? Yes. And b is equal to k minus i alpha over k plus i alpha. And so you can plug these back in and get the explicit form of the wave function. Now, what condition must the energy satisfy order that this is a solution that's continuous and derivative is continuous? Last time we found that in order to satisfy for the finite well, the continuity normalizability conditions, only certain values of the energy were allowed. Here we've imposed normalizability and continuity. What's the condition on the energy? This is a trick question. There isn't any. For any energy we could find a solution. There was no consistency condition amongst these. Any energy. So are the energy eigenvalues continuous or discrete? AUDIENCE: [INAUDIBLE]. PROFESSOR: Continuous. Anything above 0 energy is allowed for the system, it's continuous. So what does this wave function look like? Oh, I really should have done this on the clean board. What does this wave function look like? Well, it's oscillating out here, and it's decaying in here, and it's smooth. So what's the meaning of this? What's the physical meaning? Well, here's the way I want to interpret this. How does this system evolve in time? How does this wave function evolve in time? So this is psi of x. What's psi of x and t? Well, it just gets hit by an e to the minus i omega t minus omega t plus omega t minus i omega t. Everyone cool with that? So all I did is I just said that this is an energy eigenfunction with energy e, and frequency omega is equal to e upon h-bar. And saying that this is an energy eigenstate tells you that under time evolution, it evolves by rotation by an overall phase, e to the minus i omega t. And then I just multiplied the whole thing by e to the minus i omega t and distributed it. So here's the minus i omega t minus i omega t minus i omega t in the phase. So doing that, though, gives us a simple interpretation. Compared to our free particle, e to the i kx minus omega t, it's a wave moving to the right with velocity omega over k, the phase velocity. This is a right-moving contribution. So plus. And this is a left-moving contribution. Everyone cool with that? So on the left-hand side, what we have is a superposition of a wave moving to the right and a wave moving to the left. Yeah? On the right, what do we have? We have an exponentially falling function whose phase rotates. Is this a traveling wave? No. It doesn't have any crests. It just has an overall phase that rotates. And now here's one key thing. If we look at this, what's b? We did the calculation of v. v is k minus i alpha over k plus i alpha. What's the norm square root of b? Well, the norm square root of b is going to be multiply this by its complex conjugate, multiply this by its complex conjugate. But their each other's complex conjugates will cancel. The norm squared is 1. So pure phase. So this tells you b is a pure phase. So now look back at this. If b is a pure phase-- sorry-- if b is a pure phase, than this left-moving piece has the same amplitude as the right-moving piece. This is a standing wave. All it's doing is it's rotating by an overall phase. But it's a standing wave because it's a superposition of a wave moving this way and a wave moving that way with the same amplitude. Just slightly shifted. So the fact that there's a little shift tells you that it's norm squared is not constant. It has a small wave. So what we see is we get a standing wave that matches nicely onto a decaying exponential in this classically disallowed region. So what that tells you is what's the probability to get arbitrarily far to the right? 0. It's exponentially suppressed. What's the probability that your found some point on the left? Well, it's the norm squared of that, which is some standing wave. So the reflection of how good is this is a mirror. AUDIENCE: [INAUDIBLE]. PROFESSOR: It's a perfect mirror. Well, it's not-- Now we have to quibble about what you mean by perfect mirror. It reflects with a phase. It reflects with a phase beta, and I'm going to call that phase-- I want to pick conventions here that are consistent throughout. Let's call that e to the i phi. So it reflects with a phase phi, which is to say that [INAUDIBLE] squared is 1. And what happens-- I don't want to do this. So in what situation would you expect this to be a truly perfect mirror, this potential? When do you expect reflection to be truly perfect? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah, exactly. So if the height here were infinitely high, then there's no probability. You can't leak in at all. There's no exponential tail. It's just 0. So in the case that v0 is much greater than e, what do we get? We get that v0 is much greater than e. That means alpha's gigantic, and k is very, very small. If alpha is gigantic and k is very small, negligibly small, than this just becomes minus i alpha over i alpha. This just becomes minus 1 in that limit. So in the limit that the wall is really a truly hard wall, this phase becomes minus 1. What happens to a wave when it bounces off a perfect mirror? AUDIENCE: [INAUDIBLE]. PROFESSOR: Its phase is inverted, exactly. Here we see that the phase shift is pi. You get a minus 1 precisely when the barrier is infinitely high. When the barrier's a finite height, classically it would be a perfect mirror still. But quantum mechanically it's no longer a perfect mirror. There's a phase shift which indicates that the wave is extending a little bit into the material. The wave is extending a little bit into the classically disallowed region. This phase shift, called the scattering phase shift, is going to play a huge role for us. It encodes an enormous amount of the physics, as we'll see over the next few weeks. At the end of the semester it'll be important for us in our discussion of bands and solids. So at this point, though, we've got a challenge ahead of us. I'm going to be using phrases like this is the part of the wave function moving to the right, and that's the part of the wave function moving to the left. I'm going to say the wave function is a superposition of those two things. But I need a more precise version of stuff is going to the right. And this is where the probability density, phi squared, and the probability current, which you guys have constructed on the problem set, which is h-bar upon 2mi. Psi star dx i minus psi dx psi star. These guys satisfy the conservation equation which is d RHO dt. The time rate of change of the density at some particular point at time t is equal to minus the gradient, derivative with respect to x, of j. And I should really call this jx. So the current in the x direction. So this is the conservation equation, just like for the conservation of charge, and you guys should have all effectively derived this on the problem set. Yeah. AUDIENCE: Is the i under or above [INAUDIBLE]? PROFESSOR: Is the i-- oh, sorry, the i is below. That's just my horrible handwriting. So what I want to define very quickly is I want to be able to come up with an unambiguous definition of how much stuff is going left and how much stuff is going right. And the way I'm going to do that is I'm going to say that the wave function in general-- I'll do that here-- when we have a wave function at some point. It can be approximated if the potential is roughly constant at that point. It can be approximated in the following way. The wave function is going to be psi is equal to psi incident plus psi transmitted-- yeah, psi reflected on the left of my barrier. And psi transmitted on the right. Where psi incident is that right-moving part. The part that's moving towards the barrier. Psi reflected is the left-moving part that's reflected. Psi transmitted is the part that is on the right-hand side, the entire thing. So here the idea is I'm sending in something from the left. It can either reflect or it can transmit. So this is just my notation for this for these guys. And so what I wanted to measure of how much stuff is going left, how much stuff is going right. And the way to do that is to use the probability current. And to say that associated to the incident term in the wave function is an incident current, which is h-bar k upon m from a squared. So if we take this wave function, take that expression, plug it into j, this is what you get. If you take the reflected part b to the minus i kx minus omega d, and you plug it into the current expression, you get the part of-- sorry, I should call this capital J-- you get the part of the current corresponding to the reflected term, which is h-bar k with a minus k upon m v squared. And we'll get that J transmitted is 0. Something that you proved on your problem set. If the wave function is real, it must-- the current to 0. So let's think about what this is telling us quickly. So what is this [INAUDIBLE]? So what is a current? A current is the amount of stuff moving per unit time. So charge times the velocity. So what's the stuff we're interested in? It's the probability density. There's a squared. Norm a squared. That's the probability amplitude, or probability density of the right-moving piece. Just consider it in isolation. So that's the probability density. It's the amount of stuff. And this is the momentum of that wave, that's e to the i kx. There's h-bar k. That's the momentum divided by the mass, which is the classical velocity. The amount of stuff times the velocity. That a current. It's the current with which probability is flowing across a point. Similarly, Jr, that's the amount of stuff, and the velocity is minus h-bar k upon m. So again, the current is the amount of stuff times the velocity. I'm going to define-- this is just the definition, it's the reasonable choice of definition-- the transmission probability is the ratio, the norm squared of J transmitted over J incident. And here's why this is the reasonable thing-- oops-- incident. Here's why this is the reasonable quantity. This is saying how much current is moving in from the left? How much stuff is moving in from-- And how much stuff is moving off on the right, to the right? So what is this in this case? 0, although this is a general definition. And similarly, the reflection you can either write as 1 minus the transmission, because either you transmit or you reflect, the probability must sum to 1. Or you can write r as the current reflected. What fraction of the incident current is, in fact, reflected? And here this is 1. Because beta-- b is a phase, and the norm squared of a phase is 1. So on your problems set, you'll be going through the computation of various reflection and transmission amplitudes for this potential. And we'll pick up on this with the barrier uphill next week. Have a good spring break, guys.
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_14_Resonance_and_the_SMatrix.txt
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the following content is provided under a creative commons license your support will help mit opencourseware continue to offer high quality educational resources for free to make a donation or to view additional materials from hundreds of mit courses visit mit opencourseware at ocw.mit.edu so today we're going to do our last uh lecture on scattering in quantum mechanics in 1d quantum mechanics and we're going to introduce some powerful ideas in particular the phase shift and the s matrix and they're cool we'll use them for good before we get started questions from last time just now it's a band called saint germain um it's actually a guy but he's refers to himself as a band called cengermain anyway it's from an album i think called traveler physics questions [Music] anyone okay good um well bad actually i'd much referred if you had questions but i'll take that as a sign of of knowledge competence and mastery okay so yeah so so last time we talked about uh scattering past a barrier with some height l and some height v which i think we called v naught and we observed a bunch of nice things about this first off we computed the probability of transmission across this barrier oops the probability of transmission across this barrier as a function of the energy and it had a bunch of nice features first off um it asymptoted to one at high energies which makes sense things shouldn't really care if you have a little tiny barrier it went to zero at zero energy uh that i don't know how uh i think we should all find that obviously if you have extremely low energy you're just going to bounce off this very hard wall in between though there's some structure and in particular uh at certain values of the energy corresponding to certain values of the wave number in the barrier region at certain values of the energy we saw that the transmission was perfect oops this should have been here sorry the transmission was perfect at special values of the energy and that the reflection at those points was zero you transmitted perfectly but meanwhile the reflection which is one minus the transmission hit maxima at special points and those special points turn out to be half integer shifted away from the special points for perfect transmission so at certain points we have perfect transmission at certain points we have extremely efficient reflection and so one of our goals today is going to be to understand this physics the physics of resonance in scattering off of a potential okay so the asymptotes we understand classically that makes sense classically that makes sense but classically this is a really weird structure to see and notice that it's also not something that we saw when we looked at scattering off of a simple step when we looked at a simple step what we got was something that looked like zero and then this right there was no structure it was just a nice simple curve so something is happening when we have a barrier as opposed to a step so our job is going to be in some sense to answer why are they so different before i get going questions about the step barrier yeah is that just scale oh sorry um this one this that was the transmission sorry i should have drawn it separately that was the transmission as um as a function of energy across a single step and so that looked like this exactly oh oh sorry so so i just meant to compare the two i just want to think of them as two different systems in this system we have the transmission curve is a nice simple curve it has no structure in the case of a step barrier we get non-zero transmission at low energies instead of going to zero and we get this resonant structure so it's just for contrast okay so before before going into that in detail i wanna do one slight variation of this problem and i'm not going to do any of the computations i'm just going to tell you how to get the answer from the answers we've already computed so consider a barrier a square well you guys have solved the finite square well in your problem sets consider finding square well again of width l and of depth now minus v naught so it's the same thing with v goes to minus v okay and again i want to consider scattering states so i want to consider states with positive energy energy above the asymptotic potential now at this point by now we all know how to solve this we write plane waves here plane waves here we solve and plane waves in between we solve the potential in each region where it's constant and then we impose p uh continuity and of the wave function and continuity the derivative of the wave function at the matching points right so we know how to solve this problem and deduce the transmission reflection coefficients we've done it for this problem and it's exactly the same algebra and in fact it's so exactly the same algebra that we can just take the results from this one and take v to minus v and we'll get the right answer you kind of have to so if we do what we get for the transmission probability and now this is the transmission probability for a square well and again i'm going to use the same dimensionless constants g naught squared is equal to 2m l squared over h bar squared times v naught this is 1 over an energy this is an energy this is dimensionless and the dimensionless energy epsilon is equal to e over v to express my transmission probability life is better when it's dimensionless so t the transmission probability is again it's one of these horrible one overs one over 1 plus 1 over 4 epsilon epsilon plus 1. good sine squared of g naught square root of epsilon plus 1. and this is again the same so if you remember last time you can see what we got last time by now taking v to minus v again that takes epsilon to minus epsilon so we get out epsilon a one minus epsilon or epsilon minus one picking up the minus sign from this epsilon um and here we get a one minus epsilon instead of one plus epsilon that's precisely what we got last time so if you're feeling punchy at home tonight check this in some sense you're going to rederive it on the problem set okay so it looks basically the same as before we have a sine function downstairs which again will sometimes be zero the sine will occasionally be b0 when it's argument is a multiple of pi and when that sine function is zero then the transmission probability is one over one plus zero it's also known as one transmission is perfect so we again get a resonant structure and it's in fact exactly the same plot or almost exactly the same plot so um i'm going to plot transmission let's go ahead and do this transmission probability as a function again of the dimensionless energy and what we get is again oops not very well drawn resonances where transmission goes to one thanks to my beautiful artistic skills points where the resonance or the where the reflection hits a local maximum okay but we know one more thing about this system which is that in addition to having scattering states whose transmission probability are indicated by this plot we know we also have for negative energy bound states right unlike the step barrier for the step well we also have bound states so here's the transmission curve but i just want to remind you that we have energies at special values of energies we also have bound states and precisely what energies depends on the structure of the well and the depth but if the depth of the well is say v naught so this is minus one we know that the lowest bound state is always greater energy than the bottom of the well cool so epsilon remember is e over v naught and in the square well of depth v naught the lowest bound state cannot possibly have energy lower than the bottom of the well so its epsilon must have epsilon greater than one or minus one cool anyway this is just to remind you that there are bound states and um think of this like a gun put down on a table in a play in act one okay it's that dramatic so it will show up again it will come back and play well with us so i want to talk about these residences so let's understand why they're there so there are a bunch of ways of understanding why these resonances are present and let me just give you a couple so first a heuristic picture and then i want to give you a very precise computational picture of why these resonances are happening so the first is imagine we have a state where the transmission is perfect what that tells you is kl is a multiple of pi so in here so if this is the distance l of the well in here the wave is exactly one period or if it's kale is equal to two to a multiple of pi um so if kl is pi whoops i think i want do i want two good let me simplify my life and consider the case kale is two pie so when the wave comes through and becomes oh i know why shoot so for this sorry four pi so um i switched notations for you i was using in my head uh and or rather in my notes uh width of the well is 2l from um uh for reasons you'll see on the notes if you look at the notes but let's ignore that let's focus on these guys so consider the state a configuration in energy such that in the well we have exactly one period of the wave function that means that the value of the wave function at the two ends is the same and the slope is the same so whatever the energy is out here if it matches smoothly continuously and its derivative is continuous here it will match smoothly and continuously out here as well with the same amplitude and the same period inside and outside right the amplitude is the same in the phases the same that means this wave must have the same amplitude in the same period well it has to have the same period is the same energy but must have the same amplitude and the same slope at that point and that means that this wave has the same amplitude as this wave the norm squared is the transmission probability norm squared of this divided by the normal square of this that means the transmission probability has to be one what would have happened had the system instead of being perfectly periodic inside the well well let me actually let me leave that up so we have let's do it as this oh shoot i'm even getting my qualitative wave functions wrong let's try this again okay start at the top go down and then it's got a lower uh uh it's deeper inside the well so the amplitude out here the the potent the difference in energy between the energy and the potential is less which means the period is longer and the amplitude is greater but it's the way i've drawn it it's got a particular value it's got zero um zero uh uh uh derivative at this point so it's got two okay good so there's our wavefunction same thing out here but the important thing is the amplitude here has to be the same as the amplitude here because the amplitude and the amplitude in the phase were exactly as if there had been no intervening region wouldn't agree with that by contrast if we had looked at a situation where inside the well it was not the same amplitude so for example something like this came up with some slope which is different in a value which was different then this is going to match onto something with the same period but with a different amplitude than it would have over here same period because it's got the same energy but a different amplitude because it has to match on with the value and the derivative so you can think through this and pretty quickly convince yourself of the necessity of the transmission amplitude being one if this is exactly periodic again if this is exactly one period inside you can just imagine this is gone and you get a continuous wave function so the amplitude and the derivative must be the same on both sides as if there is no barrier there and when the phases uh when the phase is not same here when the wave function doesn't have exactly one period inside the well you can't do that so the amplitudes can't be the same on both sides okay but that's not a very satisfying explanation that's really an explanation about solutions to the differential equation i'm just telling you properties of second-order differential equations let's think of a more physical more quantum mechanical explanation why are we getting these resonances well i want to think about this in the same way as we thought about the boxes in the very first lecture suppose we have this square well and i know i have some amplitude here and i want to know with some i've got a wave function it's got some amplitude here it's got some momentum going this way some positive momentum and i want to ask what's the probability that i will scatter past the potential let's say to this point right here just on the other side of the potential what's the probability that i will scatter across well you say we've done this calculation we know the probability to transmit across this step potential we did that last time so that's t step and we know the probability we know what happens to the phase of the wave function or sorry we know the probability of scattering across this potential step and that's transmission uh up and so the probability that you transmit from here to here is the probability you transmit here first and then the probability that you transmit here right the product of the probabilities sound reasonable okay let's vote how many people think that this is equal to the transmission probability across the potential well any votes you have to vote one way or the other no it is not how many people vote okay yes it is okay the nose have it and that's not terribly surprising uh because of course these have no resonance structure so where did that come from if it's just that thing squared right so that probably can't be it but here's the better here's the the bigger problem why is this the wrong argument because there's reflection yeah exactly there's reflection but that's only one step in the answer of why it's the wrong answer why else uh your transmission definitions have to do far away that's true but i just want the probability if i got here with positive momentum i'm eventually going to get out to infinity so it's the same probability because it's just an e to the ik x out here the wave function is just e to the ikx so the probability is going to be the same other well the reasons of the well is very important but the first argument ignores that exactly that's also true so far the reasons we have are we need the width of the well doesn't appear that seems probably wrong okay uh the second is it's possible that you could have reflected we haven't really incorporated that in any sort of elegant way what's there other ways that's absolutely true there are other ways to transmit so you could transmit then you could reflect so we could transmit then reflect and reflect again and then transmit so we could transmit reflect reflect transmit what else do probabilities add in quantum mechanics and when you have products of events do your probabilities multiply what adds in quantum mechanics the amplitude the wave function we do not take the product of probabilities what we do is we ask what's the amplitude to get here from there and we take the amplitude norm squared to get the probability right so the correct question is what's the amplitude to get from here to here how does bless you how does the wave function of the amplitude change as you get as you move from here to here and for that think back to the two-slit experiment or think back to the boxes we asked the following question the transit the amplitude that you should transmit across this well has a bunch of components is a sum of a bunch of terms you could transmit down this well transmit down the down as well inside here you know your wavefunction is e to the i k prime l or i think i'm calling this k2 l uh yes key2 x and in moving across the well your wave function evolves by an e to the i k 2 l and then you could transmit again with some transmission amplitude so this would be the transmission down times e to the i k 2 l times the transmit up as we saw last time these are the same but i just want to keep them separate so you know which one i'm talking about so this is a contribution this is something that could contribute to the amplitude is it the only thing that could contribute to the amplitude no what else could contribute bounce right so to get from here to here i could transmit evolve transmit i could also transmit evolve reflect evolve reflect evolve transmit so there's also a term that's t e to the i k l r e to the minus i k l r e to the i k l sorry e either the plus ical because we're increasing the evolution of the phase um and then transmit finally at the end and these k's are all k2s k2 p2 but i could have done that many times but notice that each time what i'm going to do is i'm going to transmit reflect reflect transmit or transmit reflect reflect reflect reflect transmit so i'm always going to do this some even number of some number of times i do this once i do this twice i do it thrice this gives me a geometric series this is t e to the i k 2 x t times 1 plus this quantity plus this quantity squared that's a geometric series 1 over 1 plus this quantity squared or sorry 1 minus because it's a geometric sum and what is this quantity well it's r squared and remember the r's in both directions are the same so i'm just going to write it as r squared e to the 2i well r squared it's a real number but i'm going to put the abs on absolute value on anyway it's going to simplify my life you need the 2i k2l okay so this is our prediction from multiple bounces what we're doing here is we're taking seriously the superposition principle that says given any process any way that that process could happen you sum up the amplitudes and the probability is the norm squared if we have a source and we have two slits and i ask you what's the probability that you land here the probability is not the sum of the probabilities for each individual transit the probability is the square of the amplitude where the amplitude is the sum amplitude top plus amplitude bottom here there are many many slits there are many different ways this could happen you could reflect multiple times ever cool with that no by the same token we could have done the same thing for reflection but let's stick with transmission for the moment this is what we get for the transmission and again the transmission amplitudes across the step as we saw last time are the same so this is in fact t squared and this gives us a result for transmission down the potential well and if we put in the factors of if we use what uh the trench sorry if we use what the reflection and transmission amplitudes were for our stepwells the answer that this gives is 1 over e to the i k2l minus 2i upon the transmission for a step times sine of k prime l or sorry k2l now this isn't the same as the probability that we derived over here but that's because this isn't the probability this was the amplitude we just computed the total amplitude to get the probability we have to take the norm squared of the amplitude and when we take the norm squared what we get is 1 upon 1 plus 1 over 4 epsilon epsilon minus 1 or plus 1. sine squared of g naught root epsilon plus one we get the same answer sorry this is inequality oh sorry sorry sorry thank you martin we get that the transmission probability when we take the norm square of the amplitude thank you is equal to this which is the same as we got before thanks okay so this is to me this is very sad yeah please um kl equals pi doesn't work it does you just have to be careful what kl expressing it sorry what am i um because i'm an idiot because i got a factor of two wrong thank you thank you matt thank you um answer analysis it's a wonderful thing um okay so this gives us this has something really nice for us why are we getting a resonance at special values why are we getting a resonance of special values of the energy what's happening well in this process in this quantum mechanical process of multiple interactions multiple scatterings there are many terms in the amplitude for transmitting there are terms that involve no reflection there are terms that involve two reflections they're terms that involve four reflections and they all come with an actual magnitude and a phase and when the phase is the same they add constructively and when the phases are not the same they interfere and when the phases are exactly off they interfere destructively and that is why you're getting a resonance multiple terms in your superposition interfere with each other something that does not happen classically classically the probabilities are products quantum mechanically we have superposition and probabilities are the squares of the amplitude and we get interference effects and the probabilities cool so to me this is a nice sort of glorious version of the two-slit experiment um and we're gonna bump up again into it later when we talk about the physics of solids uh in the real world okay questions at this point yeah you said earlier yeah what's special about that so what's special about that is so at this point where kale is in pi when kale is is n pi we get perfect transmission when kl is equal to n plus one half pi the reflection is as good as it gets okay so what that's really doing is it's saying when is this largest locally largest okay so that's these special points when we're the transmission is as small as possible which means the reflection which is one minus the transmission probabilities as large as possible and you can get that again from from these expressions other questions okay so so stopping so i want to think a little more about these square barriers and in particular in thinking about the square well barrier what we've been talking about all along are monochromatic wave packers we've been talking about plane waves just simple e to the ikx but you can't actually build uh you can't put a single particle in a state which is a plane wave right it's not normalizable what we really mean at the end of the day when we talk about single particles is we put them in some well localized wave packet which at time zero let's say is at position x naught or maybe well x naught which in this case is negative and which has some well-defined average momentum cannot i'll say it's the expectation value of momentum in this wave package and the question we really want to ask when we talk about scattering is what happens to this beast as it hits the weight as it hits the barrier which i'm going to put the left-hand side at zero on the right-hand side at l and let the depth be minus v-naught again what happens is this incident wave packet hits the barrier and then scatters off well we know what would happen if it was a plane wave but a plane wave wouldn't be localized so this is the question i want to ask and i want to use the results we already have now here's the key thing if we had consider for to begin with just a wave packet for a free particle centered at x naught and with momentum k naught okay just for a free particle we know how to write this and if this is um we can put the system for example we can take our wave function at time 0 to be a gaussian some normalization times e to the minus x minus x naught squared over 2a squared and we want to give it some momentum k naught and you know how to do that after the last problem set e to the i k naught x everyone call that so there's our initial wave function and we want to know how it evolves with time and we know how to do that to evolve in time we first expand it in energy eigenstates so psi of x naught is equal to well the energy i can states in this case are plane waves dk e to the ikx over root 2 pi times some coefficients f of k the expansion coefficients but these are just the fourier transform of our initial gaussian wave packet and we know what the form of f 0 is f 0 or sorry f of k is equal to well it's a gaussian of width 1 upon alpha e to the minus k minus k naught because it has momentum k naught so it's centered around k naught squared over 2 times a squared and the a goes upstairs and the position of the fourier of the uh the initial wave packet is encoded in the fourier transform and i'm going to put a normalization here which i'm not going to worry about with an overall phase e to the i um minus i k x naught so in just the same way that adding on a phase e to the aik not x in the position space wave function tells you what the expectation value of momentum is tacking on the phase and we can get this from just fourier transform attacking on the phase e to the ik x naught tells you in the fourier transform tells you the center point the central position of the wave pack okay so this you did on the problem set and this is a trivial momentum space version of the same thing so here's our wave packet expanded in plane waves which are energy eigenstates and the statement that these are energy eigenstates is equivalent to the statement that under time evolution they do nothing but rotate by a phase so if we want to know what the wave function is a function of time psi of x and t is equal to the integral dk the fourier mode uh f of k i'll write it out actually i'm going to write this out explicitly f of k 1 over root 2 pi e to the i k x minus omega t where omega of course is a function of k it's a free particle so for our free particle uh h bar squared omega squared oh sorry h bar squared k squared upon 2m is equal to uh h bar omega okay so this is we've done this before but now what i want to do is i'm going to take exactly the same system and i want to add at position 0 a well of depth v naught and width l how is our story going to change well we want our initial wave packet which is you know up to us to choose we want our initial wave packet to be the same we want to start with the gaussian far far far away from the barrier we want it to be well localized in position and well localized in momentum space not perfectly localized of course it's a finite gaussian to satisfy the uncertainty principle but it's well localized so how does the story change well this doesn't change at all it's the same wave function however when we expand in energy eigenstates the energy eigenstates are no longer the energy eigenstates are no longer simple plane waves the energy eigen states as we know for the system take a different form for the square well and positive energy scattering states the plane wave or the energy eigenstates which i will label by k just because i'm going to call e is equal to h bar squared k squared upon 2 m which is the energy well the energy is constant which is the k asymptotically far away from the potential the wave function i can write as 1 over root 2 pi times e to the ikx when we're on the left hand side but left being on the left-hand side is equivalent to multiplying by a theta function of minus x minus x this is the function which is zero when its argument is negative and positive when its um argument is and one when its argument is positive plus we have the reflected term which has an amplitude r which is again a function of k this is the reflection amplitude also known as c upon a e to the minus psi kx again on the left hand side theta of minus x plus a transmission amplitude whose norm squared is transmission probability e to the ikx when we're on the right uh x whoops theta of x okay so this is just a slightly different notation than what we usually write with left and right separated so what we want to do now is we want to decompose our wavefunction in terms of the actual energy eigenstates and so the way we're going to do that and having done that having expanded our wave function in this eigens in this basis um we can determine the time evolution in the following way first we compute we expand the wave function at time 0 as an integral dk and i'm going to pull the root 2 pi out and we have some for a transform which i'm not going to call f tilde because it's slightly different than the f we used before but it's what it's what the expansion coefficients have to be f tilde of um yeah sorry f tilde of k times this beast phi sub k 5k good and let me actually write this out in terms of the let me actually take this product and write it out in terms of the three terms so those three terms are going to be f of k again tilde e to the i kx theta minus x plus f tilde r e to the minus ikx theta minus x plus f tilde uh t e to the i k x theta of x okay so let's look at these terms this is this wave function now oh and sorry finally we want to look at the time evolution but this we started out as a superposition of states with definite energy labeled by k so we know the time evolution is e to the i k x minus omega t e to the i minus i k x plus omega t that's minus i omega t and k x minus omega t so we can immediately from from this time evolving wave function identify these two terms as terms with with a central peak moving to the right and this a central peak moving to the left kx plus omega t everyone cool with that questions or just where this came from oh good good good okay so this is just a notational thing so usually when i say a uh phi sub k is equal to on the left e to the ikx uh with some overall amplitude a let's say 1 over root 2 pi e to the ikx plus c over a e to the minus ikx right on the left and i'm just going to call this because this is the reflected wave i'm just going to call this r okay and then on the right we have e to the ikx there's only a wave traveling to the right and the coefficient is the transmission amplitude so this is what we normally write but then i'm using the function the theta function so called theta function theta of x is defined as zero when x is less than zero and one when x is greater than zero okay so this is a function one so this allows me using theta function allows me to write this thing as a single function without having to goose around with lots of terms is that cool great i will just take into account because otherwise we should have something excellent excellent thank you so much so here what i want to do is thank you what i want to do is i want to think of the wave function so this is a good description when the particle is far far away and this is a good description when the particle is not in the potential so i want to just imagine this sorry thank you i totally glossed over this step i want to imagine this as a potential where all of the matching is implemented at x equals zero okay um so when i write it in this fashion but the better way to another equivalent way to think about this is this is a good description of the wave function when we're not inside the well so for the purposes of the rest of the analysis that i'm going to do this is exactly what the form of the wave function is when we're not inside the well okay and then let's just not use this to ask about questions inside the well when i write it in this theta form or for that matter when i write it in this form this is not the the form what i mean is left of the well and right of the well and inside t e to the ikx inside it's doing something else but we don't want to ask questions about it okay cool that's just going to simplify my life root of capital yeah it's the square root of capital t but you've got to be careful because there can be a phase remember this is the amplitude and what it really is is this reflection is b over a and these guys are complex numbers and it's true that b over a norm squared is the transmission probability but b over a has a phase and that quantity i'm going to call t and we'll interpret that in more detail in a few minutes okay okay so it's of this form and i just want to look at each of these three terms and in particular i want to focus on them at time 0. so at t equals 0 what does this look like well that first term is integral dk of f of k e to the ikx minus omega t okay theta minus x notice that theta minus x is independent of k it's independent of the integral so this is just a function times theta of minus x okay and this function was constructed to give us the initial gaussian so this is just oh sorry and this was at t equals 0. so this is just our gaussian at position x naught centered around value k naught at time equals zero theta of minus x okay this is just saying that um sorry at x at position x sorry of x naught so this all this function is is the this is the fourier transform of our gaussian and we're undoing the fourier transform so this is just giving us our gaussian back and as long as the particle is far away from the well so here's the well and here's our wave packet uh that theta function is totally irrelevant because the gaussian makes it zero away from the center of the gaussian anyway and so just as a quick question if we look at this as a function of t so we put back in the minus omega t so if we put back in the time dependence this is a wave that's moving to the right and we need to make to be more precise about that what does it mean to say it's a wave moving to the right well this is an envelope on this set of plane waves and the envelope by construction was well localized around position x but it was also well localized in momentum and in particular the fourier transform is well localized around the momentum k naught and so using using the method of stationary phase or just asking where is the where is the phase constant stationary we get that the center of the the peaks the the central peak of the wave function satisfies the equation ddk naught if you're not familiar with stationary phase let the restoration instructors uh know and they will discuss it for you um so this the the points of stationary phase of this uh this superposition of this wave packet lie at the position ddk naught of kx sorry ddk of kx minus omega of kt evaluated at the peak k naught of the distribution but ddk of kx minus omega t is for the first term x and for the second term ddk of omega well we know that omega we're in the free particle regime um omega where did it go um well we all know what it is it's h bar h bar k squared upon 2 m we take a derivative with respect to k we get h bar k over m the twos cancel so x minus h bar k over m t and a place where this point where this phase is zero where the phase is stationary moves over time as x is equal to h bar k over m t but h bar k over m that's the class that's the momentum p over m that's the classical velocity so this is v naught t the velocity associated with that momentum so this is good we've done the right job of setting up our wave packet we built a gaussian that was far away that was moving in with a fixed velocity towards the barrier and now we want to know what happens after it collides off the barrier cool so what we really want to ask is that late times what does the wave function look like well at late times the position of this so again we're focusing on the first term the position of this wave packet at late times at positive t is positive and when it's positive then this theta function kills its contribution to the overall wave function right this theta is now theta of minus a positive number and this gaussian is gone what is it replaced by well these two terms aren't necessarily zero in particular this one is moving to the left so x is as time goes forward x is moving further and further to the left and so this theta function starts turning on and similarly as x goes positive this theta function starts turning on as well let's focus on this third term transmitted term so let's focus on this third term so in particular that term looks like integral dk over root 2 pi f times the transmission amplitude times e to the i kx minus uh omega t okay and there's an overall theta of x outside but for late times where the center of the wave packet for the transmitted wave packet should be positive the state is just going to be one so we can safely ignore it it's just saying we're far off to the right and now i want to i want to do one last thing this was an overall this was an overall envelope this t was our scattering amplitude and i want to write it in the following form i want to write it as root t e to the minus i phi where phi is okay so what this is saying is that indeed as was pointed out earlier the norm squared of this coefficient t is the transmission probability but it has a phase and what i want to know is what does this phase mean what information is contained in this phase and that's what we're about to find so let's put that in here we have root t and minus phi okay where omega and k and phi are both functions of k because the transmission amplitudes depend on the momentum or the energy okay and now i again want to know how does this wave packet move if i look at this wave packet how does it move on mass how does it has a group of waves how does this wave packet move in particular with y velocity so i'll do again make an argument by stationary phase i'll look at a point of of phase equals 0 and ask how it moves over time and the point of stationary phase is again given by ddk of the phase kx minus omega of kt minus 5k is equal to zero evaluated at k0 which is where our envelope is sharply peaked so this expression is equal to this is again x minus d omega dk that's the classical velocity v naught t minus d phi d k but just as a note d phi d k is equal to d omega d k d phi d omega but this is equal to this is just the chain rule the omega dk that's the classical velocity d omega or defined the omega that's d phi d e times h bar i just multiplied by h bar on the top and bottom so this is equal to x minus v naught t plus h bar d phi d e so first off let's just make sure that the units make sense that's a length that's a velocity so they said better have units of time time that's good so h bar times d phase over d e well a phase is dimensionless energy is units of energy h is energy times time so this dimensionally works out so this is zero so the claim is that the point of stationary phase has this derivative equal to zero so setting this equal to zero tells us that the the peak of the wave function moves according to this equation and so now so this is really satisfying this should be really satisfying for a couple of reasons first off it tells you that the peak of the transmitted wave the peak of the transmitted wave packet not just a plane wave the peak of the actual wave packet well localized moves with overall velocity v naught the constant velocity that we started with and that's good if it was moving with some other velocity we would have lost energy somehow that would be not so sensible so this wave packet is moving with an overall velocity v naught however it doesn't move uh at the same it doesn't move along just as the original wave packet's peak had it moves as if shifted in time okay so the phase and more to the point the gradient of the phase with respect to energy the rate of change with energy of the phase times h-bar is giving us a phase a shift in the time of where the wave packet is so what does that mean so let's be precise about this so so let's interpret this and here's the interpretation i want to give you so in the absence so consider it classically consider classically the system classically we have an object with some energy and it comes along it rolls along and it finds a barrier a potential well and what happens when it gets into the potential well it speeds up right because it's got a lot more energy relative to the potential so it goes much faster in here and it gets to the other side it slows down again so if i had taken a particle with velocity and so let's call this position zero let's say it gets to this wall at time zero so it's moving with x is equal to v naught t if there had been no barrier there then at subsequent times it would get out here in a time x that distance over v naught right however imagine this well was extraordinarily deep if this well were extraordinarily deep what would happen well basically in here its velocity is arbitrarily large and it would just immediately jump across this well yeah so its velocity so this would be a perfectly good description of the of the motion before it gets to the well but after it leaves the well the position is going to be v naught t plus well what's the time shift it the time shift is the time that we didn't need to cross this gap and how much would that time have been that time would have been well it's the distance divided by the velocity right so that's the time we didn't need so t plus it moves as if it's at a later time t plus l over v naught cool so if we had a really deep well and we watched a particle move we would watch it move x naught x is v not t v not t v not t v naught t plus l over v v naught t plus over v okay so it's the time that we made up by being deep in the well so there's a classical picture because it goes faster inside so comparing these here we've done a calculation of the time shift due to the quantum particle quantum mechanically transverse transiting the potential well yeah so let's compare these the classical prediction this says the classical prediction is um the time it took delta t classical is equal to l over v naught and the question is is this the same so one way to phrase this question is is the same as h bar d phi d e evaluated at uh k naught okay and from our results last time for the amplitude c over a we get that phi is equal to which is just the minus the uh the argument or the phase of c over a or of the transmission coefficient little transition amplitude little t phi turns out to be equal to k2 l minus arctan of k1 squared plus k2 squared over 2 k1 k2 tan of k2l okay i look at this and it doesn't tell me all that much it's a little bit bewildering so let's unpack this what we really want to know is is this close to the classical result okay so here's a quick way to check this is we know this expression is going to simplify near resonance where the sign vanishes so let's look just for simplicity near the resonance and in particular let's look near the resonance k2 l is equal to n pi then it turns out that a quick calculation gives you that h bar d phi d e at this value of k at that value of the energy goes as l over 2 v naught times 1 plus whoops 1 plus the energy over the depth of the well v naught now remember the classical approximation was l over v naught we just did this very quickly we did it assuming an arbitrarily deep well so v naught is arbitrarily larger in magnitude than e so this term is negligible so where we should compare that very simple naive classical result is here l over two v naught and what we see is that the quantum mechanical result gives a time shift which is down by a factor of two okay so what's going on well apparently the things slowed down inside the time the time that it's that it took us to cross was greater than you would have naively guessed by making it arbitrarily deep and we can make that a little more sharp by plotting as a function of e over v naught the uh the actual phase shift so the classical prediction if you do better job than saying it's infinitely deep the classical prediction looks something like this okay and this is for delta t the time shift classical when you look at the correct quantum mechanical result here's what you find oops where the difference is a factor of two one-half the height down and again one-half a height down so this is that factor of two downstairs so the wave packet goes actually a little bit faster than it would than the classical prediction would guess except near resonance and these are at the resonant values of the momentum at the resonant values of the momentum it takes much longer to get across instead of going a little bit faster than the classic result it goes a factor of two slower than the classical result and so now i ask you the question why is it going more slowly why does it spend so much more time inside the well quantum mechanically than it would have classically why is the particle effectively taking so much longer to transit the well near resonance yeah exactly so the classical particle just goes across the quantum mechanical particle has a superposition of contributions to its amplitude where it transits transit bounce bounce transit transit bounce bounce bounce bounce transit and now you can ask how much time was spent by each of those imaginary particles imaginarily moving across and if you're careful about how you set up that question you can recover this factor of two which is kind of beautiful but the important thing here is when you're hitting resonance the multiple scattering processes are important they're not canceling out they're not at random phase they're not interfering destructively with each other they're interfering constructively and you get perfect transmission precisely because of the constructive interference of an infinite number of contributions to the quantum mechanical amplitude okay and this is again we're seeing the same thing in this annoying slowdown and this tells us another thing though which is that the phase the scattering phase the phase in the transmission amplitude contains an awful lot of the physics of the system it's telling us about how long it takes for the wave packet to transit across the potential effectively yeah sorry the vertical axis here is the time that it takes the the shift in the time due to the fact that it went across this well and went a little bit faster inside okay so empirically what it means is when you get out very far away and you watch the velocity the motion of the wave packet and you ask how long has it been since it got to the barrier in the first place it took less time than you would have guessed by knowing that its velocity is v naught and the amount of time less is this much time that answer your question good of uh one yeah it bounces around twice um uh so so the the classical amplitude is so when we compare the classical length to the limit that v naught goes to infinity so the the comparison is just a factor of two okay it's more complicated out here and i guess it's not it's it in the limit that v naught is large this is exactly one half well it's got the yeah it's got the the resonances are leading to this extra factor of one half um and so as you i i don't i have to say i don't remember exactly whether as you um include the the subleading terms of one over v naught whether it stays one half or whether it doesn't but in the limit that v naught is large it remains either close to one half or exactly one if i just don't remember exactly the important thing is that there's a sharp dip it takes much longer to transit and so you get less bonus time as it were uh you've you've gained less time in the quantum mechanical model than the classical model and when you do the experiment you get the quantum result right so that's uh that's the crucial point other questions okay so the phase contains an awful lot of the physics so i want to generalize this whole story in a very particular way and this way of reorganizing the scattering in 1d so what we're doing right now is we're studying scattering problems in one dimension but we live in three dimensions the story's going to be more complicated in three dimensions it's going to be more complicated in two dimensions but the basic ideas are all the same it's just the details are going to be different and one thing that turns out to be very useful in organizing scattering both in in one dimension and in three dimensions something called the scattering matrix and i'm going to talk about that now in three dimensions it's almost i mean it's essential but even in one dimensions where it's usually not used it's a very powerful way to organize our knowledge of the system as encoded by the scattering data so here's the basic idea as we discussed before what we really want to do in the ideal world is take some unknown potential in some bounded region some region and outside we have the potential is constant modula my bad artistic skills so potential is constant out here and the potential could be some horrible thing in here that we don't happen to know and we want to read off of the scattering process we want to be able to do something about the potential so for example we can deduce the um the uh the energy of our step um or we can deduce the energy by looking so good we can reduce the energy by looking at the um the position of the barriers and we can disentangle the position of the energy and the uh the width the depth and the width by looking at the phase shift by looking at the time delay so we can deduce all the parameters of our potential by looking at the resonance points and the phase shifts i want to do this more generally for a general potential and to set that up we need to be a bit more general than than we've been okay so in general if we solve this potential as we talked about before we have a and d oh sorry a plus b i e to the i k x minus i k x out here and out here we have c e to the i k x and d e to the minus sign and i'm not going to ask what happens inside now we can do as discuss two kinds of scattering experiments we can send things in from the left in which case a is non-zero and then things can either transmit or reflect but nothing's going to come in from infinity so d is zero or we could do the same in reverse send things in from here and that corresponds to a zero nothing coming in this way but d non-zero and more generally we can ask the following question look suppose i send some amount of stuff in from the left and i send some amount of stuff in from the right d then that will tell me how much stuff will be going out to the right and how much stuff will be going out to the left yeah if you tell me how much is coming in i will tell you how much is coming out bc so if you could solve this problem the answer is just some relation some pair of linear relations between these and we can write this as a matrix which i will call s11 s12 s21 s22 okay what this matrix is doing is it takes the amplitude you're sending in from the left or right and tells you the amplitude coming out to the left or to the right yes how would we know this relation is linear uh if you double the amount of stuff coming in then you must double the amount of stuff going out or probability is not conserved also we've derived these relations you know how the relations work the relations work by satisfying a series of linear equations uh between the various coefficients such that you have continuity and differentiability at all the matching points but the crucial thing of linearity is you need probabilities conserved and time evolution is linear so other questions okay so this is just some stupid matrix and we call it not surprisingly the s matrix in all of its majesty um and the basic idea is this for scattering problems if you if someone tells you the s matrix and in particular if someone tells you how all the coefficients of the s matrix vary with energy then you've completely solved any scattering problem you tell me what a and d are great i'll tell you exactly what b and c are mode by mode and i can do this for a superposition so this allows you to completely solve any scattering problem in quantum mechanics once you know it so it suffices to know s to solve all scattering problems so let's i want to now spend just a little bit of time thinking about what properties the s matrix and its components must satisfy what things what properties must the s matrix satisfy in order to uh you know jive well with the rest of the rules of quantum mechanics and i'm not going to study any particular system i just want to ask general questions so the first thing that must be true is that stuff doesn't just doesn't disappear i guess disapperate it's probably the appropriate so stuff doesn't leak out of the world so whatever goes in must come out so what that means is norm of a squared plus norm of d squared which is the probability density in probe and probability density out must be equal to b squared plus c squared okay everyone agree with that uh why can't stuff stay in the potential that's a good ques so um so yeah it's so it's a stationary state is so if we're looking at at fixed energy eigenstates we know that we're in a stationary state so whatever the amplitude um going into the middle is it must also be coming out of the middle right so there's another way to say this which is that um let's think about it not in terms of energy individual energy i can say so let's think about it in terms of wave packets so if we take a wave packet of stuff and we send in that wave packet it has some momentum right um oh this is going to get delicate technical let me just stick with the first the first statement which is that if stuff is going in then it has to also be coming out by the fact that this is an energy eigenstate the overall probability distribution is not changing in time so if if stuff went in and it didn't come out that would mean it's staying there that would mean that the probability density is changing in time that's not what happens in an energy eigenstate the probability distribution is time independent everyone call that okay so um good so uh so let's think though about what this is i can write this in the following nice way i can write this as uh a complex conjugate d complex con whoops d complex conjugate a d and on the right hand side this is equal to uh b complex conjugate c complex conjugate bc i have done nothing other than write this out in some suggested form but b and c are equal to the s matrix so bc is the s matrix times a and b comics conjugate c complex conjugate is the transpose complex conjugate so this is equal to a complex conjugate d complex conjugate s transpose complex conjugate also known as adjoint and this is s on a d yeah but this has to be equal to this for any a and d so what must be true of s dagger s as a matrix it's got to be the identity as a matrix in order for this to be true for all a and e ah that's cool stuff doesn't disappear s is a unitary matrix so s is unitary matrix its inverse is its adjoint okay and you'll prove this in a little more um uh elegance um you'll you'll study this in a little more detail on the problem set you showed uh the definition of unitary you studied the definition of unitary and the last problem is that okay so that's the first thing about s and it turns out to be completely general any time whether you're in one dimension or two dimension or three if you send stuff in it should not get stuck it should come out and when it does come out uh the statement that it comes out for energy eigenstates is the statement that s is unitary matrix questions on that okay so a consequence of that is that the the eigenvalues of s are phases pure phases so i can write s i'll write them as s1 e to the i uh phi 1 and s2 is equal to i phi 2. okay so the statement that s is a unitary matrix leads to constraints on the coefficients okay and you're going to derive these on your problem set i'm just going to list them now so the first is that the magnitude of s1 1 is equal to the magnitude of s22 the magnitude of s12 is equal to the magnitude of s21 oops good and more importantly s12 norm squared plus s one one squared is equal to one and finally s11 s2 s12 complex conjugate plus two one s two two complex conjugate uh is equal to zero so what do these mean what are these conditions telling us what are these can so they're they're telling us of course they're a consequence of conservation of probability but they have another meaning and let's to get that other meaning let's look at the definition of of uh the transmission amplitudes so in particular consider the case that we send stuff in from the left and nothing in from the right so that corresponds to d equals zero so when d equals zero what does this tell us it tells us that b is equal to and a equals one for normalization so b is equal to well d is equal to zero so it's just s11 a so b over a is s11 and similarly c over a is s21 but c over a is the thing that we've been calling the transmission amplitude little t and this is the reflection amplitude little r so this is the reflection amplitude if we send in stuff and bounces off reflex to the left and this is the transmission amplitude for transmitting to the right everyone cool with that and by the same token so this is reflection to the left this is transmission to the right by the same token this is going to be transmission to the left and reflection to the right so now let's look at these conditions s11 is reflection this is going to be reflection this says that the reflection to the left is equal to the reflection to the right in magnitude before what we saw was for the simple step the reflection was equal to the reflection the reflection amplitude was equal from left to right not just the magnitude but the actual value was equal from the left and the right it was a little bit of a cheat because they were real and we saw that that was a consequence of just being the step we didn't know anything more about it but now we see that on general grounds on conservation and probability grounds the magnitude of the reflection to the left and to the right for any potential had better be equal and similarly the magnitude of the transmission for the left and the transmission to the right had better be equal all other things being uh you know if you're sending in from the left and then transmitting to the left or sending in from the right and transmitting to the right and what does this one tell us well s12 and s11 that tells us that t squared little t squared which is the total probability trans to transmit plus little r squared norm squared is the total probability to reflect is equal to one and we saw this last time too and this was the earlier definition of nothing gets stuck and this one you'll study on your problem set it's a little more subtle okay questions yeah one more time yeah i got s's okay good so the way we got s was unitary first off this is just the definition of s s is the matrix that g for any energy relates a and d the in going amplitudes to the outgoing amplitudes b and c just the definition meanwhile i claim that stuff doesn't go away nothing gets stuck nothing gets nothing disappears so the total probability density of stuff going in must be equal to the total probability density of stuff going out but the probability this the probability of stuff going in can be expressed as this row vector times this column vector and on the out this row vector times this column vector and now we use the definition of the s matrix this column vector is equal to s times this row vector yeah b c is equal to s times a d and when i take the transpose complex conjugate i get a d transpose complex conjugate s transpose complex conjugate but that's yes adjoint but in order for this to be true for any matrix any vectors a and d this must be it must be that as stagger s is the unitary is the identity but that's the definition of a unitary matrix cool others okay you're going to prove a variety of things on the problem set about the um about the scattering matrix and its coefficients but i want to show you two properties of it the first is reasonably tame and it'll make a little sharper the step result we got earlier that the reflection in both directions off the step potential is in fact exactly the same so where did i put this good so suppose suppose our system is time reversal invariant so if t goes to minus t nothing changes this would not be true for example if we had electric currents in our system because as we take t to minus t then the current reverses so if their current shows up in the potential or if a magnetic field due to a current shows up in the potential energy then as we change t to minus t we change the direction of the current we change the direction of the magnetic field so in simple systems where we have time reversal invariants for example electrostatics but not for example magnetostatics suppose we have time reversal and variance then what you've shown on a previous problem set is that psi is a solution then psi star psi complex conjugate is also a solution and using these what you'll what you can show and i'm not going to go through the steps for this well okay that's easy so if we do the time reversal the wave function on looking on the left or on the right so comparing these guys um so what changes well under time reversal we get a solution so given this solution we have another solution a star e to the minus i k x b star e to the plus i k x see minus star star plus okay so we can run exactly the same game now with this amplitude and when you put these conditions when you put the conditions together which you'll do on the problem set um so this another way to say this is the same solution with k to minus k and with a and b ex replaced by their complex conjugates and c and d replaced by each other's complex conjugates then this implies that it must be true that a uh and d which are now the outgoing guys because we've time reversed is equal to s and i'll write this out explicitly s11 s12 s21 s22 b star and c star okay and so these together give you that i should have done this over here therefore s complex conjugate s is equal to one okay so if s complex conjugate s is equal to one then s inverse is equal to s transpose just putting this on the right so s transpose is equal to s inverse is equal to s adjoint because s is also unitary so time reversal and variance implies for example that s is equal to whoops um uh that s dagger is equal to s r s is equal to s transpose stem reverse this is what i wanted to write here it gives us that s is equal to s transpose and in particular this tells us that s21 is equal to s12 the off diagonal terms are equal not just in magnitude which was insured by unitarity but if in addition to uh to being a unitary system which of course it should be if in addition it's time reversal invariant then we see that the off-diagonal terms are equal not just in magnitude but also in phase and as you know the phase is important the phase contains physics it tells you about time delays and shifts on in the scattering process so the phases are the same that statement is not a trivial one it contains physics so when the system is time reversal invariant the phases are the the phases as well as the amplitudes are the same and you'll derive a series of related conditions or consequences for the s matrix from various properties of the system for example parity if you can if you have a symmetric potential but now i want to in the last few minutes i just want to tell you a really lovely thing so it should be pretty clear at this point that all the information about scattering is contained in the s matrix and its dependence on the energy if you know what the incident amplitudes are you know what the outgoing amplitudes are and that's cool because you can measure this right you can take a potential you can literally just send in a beam of particles you can ask how likely are they to get out and more importantly if i build a wave packet on average what's the time delay or acceleration of the transmitted wave pack and that way i can measure the phase as well i can measure both the transmission probabilities and the phases release the gradient to the phase with energy hold on go ahead is there a special condition that we can impose to see the resonance excellent question hold on to your question for a second yeah so there's an enormous amount of the physics of scattering contained in the s matrix and you can measure the s matrix and you can measure its dependence on the energy you can measure the coefficients s12 and s22 their phases and their amplitudes as a function of energy and you can plot them and here's what i want to convince you of if you plot those and look at how the functions behave as functions of energy and ask how do those functions extend to negative energy by just drawing the line continuing the lines you can derive the energy of any bound states in the system too knowledge of the scattering is enough to determine the bound state energies of a system and let me show you that and this is one of the coolest things in quantum mechanics so here's how this works we have from the definition of the s matrix that bc is equal to the s matrix on a d right where the wave function let me just put this back in the original form is c d a b e to the i k x e to the minus i k x and e to the plus i k x e to the minus i k x yeah so that's the definition of the s matrix the s matrix at a given energy e is this relation is a coefficient relation matrix between the in-going outgoing or more to the point a and d and in all of this i've assumed that the energy was positive that the k1 and k2 are are positive and real but now let's ask the question what would have happened if in the whole process i had taken um take the energy less than zero so if the energy were less than zero instead of k k would be replaced by i alpha let's think about what that does if k is i alpha this is e to the i k is minus alpha and minus ik is plus alpha plus alpha similarly i k times i that gives me a minus alpha and this gives me a plus alpha yeah so as equations they're the same equations with k replaced by i alpha yeah and now what must be true for this state for these states to be normalizable what must be true for example of a a must be zero because at minus infinity this diverges it's not normalizable so in order to be a bouncy in order to have a physical state a must be zero what about d same reason it's got to be zero at positive infinity these guys are convergent so c and b can be non-zero so now here's my question can i scatter a state from a b we know that these relations must be true because all these relations are encoding all these relations are encoding is how a solution here matches to a solution here through a potential in between with continuity continuity of the derivative and anything else that's true of that potential inside all that s is doing from that point of view is telling me how these coefficients match onto these coefficients yes now what for a bound state so if we have e less than zero what must be true we must be true that a d is equal to zero and in particular zero zero right so what are b and c oh well a matrix times 0 is equal to unless unless the matrix itself is diverging and then you have to be more careful but let's be naive for the moment if a is 0 0 then in order for b and c to be non-zero s must have a pole s must go like one over zero s must diverge at some special value of the energy okay well that's easy that tells you that if you look at any particular coefficient in s any of the matrix elements of s the numerator can be whatever you want some finite number but the denominator had better be zero so let's look at the denominator if i compute s21 oh actually let me do this over here let's compute ah no it's all filled damn let's do it over here so if i look at s21 for the barrier or for the potential well scattering off the potential well that we looked at at the beginning of today's lecture this guy and now i'm going to look at s21 one of the coefficients of this guys also known as the scattering amplitude t for the well this is equal to and it's got awful expression 2 k1 k2 e to the ik k2 l over 2 1 k 2 cos of k prime l minus i k squared plus k 1 squared plus k 2 squared times sine of k 2 l okay this is some horrible thing but now i ask the condition look when does this have a pole for what values of the energy does this when the energy is continued to be negative for what values does this have a pole or does the denominator have a zero and the answer is if you take this and you massage the equation this is equal to 0 a little bit you get the following expression k2 l upon 2 tangent of k2 l upon 2 is equal to k1 l upon 2. this is the condition for the bound state energies of the square well and we've computed it using knowledge only of the scattering states if you took particles and a square well and roll the particles across the square well potential and measured as a function of energy the scattering amplitude the transmission amplitude in particular s21 an element to the s matrix and you plotted it as a function of energy and then you approximated that by a function of energy that satisfies the basic properties of unitarity what you would find is that when you then extend that function in mathematica to minus a particular value of the energy the denominator diverges at that energy you know that there will be a bound state and so from scattering you've determined the existence of a bound state and this is how we find an awful lot of the particles that we actually deduce must exist in the real world we'll pick up next time [Applause]
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_19_Identical_Particles.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right, so for the next six lectures, including today, we're going to finish off the course with the application of everything we've studied so far to a couple of ideas. The first being the existence of solids-- why we have solids and why we have conductivity in solids, which is basic properties of materials. In particular, the story I want you guys to leave the course with is an understanding of why diamonds are transparent and why copper isn't, which is sort of a crude fact about the world. But we can explain it from first principles, which is pretty awesome. The second is we're going to come back to this idea of spin and of 1/2 integral angular momentum. This intrinsic angular momentum of the electron. And we're going to use that to motivate a couple of ideas. First off, we're going to tie back to Bell's inequality from the very beginning, and then we're going to do a little bit on quantum computation. That will be the last two lectures. So before I get started though with today, two things. First, I'm going to ask for questions. But the second is I got some really good questions about hydrogen, so I want to wrap up with one last comment on hydrogen because it's entertaining. But before I get started on that, questions about everything up till now? Yeah. AUDIENCE: So in the [INAUDIBLE] there's someting called spirit orbit couplings, another edition, the [INAUDIBLE],, that we haven't talked about, in addition throughout this egression. Is that something we can do with our knowledge now? Or is something [INAUDIBLE]? PROFESSOR: It absolutely is. So the question is-- and this is a very good question-- the question is, look, if you look on Wikipedia about-- seriously, this is a rational thing to do. If you look at Wikipedia to learn something about hydrogen, what you discover is everything we've talked about, including the fine structure, including the Zeeman effect. But you see that there are a couple of other effects. So one effect, for example, is when you look at the Zeeman effect a little more carefully than we did, there's a second term in the Zeeman effect, which is an induced magnetic moment. So even when the angular momentum is smaller or even zero, there's still a contribution to the energy from the externally imposed magnetic field. So there are lots of other corrections. One of them in particular is something called spin orbit coupling. And so the question was, what is that? And do we know enough to explain what spin orbit coupling is? So let me give you a very short answer to that question. So we know that if we write down the energy operator for our system, we have the energy operator-- so the full energy operator for hydrogen, I'll say, is the energy operator for coulomb plus a bunch of corrections. So for example, we had the relativistic correction plus a term that was a coefficient times p to the fourth. And this is on your problem set. And this came from relativistic corrections. I shouldn't call this c. I'll call it beta, maybe. Some coefficient. And there are a whole bunch of other contributions. For example, there's the term we studied that contributed to the Zeeman effect. If there's an external magnetic field, there's a B dot-- magnetic moment of the system. And if the electron is in a state with finite angular momentum, the magnetic moment is some dimensionful coefficient, which we usually call the Bohr magneton just because-- whatever-- we liked the guy. Times the angular momentum. But there can be additional contributions. There are a whole bunch of additional terms in the true potential for hydrogen. One of them takes the following form. It's a constant plus-- maybe I'll call it kappa times spin dotted into the angular momentum of the electron, of the electron bound to the hydrogen system. So where this comes from is a sort of beautiful story. So first off, could it be there? Sure. This is a term that could exist with some coefficient. Why not? It turns out that you can derive its existence from a study of the relativistic version of the hydrogen system. We're not going to study that in any detail in 804, but the basic idea and what this does is totally amenable to 804-level analysis. This is just saying that when you have some orbital momentum, or when you have some spin angular momentum, both of those corresponds to angular momentum of an object that carries charge. And so it's not unreasonable that there's going to be some interaction between the two where the magnetic moments-- those two magnetic moments either want to be aligned or anti-aligned depending on the sign of this coefficient. However, we haven't talked about spin in any great detail yet. We're going to do that in the last week of the course. So I'm going to defer talking about this in any detail. And we're only going to discuss it briefly at the end for a couple of weeks. But absolutely, these additional couplings are important for hydrogen and they're also things you can study at this level, at 804 level. I should also emphasize that there are a whole bunch of other corrections. There are lots of terms. One set of terms, an infinite number of terms, are the further sub-leading relativistic corrections of kinetic energy. And there are a whole bunch of other corrections, so I'm going to leave those out for the moment. But I want to emphasize to everyone that we're building models of hydrogen, and they're all approximations. Yeah. AUDIENCE: So it's a question regarding this [INAUDIBLE].. PROFESSOR: Yes. There is. So on the problem set, you're asked to estimate the correction to the energy. I wrote down the answer in lecture the other day. You're asked to compute, in particular, the l dependence. But to estimate the magnitude or to estimate the value of the shift in the energy eigenvalues to hydrogen from the first relativistic correction is p to the fourth correction. And I expect that this will be covered in your recitation in some detail, but there the question was, is there a trick to make this a little easier? There are a couple of good physical arguments, which I'm not going to tell you. But there are nice ways to do this. But at the end of the day, you do not want to end up doing the following computation. You don't want to take the expectation value of p to the fourth and write this as the sum integral of psi complex conjugate derivative to the fourth psi. If you attempt to do this integral, you will weep. This is not the way you want to do it. And here's what you want to do. And I'm not going to tell you how to get here, but you want to reduce the calculation of this expectation value to the calculation of the expectation value of 1 over r and of 1 over r squared. And it turns out that if you know these expectation values, that entirely suffices to compute the expectation value of p to the fourth if you take into account the fact that the energy eigenfunctions satisfy the original energy eigenvalue equation. So p squared upon 2m plus v of x of v of r is equal to e when acting on those wave functions. So for computing these guys, you could try to do this brute force, not unrelated to problem, I believe, two on Rydberg atoms on the problem set. But there are actually sneakier ways to do these expectation values as well. And I will leave that to you. But let me emphasize that you can do a direct brute force calculation, but you don't need to. And I would encourage you to try to find an efficient, indirect way to do these calculations. Did that answer your question? OK. Anything else? Yeah. AUDIENCE: So is [INAUDIBLE] a time dependence to an operator with the [INAUDIBLE] being [INAUDIBLE]?? PROFESSOR: Does adding-- well, it depends on how you introduce the time dependents. AUDIENCE: Can a operator [INAUDIBLE]?? PROFESSOR: Yes. Yeah, absolutely. What's the reason for the question? AUDIENCE: One of the problems here. PROFESSOR: One of the problems in the problem set? AUDIENCE: Yeah. It's like the last part of four. PROFESSOR: Remind me which problem this is, I don't remember. AUDIENCE: [INAUDIBLE]. PROFESSOR: Oh. Oh, oh! Ha, ha, ha. Sorry, I really like that problem. Let me rephrase your question in the following way. Is it intrinsically non-Hermitian to have time dependence in a system? So no, you can have-- so what does Hermitian mean? Physically, what does it mean for the energy to be Hermitian for example? What it means for the energy operator to be Hermitian is that time evolution, which is represented by the oper-- so let me phrase it this way. So if we know that the energy operator is Hermitian, what is that telling us? Well, it tells us a lot of things. It tells us the energy eigenvalues are real. That's good. So E dagger is equal to E. But it tell us something else. Remember that the Schrodinger equation is that ih bar dt on psi is equal to E on psi. And we use this to argue that the general solution to the Schrodinger equation can be written in the following elegant way, psi of t is equal to e to the minus i upon h bar tE on psi at 0, where this was the evolution operator u sub t. Yeah? And this is a unitary operator. And the way they say its unitary goes back to the exam. The way to say it's unitary is this is e to the i Hermitian operator. t is a real number, h is a real number. So this is e to the i Hermitian operator. And anything of the form e to the i Hermitian operator is unitary. It's adjoint to its inverse. What that tells you is that since this is unitary, it preserves the magnitude, or the norm, of the wave function. So probability is conserved. What would it mean for the energy operator to be not Hermitian? Well, it would mean a lot of things. One thing it would mean is that the energy eigenvalues are no longer real. That's a little weird. But the much more troubling thing is that the energy would no longer be-- sorry, the probability would no longer be conserved. The evolution operator, which is the solution to the Schrodinger equation, would no longer be a unitary operator. And the probability, the norm or the wave function, would no longer be conserved. So the question at the end of-- the last question in problem four is really asking, in the system you're thinking about is, probability conserved? And that's the question that you should be asking yourself when you finish up problem four. Good? Yeah. AUDIENCE: When we solve the Schrodinger equation in that way, does this solution come from the fact that E is Hermitian or that E is like that nice little [INAUDIBLE],, that E is time dependent. And that if E was time dependent, why couldn't the coefficients for the probability also be? PROFESSOR: Absolutely. So here I was just focusing on the Hermiticity. And in solving this, I'm assuming that the energy is time independent. If the energy is not time independent, then this is not the right answer as you're pointing out. So in fact, you have to do an integral and you have to time order things, and it's a complicated story. But this is not the solution. But even if we have a time independent system, if the energy operator is not Hermitian, this is not unitary. So indeed, you're absolutely right that if the energy operator is time dependent, the story's more complicated than just this. But we already see the problem that's salient for problem four at this stage with a time independent operator. Yeah. AUDIENCE: Just one question please. So if I had some sort of system which is [INAUDIBLE],, some of my particles are weak in the other process. PROFESSOR: Yeah. AUDIENCE: And now the probability is not conserved. Saying that this thing is not [INAUDIBLE],, the issue is, I think, which is also extremely disturbing for [INAUDIBLE] complex in energy, [INAUDIBLE],, what is that? PROFESSOR: Yeah. AUDIENCE: That seems even worse. PROFESSOR: Well, I guess it's a matter of taste. I would say that they're the same thing in the following sense. Suppose I have complex energy eigenvalues. Well, I know that if I have a stationary state, then that stationary state as a function of time is equal to the stationary state at time 0 times e-- so this is an energy eigenfunction rather than-- is e to the minus i the energy times t over h bar? Or this is the energy eigenvalue. Now, imagine e is complex. Sorry, first imagine e is real. That's not hard to imagine because it's usually the case. And if that's true, what happens to the wave function as we evolve through time? It rotates by phase. Now, if e is complex, imagine e is of the form e-- let e be complex. So e is going to be E real minus i gamma, some imaginary piece. So what that's going to give me is an e to the minus gamma t. And that's a decaying thing that when you get to norm squared gives you a decaying envelope over time. It's going to give you an exponential decay. This is going to be equal to e to the minus ie real t over h bar times e to be minus gamma t over h bar phi of 0. And when we take the norm squared, the phase goes away, but this doesn't. So having loss of probability is having a complex energy eigenvalue. Yeah. AUDIENCE: Why are we talking about time independent operators if there's an electromagnetic field and stuff is clearly going on between the avenue of the electromagnetic field? PROFESSOR: This is a very good question. And that actually leads me into the thing I wanted to talk about first. So I got a bunch of questions-- thanks for that question. I got a bunch of questions over the past few days about the magnetic moment of the hydrogen system and what a strange idea that is. So let me talk about that for a second. And when I'm done with this little spiel, tell me if I've answered your question. Well, let's just leave this up. OK. So in particular, let's think about this term for a moment. Where this term came from last time was we said, look, we turn on an external magnetic field. Someone turns on the switch and current runs through an electromagnet and we get a uniform magnetic field in our fiducial volume, where we're doing the experiment. And the electron system, if it carries some angular momentum, it also has a charge, angular momentum charge. That means it's got a magnetic moment. That's how much magnetic moment. So this is saying that the magnetic moment of the electron bound to the proton with angular momentum wants to anti-align with the magnetic field. Or align in this case because I put the wrong sign. But here's something really upsetting about that. So what I just said sounds crazy if you think about it in the following way. Take an electron in the coulomb system, just straight up coulomb. Take an electron in the coulomb system, put it in, say, the ground state, lowest energy state. Is it moving? This is problem four. So is it moving? The expectation value of position doesn't change in time. It's a stationary state. Expectation value of nothing changes in time. OK, fine, but it's the ground state that carries your angular momentum. That's not so upsetting. Go to the first excited state with angular momentum. The n equals 2, l equals 1, m equals 1 state. So it's got as much angular momentum in lz as possible. Is that thing moving? Yeah. It's not moving at all. And yet, we're saying there's a current associated with it, an electric current. And that electric current is inducing a magnetic moment, a la right-hand rule. Maybe via Biot-Savart, if you want to be fancy. And that magnetic moment-- a little current loop-- is leading to this interaction, the Zeeman's interaction. We know that it's true because experiments show the Zeeman splitting. So it is definitely true that there is a magnetic moment of this thing. But it's not moving, how can that be? So how can there be a current if nothing is moving? AUDIENCE: It is, in some sense moving, [INAUDIBLE].. Why is the expectation value of the [INAUDIBLE] to stop moving? PROFESSOR: What's the expectation value of the momentum? AUDIENCE: I think it's also probability. PROFESSOR: Yeah, it's 0. AUDIENCE: I mean, first of all, the world isn't classical. You can't use classical intuition. PROFESSOR: OK, true. AUDIENCE: Second of all, let's suppose you have a uniform ring of classical charge and set it spinning. PROFESSOR: Yes. AUDIENCE: That ring is as a ring, not moving, but there's still a current associated with it. PROFESSOR: So you're saying that the electron is a ring? AUDIENCE: It might make sense if you give it that [INAUDIBLE].. PROFESSOR: I like this. No, this is good. It's wrong, but it's good. And the reason it's good is that you're really pushing your assumptions to try to figure out how the experimental data can possibly match. And you're saying, look, we have to just reject our intuition. Our intuition is clearly leading us astray. And I like that. That's correct. So what we're going to do in the next few minutes is work through that and try to find the best way to phrase that. Your strategy is the correct one. So let me rephrase that slightly. Look, if you have a classical distribution of charge, that distribution of charge could be a stationary distribution-- a distribution, which as a distribution of charge doesn't change a dime. But each individual charge in that distribution is itself moving. The problem here is that we just have the one electron. If you ever look, you will find the electron at a spot. But what you're really saying is, look, it's a mistake to think about the electron having a definite position in the first place. You just shouldn't think about that. The best you can say is that it has some probability of being in any spot. So let's work with that. Let's take that idea and let's push it. Let's see how far we can take this idea that the electron has some probability [INAUDIBLE]. So suppose we have an electron in a stationary state of the coulomb potential. The stationary states are labeled by three integers-- n, l, and m. And we'd written their wave functions. There are, of course, an infinite number of ways to write in different notation. There are an infinite number of fonts one could use for the normalization constant 1/r R nl of little r, Y lm of theta and phi. And I'm going to rewrite this. I'm going to expand this out slightly. This is N 1/r R of R and l. And Ylm, remember, was of the form Pl of cosine theta times e to the im phi. Everyone cool with that? So I just wrote the spherical harmonic as a polynomial in cosine theta times an exponential in phi. AUDIENCE: Where did the 1/r come from, on the left? PROFESSOR: The 1/r was-- so most people call capital R the whole thing. But this is the pulling out to 1/r to simplify the radio wave equation. And the reason I prefer writing it this way is just that this guy satisfies a simple 1d Schrodinger equation. AUDIENCE: That was the u of r? PROFESSOR: That was a thing that we called u of r, and then I confused the hell out of everyone by calling three different things on the board u, which was sort of unnecessary. So this is the artist previously known as u. The dude can sing. OK, so the first question I want to ask is, so this is a stationary state. Is the electron moving? No, not in any conventional sense. If you compute the expectation value of the position and you take its time derivative, this is zero. We could do this either by calculating it or just by observing that it's a stationary state. And on principle, it can't change in time. So this guy is not moving in any conventional sense. So why is there an electric current? Why do we get the Zeeman magnetic moment, the Bohr magneton? So as was pointed out, look, this is quantum mechanics. It's not classical mechanics. And in quantum mechanics, the electron isn't at any point. Rather, there's some probability density. And the probability density that we find the electron at some point r is psi squared l m or r squared. A familiar beast. And meanwhile, a wonderful thing about the probability distribution in quantum mechanics that we've already discussed here is that it's conserved. The time derivative of the probability density or r-- remember, this is a density, not a probability. The time rate of change of the probability distribution is minus the gradient-- the divergence of the probability current, where the probability current j is equal to h bar over the mass, which I'm going to call the mass mu. Oh, gee, I don't want-- I'm going to call the mass capital M. That's the mass of the electron, which is a little strange because it's a very small quantity. But anyway, h over capital M of the imaginary part of psi complex conjugate gradient psi. And you showed on a problem set long ago that this is true if j takes this form by virtue of the Schrodinger equation. Now, we usually write this imaginary 1 over 2i times psi gradient-- or psi star gradient psi minus psi gradient psi star, but that's equal to the imaginary part of the first term. The thing you want to emphasize is the imaginary part. So we have this current. In our system, the position expectation value is time independent. And indeed, it's a stationary state. So beyond the position expectation value being time independent, the probability density itself is time independent because the wave function evolves by an overall phase and the probability density is the norm squared. So the phase goes away. So in our system, in an electron in this state in the coulomb potential, the time rate of change of the probability density-- let me actually do this over here. So in the stationary state, psi l, n, m, the time rate of change of the density is 0. And this tells us by the conservation equation that the divergence of the current is also 0. Does that tell us that the current is 0? AUDIENCE: No. PROFESSOR: Right. So what's the current? Well, we've written everything here in spherical coordinates. And the current is given in terms of the gradient operator. So let me remind you quickly of what the gradient operator is in spherical coordinates. It has three components-- a radial component, a theta component, and then a phi component. And the radial part is just D r. The theta component is 1 over r D theta. And the phi around the equator component is equal to 1 over r sine theta D phi. Everyone cool with that? So what's j? Well, j is going to have a component, j in the radial direction. The current in the radial direction. And intuitively, what should that be? Is there stuff going out or in? There shouldn't be. It's hydrogen. It's not doing this. So this should be 0. And we can just quickly see that this is h bar upon M imaginary part of-- well, this is the r component, it's going to be the derivative in the radial direction. But the derivative in the radial direction is going to be real. The derivative in the radial direction is going to be real. So when we take the norm squared, we don't pick up anything. We pick up an overall coefficient. It's going to be strictly real. The phase, e to the i m phi cancels out because this is the complex conjugate. So the imaginary part of this thing is going to be 0. OK? So J r is 0. And similarly, J theta p is a real function. When we take a derivative, we get a real function. And then when we multiply by its complex conjugate, again, the phase cancels out and we just get a whole bunch of real stuff whose imaginary piece is 0. So J theta is 0. But J phi is a cool one. In my head, I was just thinking J 5. OK, at least someone got that. So J phi, however, is not going to be 0 for the following reason. So what is it equal to? It's equal to h bar upon M-- the mass, M-- times the imaginary part of-- well, psi star, which is psi complex conjugate. And then the gradient with respect to phi. The phi component is 1 over r sine theta. And then derivative with respect to phi. The derivative with respect to phi pulls down a i m. Oh, look-- times psi, which is equal to h bar upon m, times-- well, psi is norm squared. That's real. Psi squared. 1/r sine theta. That's real, so I can pull that out. And we are left with imaginary part of im, which is just m, which is not 0. And in particular, it's proportional to h bar m. Everyone see that? So what this is saying is that the current, the probability current, for an electron in the stationary state psi nlm of the Coulomb potential is equal to norm psi squared over m r sine theta h bar m in the phi direction. The phi uni-vector. Cool? So nothing is moving, but there's a current. What is moving? What is the thing of whose current-- of whom this is the current? AUDIENCE: Probability density. PROFESSOR: The probability density. The probability density is rotating. So what this is telling us is that it's true the system is stationary. But I want to know-- I'm looking down at the equatorial plane, OK. So this plot is going to be looking down on the equatorial plane of hydrogen. Here is the origin, the center of the potential. And what this is telling-- so I'm going to draw these vectors. They're in e phi direction. OK. But their magnitude falls off with 1 over r and the norm square of the wave function, which also is falling off exponentially. And it goes to 0 at the origin as r to the l plus 1. And m must be no bigger than l. So in order for m to be non-zero, l must be non-zero. Which means this must go like r to the something greater than m plus 1. So the r's cancel out. And we have that it vanishes. So the contribution vanishes at the origin, is small near the origin, is largest at some radius, and then falls off again. So the magnitude of the arrow here is meant to indicate the magnitude of the flux. So what we see is that we have a probability distribution where the probability, as a distribution, is time independent. Right? The probability distribution is, in fact, time independent. But there's a current which is a persistent, stationary current. Everyone cool with that? So if I ask you, where is the particle? Well, the probability density tells you that. And if I ask you, what's the current? The electric current, minus e times j. At a point. And this is really the current density and the probability current density. Now what is the consequence of the statement, the probability density itself is not changing in time. It's that there is zero divergence of this current. And indeed, there is zero divergence. Every little bit going into a spot is matched by some equivalent amount going out. There's zero divergence. Cool? And so we see that while it's true that nothing is moving, it is also true that there is a non-trivial electric current. And when you use the Biot-Savart law to sum up the contributions from each little bit of this current, what you get is-- right hand rule-- a magnetic field, a magnetic moment. And following nothing but the Biot-Savart law and using what you know about the wave functions, this gives us that the magnetic moment is equal to the Bohr magneton, b times m, z. OK? Everyone cool with that? Yeah? AUDIENCE: So when the perimeter intuitively is [INAUDIBLE],, still make all these other expectation values? PROFESSOR: Yeah. AUDIENCE: Why is that an intuitive force? Or maybe it's not. PROFESSOR: Well, here's one way to say it. There are two-- let me rephrase that question. Let me give you a different question to ask. And let's think about the answers to that question and I hope that will answer your question, if it doesn't ask it again. So here's the question I would suggest that you ask. Or that someone ask. I'll ask it. And I just totally lost my train of thought. That's totally me. What was the question I wanted you to ask? Wow, that's amazing. I completely just in the blink of an eye totally lost my train of -- ah, yes. So here, it was crucial in this calculation that the current was given by the imaginary part of the gradient. And as you showed in a problem set a while back, anytime you have a wave function which is real, then the current will vanish. Here it was crucial-- in order to get a non-vanishing current-- it was crucial that the wave function was not real. It had a phase. But you also showed on a problem set that you can always take your energy eigenfunctions-- for bounce-- you can always take your energy eigenfunctions and make them real. You can always construct a basis of energy eigenfunctions which are strictly real. Sorry? AUDIENCE: In one dimension, right? PROFESSOR: Yeah well-- AUDIENCE: Aren't you in three also? PROFESSOR: What did you use for that proof? AUDIENCE: Just used linear combination of the two. PROFESSOR: Yeah. You just use the energy eigenvalue equation, hermiticity, and the linear, the energy operator. And that's perfectly true in any number of dimensions. AUDIENCE: OK. PROFESSOR: So that sounds crazy. So first off, why do we have a not real energy eigenfunction? Can we construct purely real energy eigenfunctions? If we can, those energy eigenfunctions would appear to have no current. So they would have no magnetic moment. How do these fit together? Can you construct energy eigenfunctions that are pure real for this system, for the Coulomb potential? Let me give you a hint. In 1d for a free particle, what are the energy eigenfunctions? 1d, AUDIENCE: [INTERPOSING VOICES]. PROFESSOR: E to the ikx, right? That's not real. But there's another eigenfunction, you get the minus ikx. And if you take the sum of those, you get e to the ikx plus e to the minus ikx-- divide by 2 for fun-- and that gives you cosine of kx. That's real. But there's a second one, which is sine of kx, which is also real. Now you can take linear combinations of them with i's and get the exponentials back. But let's just take the real part, right? Now, do those carry any momentum? What's the momentum expectation value for cosine of kx? AUDIENCE: 0. PROFESSOR: 0, because you get confirmation from plus k and one from minus k. They exactly cancel. Now look at this solution. Can you construct a real energy eigenfunction of the Coulomb potential? AUDIENCE: Sure. PROFESSOR: How. AUDIENCE: [INAUDIBLE]. PROFESSOR: Great.. Let's take this and its complex conjugate. So what that would be? Well, if we take this, psi nlm. And its complex conjugate, well, what is its complex conjugate? These are real. That just gives me a minus. So that's the state plus psi and l minus m, which is also an allowed state with the same energy by rotational invariance. And this is thus the state psi nl. And it doesn't have a definite lz angular momentum, right? So I'll just call it psi sub nl unhappy face. It is not an lz eigenfunction. But it is an energy eigenfunction. And it could also have built a minus with divide by 2i for fun. I could have built the sine or the cosine of phi. One of which showed up on your exam. So I can find a basis of these states, sine and cosine, instead of, give me the im phi, I need the minus im phi. And those would have been perfectly real. And in those situations, what would the current have been? AUDIENCE: [INAUDIBLE]. PROFESSOR: If I put the system in this stationary state, what would the current have been? AUDIENCE: 0. PROFESSOR: Identically 0. However, that wouldn't be a state with a definite angular momentum. So it wouldn't be surprising that it has zero current. It's got some contribution. It's a super position of having some angular momentum and having the opposite angular momentum. And the expectation value, the expected value, is zero. I can also study states with a definite value of the angular momentum. Those are not real. There's nothing wrong with that. I could have constructed states which don't have a definite angular momentum but are real. But instead, I want to work with states that have a definite angular momentum. That cool? So there's no tension. There's no contradiction. It's just that, if we're interested in finding states that have a definite energy and a definite lz angular momentum, that is not going to be a real function. And nothing tells us that it has to be. And when you have a non-trivial lz, when you have a non-trivial angular momentum, just like a classical particle with nontrivial angular momentum that has charge, we find that there's a current. And thus a magnetic moment. So the important thing with having a definite nonzero momentum, or in this case angular momentum, to give us the current. Yeah? Questions about this? Yeah? AUDIENCE: Using the same logic, it kind of looks like we're taking it a charged particle and moving it around in circles. Like, where we have a distribution and we're spinning. PROFESSOR: Yeah. AUDIENCE: With spin a distribution of charge, we're accelerating the charged particles in it. PROFESSOR: Yes. AUDIENCE: Individually. PROFESSOR: Yes. AUDIENCE: If we accelerate them, they emit radiation. PROFESSOR: Yes. AUDIENCE: Hydrogen atoms don't emit radiation. PROFESSOR: Yes. AUDIENCE: How does that go? PROFESSOR: That sounds a lot like problem four on your problem set. AUDIENCE: [INAUDIBLE] PROFESSOR: Ah, OK. Well, that's a great question. And you should think about it. How could I say-- you read my mind. Yes. Struggling with this is exactly the point of one of the problems on your problem set. And ask me that again after the problem set has been turned in, and I'll give you a happy disquisition on it. But I want you to struggle with it. Because it's hard and interesting question. Yeah? AUDIENCE: So I don't think this answers my question necessarily about the radiation. Because, analogously, the classical enm, this would be the magnetostatic case, where you just have a static-- PROFESSOR: Precisely. Precisely. AUDIENCE: But with radiation, you have, quickly oscillating fields and the transition times are really small and the frequencies of the wave functions are really fast. So I still don't know if that's the same. PROFESSOR: So this question is unfortunately a linear combination of a really good independent question and that question. So ask me after the lecture and I'll talk to you about the part that's linearly independent. OK. I'm on thin ice here. OK. Anything else before we dispense with hydrogen? OK. The question that you all keep coming to about the-- well, problem four on the-- I think it's problem four on the problem set. Maybe it's problem two. It involves almost no computation, but it's the most intellectually challenging problem on the exam-- on the problem set. So take it seriously, even though it's calculation-free. OK. Yeah? AUDIENCE: Due to this proton in the [INAUDIBLE].. PROFESSOR: Yeah. Yeah. It's so that the-- it's not by symmetry, because there isn't a symmetry. The proton is 2,000 times heavier than the electron. But-- let me make sure I'm understanding your question. Your question is, we found that the electron is in bound states in the Coulomb potential. But in hydrogen, you have two parts, you have an electron and a proton. So is the electron in that state and the proton is just free and cruising on its own thing? That's exactly the topic of the next 45 minutes. OK. Good question. So with that insight, let me turn now to the question of identical particles or multiple particles. OK. So we're done with Coulomb for the moment. Pretty much for the rest of the course. So I want to move on to the following question. Suppose I have a system-- we've spent a lot of time thinking about a particle in a potential. I would like to think about multiple particles for a minute. And neat things happen for multiple particles that don't happen for individual, isolated particles. So let's think about what those-- let's think about the physics of multiple particles. So in particular, classically-- in classical mechanics-- if I have two particles, what is the information I have to specify? I have to specify the state. I have to specify the position of the first particle. X1 and its momentum, p1. And then the position of the second particle x2 and p2. I'm going to omit vectors over everything, but everything is in the appropriate number of dimensions. So in classical mechanics-- in quantum mechanics, the state of the system is specified by a wave function, which is a function of the positions-- or the degrees of freedom, let's say x1 and x2. And x1 and x2 and p1 and p2 are promoted to operators representing these observables. And the wave function is a function-- now let me just quickly tell you what this notation means. What this notation means is x1 is some number, like 7. And the quantity in the first spot, this means that the first particle indicated by the first slot, is at x1. And the second particle is at x2. So whenever I write a wave function, what I mean is this is the probability amplitude-- the thing whose norm squared is probability-- to find the first particle at this value and the second particle of that value. So the 1 and 2 label the points, not the particles. The particle is labeled by the position inside this wave function. Everyone cool with that? OK. So that's just a notation. So the quantum mechanical description of two particles is a wave function of both positions and operators representing the position and the coordinates of the particles. And the probability-- actually, let's go ahead and finish up here-- and the probability density to find the first particle. First at x1, and the second at x2. It's just the norm squared of the amplitude, psi x1, x2, norm squared. Just as usual. That's the probability density. All right? Everyone cool with that? And what are x1, x2, commutator? What is this commutator? AUDIENCE: [INAUDIBLE]. PROFESSOR: Can you know the position of the first particle and the position of the second particle? Simultaneously? Sure. I'm here. You're there. 0. And x1 with p1-- just work in one dimension for the moment-- is equal to? AUDIENCE: [INAUDIBLE]. PROFESSOR: Our h bar. And the same would have been true of x2 and p2. And what about x1 with p2? 0. These are independent quantities. OK. So it's more or less as you'd expect. So just to be explicit about this, let's do an example. Two free particles. So what's the energy operator? Well, the energy operator is equal to p squared upon 2m for the first particle, 1, plus p squared 2 upon 2m 2. Second particle, its mass is m2. The first particle has mass m1. And the system is free, so this is plus 0. Yeah? AUDIENCE: So for the [INAUDIBLE] exponentially, is that true [INAUDIBLE] particles [INAUDIBLE]?? PROFESSOR: Yeah. OK. So the question is, x1, x2, commutator equals 0? Is that true even if there are forces between the particles? So what are the forces between particles-- what are they going to contribute to? Yeah, the potential. And that's going to show up in the energy operator. So that will certainly matter and it will change what the energy eigenvalues are. But it won't tell what can or can't be measured. It won't tell you what properties the system can have or not. Good. Yeah? AUDIENCE: What's with the particles in the state in which measuring the position of one makes the position of the other one in certainty? PROFESSOR: We'll talk about things like that in a minute. AUDIENCE: Does it screw this up? PROFESSOR: No. This is-- do the commutation relations care what state you're in? They're relations amongst operators. So they're independent of the state always. OK. Yeah? AUDIENCE: For the two free particles, are we assuming that they're not charged particles? PROFESSOR: Yeah. I'm assuming that there are no interactions whatsoever. Just totally free particles. Potential is zero. V of x1, x2 equals 0. So they're not charged. They're just totally uninteresting. No interactions, no forces, no potential. OK. So we know how to solve this problem because we can use separation. Psi of x1, x2 can be written as-- well, we know what the solutions of p1 upon 2m is equal to er. So we can write this as chi of x1. And actually, I'm going to call the positions of these guys a and b. Because the subscriptions are just terribly misleading. So the first point is called a, the second point is called b. So a and b. And this is p1 and p2 the momentum of the two guys. Chi of a and phi of b. So all I'm doing here is I'm using separation. Pa, pb. So here's, instead of calling the positions x1 and x2, I'm going to call them a and b just because it's going to be easier to write and make things clear. So I'm using separation. And then p1 squared, or pa squared, upon 2m on cih of a is equal to ea, chi a. And ditto for b. This tells us that chi of a is equal to e to the sub-coefficient c, e to the i k a. Everyone cool with that? A is just replacing x1. So it's just the position. So this is just saying that the solution to a single free particle is a plane wave. And because it's just the sum of the momentum squared, we can separate the equation and the same thing obtains for psi. So similarly for phi. Ditto. Phi of b is equal to the e to the i k sub b. I'll call this k sub a to distinguish them. So the wave function's a basis for the wave functions for two particles, psi of a b with energy e is equal to e to the i ka A plus kb B, Where E is equal to h bar squared on 2m K sub a squared plus k sub b squared. Yeah? For a system of two free particles, is every wave function of the form chi of a times phi of b? If you have a system that's separable, is every wave function, is every solution itself, separated? AUDIENCE: No. PROFESSOR: No. Because we can have arbitrary superpositions of forms of this type. So we get superpositions of plane waves as long as the energies of each plane wave in the superposition are equal to e-- the total energy is equal to e-- I can just superpose them and I still have an energy eigenfunction. Everyone cool with that? OK. So nothing shocking here. But I did this example so that we'd have the notation of chi and phi for the two states. OK. So now let me come to this question of what about the proton? Well suppose I have a system now which is not a free particle. It's two particles, a and b. And e is equal to pa squared over 2a plus pb squared upon 2mb. Plus a potential that depends only on a minus b. OK? So this is, for example, what happens in the Coulomb potential when you include the proton having a finite mass instead of being infinitely massive and stuck still. So now we can do exactly the same thing we did in classical mechanics when you have a potential that only depends on the distance between two things. I can reorganize degrees of freedom into the center of mass position. R is equal to 1 over ma plus mb of ma, a, plus mb, b. So that's the center of mass motion. And the relative distance is equal to 1 over-- whoops. I don't need that. A minus b. And then if you do this and you write out the energy operator, e is equal to minus h bar squared upon 2 capital M, total mass. Dr squared Plus 1 minus h bar squared over 2 mu d little r squared plus v or r. OK? So this is exactly what happens in classical mechanics. You work in terms of the center of mass coordinate and the relative coordinate. The relative coordinate becomes effectively an independent degree of freedom with a potential, which is the central potential. And the center of mass coordinate is a free particle. So if we have a proton and our electron and they're attracted to each other by Coulomb there's a center of mass motion. And then they do together whatever they do together, a la Coulomb potential. Everyone cool with that? The only difference is that the mass in the Coulomb potential is not the mass of the bare electron. But it's the geometric mean mu is equal to ma and b over ma plus mb. Now for a proton and an electron, if ma is the proton and mb is the electron, then this is proton electron over proton plus electron. But proton is about 2,000 times the mass of the electron. So this is basically the mass of the proton and they factor out. So the effective reduced mass is roughly equal to the electron mass. Corrections of a part in 1,000. OK? So to answer your question, is the proton also in some complicated state described it? Well in fact, neither the electron nor the proton are described by the Coulomb potential. But the relative position, the relative radial distance between them, is controlled by the Coulomb potential and the center of mass degree of freedom is a free particle. Does that make sense? OK. Good. OK. So, so much for that example. The free particle and the central potential. Here's the much more interesting that happens when we have multiple particles. Yeah? AUDIENCE: Can you explain what you mean by the [INAUDIBLE]?? PROFESSOR: Oh, yeah, sorry. This is the-- I generated the gradient with respect to r. So if r is a vector, this is the gradient with respect to r, norm squared. And this is the gradient with respect to the relative coordinate. So for example, this is-- if we're in one dimension, so this is strictly one dimensional. Then this is just the derivative with respect to r squared. And this is derivative with respect to little r squared. Just the gradient squared. Did that answer your question? AUDIENCE: Yeah. What about in hydrogen? PROFESSOR: Well then it's gradient operator. The thing that takes the function gives you a vector which is the directional derivative of that-- AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah. Exactly. In the r direction. AUDIENCE: Is that subraction right there? PROFESSOR: Sorry? AUDIENCE: Is that a subtraction between-- PROFESSOR: A subtraction. Where? AUDIENCE: Between the first and second [INAUDIBLE].. PROFESSOR: Minus h bar squared upon 2m, dr squared. Minus h bar squared over 2 mu, dr-- this? AUDIENCE: OK, so that's-- those are separate terms, not multiplied? PROFESSOR: Yeah. They're not multiplied, right? It's just the sum. So the energy is kinetic energy of the relative-- of the center of mass. Kinetic energy of the relative degree of freedom. And then potential energy. Which is exactly what happens in classical mechanics. Yeah? AUDIENCE: What's that circle under the first term? PROFESSOR: This one? M? It's the total mass. Ma plus mb. And this one is the reduced mass ma mbu upon ma [INAUDIBLE].. OK. So here's a cool thing that happens with multiple particles that didn't happen previously. Suppose we have identical particles. So in particular, imagine I have two billiard balls. So I have two billiard balls and I shoot-- I send one in from one side. And I send in the other in from the other side. And then they collide and there's some horrible chaos that happens. And one goes flying out to this position, a. And the other goes flying out to this position, b. OK? Now here's my question. Which ball went to a and which ball went to b? Well if we did this experiment, that would be easy to answer. Because we could paint a little 1 on this one and a little 2 on this one and they'd go flying out. And then at the end, when you catch the ball at a and you catch the ball at b, you can grab them and look at them and say aha, this one in my left hand which I got from a has 1 on it, and this one has 2 on it, and I'm done. The other way we could have observed which ball went where is we could've taken a high speed film of this and watched frame by frame and said aha, particle 1, particle 1, particle 1, particle 1. Particle 2, 2, 2, 2. Right? We could have just followed the paths. And we haven't done anything to the experiment. We just took a film. We haven't messed with it. We don't change the results of the experiment. We just watch. Right? Perfectly doable classically. Quantum mechanically, is this doable? AUDIENCE: No. PROFESSOR: No. Because first off, if you watch carefully along and figure out did it go through this slit, did it go through that slit? You know you change the results. 100% white versus 50-50. If you go back to the boxes. And meanwhile, if they're truly identical particles like electrons, there's no way to paint anything on the damn particle. They're just electrons. And they're completely, as far as anyone's ever been able to tell, completely and utterly identical. They cannot be distinguished in any way whatsoever. So you can't do the thing where you grab the one from a and grab the one from b and say aha, this one had the 1 on it. They're indistinguishable. Yeah. AUDIENCE: [INAUDIBLE] PROFESSOR: We're going to come to the Pauli exclusion principle. Hold on to that. Hold on to that question. We're going to come-- we're going to get there. OK. So we can't-- if we have truly identical particles, what we mean by that is there's no way to run this experiment and determine which particle ended up-- which of these two particles went to a and which went to b. Everyone cool with that? They're identical. And what I want to understand is, what are the consequences? So the first and basic consequence of this is that the probability that the first particle ends up at a and the second particle ends up at b must be equal to the probability that the first particle ends up at b and the second particle ends up at a. Because you can't tell which is which. If you can't tell, it must be that those probabilities are equal. Because if they weren't equal, you effectively have skewed the results and they're distinguishable. These are totally indistinguishable. So the probability that the first particle ends up at a. second at b, must be equal to the probability the first particle ends up at b and the second ends up at a, because you cannot tell the difference. This is what it means to be identical. That equals sign is what I mean by saying I have identical particles. Cool? So let's find out what the consequences of this are. Define the following operator. And Dave Larson, if you're watching, this is the [INAUDIBLE] [INAUDIBLE] p. So I'm going to call this operator script-y p sub 1,2. It's the operator-- so it's got a little hat on it, one more offense-- which takes the first particle and the second particle and swaps them. OK? So-- and I guess I don't even need the 1, 2. This p operator swaps particle 1 and particle 2. So for example, it takes probability of a to b to probability of b to a. But more importantly, this swapping operation takes the wave function, the amplitude, of a and b, and it swaps a and b, psi of ba. Now it's clear that the probability-- that the swapping operation does nothing to the probability, because the fact that they're identical particles means that these are equal to each other. So it hasn't changed the answer. But just because the probabilities are equal to each other, does that tell you that the wave function is invariant under swapping the particles? AUDIENCE: No. PROFESSOR: No. It doesn't have to be invariant. The important thing is that the norm squared of psi. So in principle, this could be equal to some phase, e to the i of phi sub ab. Let me call it theta sub ab. Psi of a,b. And if this was the case, that there was a phase that we got here, when we take the norm squared, the probability remains the same. OK? On the other hand, we know something else. We know that if we take the wave function psi ab. And we swap it and then we swap it again, what do we get? I have ab. But that means this is equal to e to the 2i theta ab. So swapping twice had better give me 1, so this had better be equal to 1. Let me write this slightly differently. This is e to the i theta squared is equal to 1. So what must be true of e to the i theta of our phase? It's a number that squares to 1. So it could be one of two values. It could be 1 or it could be minus 1. That's it. So what that tells us is-- psi ab. So that tells us that p-- whoops. P on psi ab is equal plus or minus psi ba. Sorry, ab. Another way to say this is just that-- another way to say this is that p squared acting on psi is just psi again. So the eigenvalues of p have to be plus or minus 1. Here they are. So in fact, this is-- let me phrase this in a little more correct way. This tells us that the eigenvalues of p are plus and minus 1. Yeah? AUDIENCE: I'm trying to figure out how it can be minus 1? If p squared is psi then it [INAUDIBLE] has to be psi [INAUDIBLE]? PROFESSOR: Yes. AUDIENCE: How could we add that other [? p? ?] PROFESSOR: Good. OK. So let's check this quickly. So the question is how can it be minus 1. How can that-- that doesn't-- that would seem to violate our calculations. So what this is saying is that, we know if we take p on, let's say, p along psi of ab. Let's say-- let psi be an eigenfunction of p. OK? So if psi if an eigenfunction of p, with eigenvalue minus 1, then this p on psi is equal to minus psi. Yeah? Now the probability is equal to psi squared. And so p on psi-- sorry, p on psi squared is equal to minus psi squared. We take each side, it goes to minus psi. So this is just equal to psi squared. Yeah? Who asked the question? Sorry. Right. OK. So it leaves the norm squared invariant. So it's OK to have a minus 1 eigenvalue under p, because that doesn't change the probability distribution. The probability distribution is left invariant. However, if we take p squared on psi, that's equal to p on p on psi, is equal to the p on minus psi, which is equal to minus minus psi, which is equal to psi. OK? So this is the statement that the square acts as the identity. Did that answer your question? AUDIENCE: [INAUDIBLE]. PROFESSOR: OK. OK. AUDIENCE: Along the [INAUDIBLE],, we need p squared on side B to go back to side B. It would preserve the probability regardless if we had had the two [INAUDIBLE].. PROFESSOR: Good. Why did p squared have to be the identity? Because what is p doing? p takes two particles and it swaps them. AUDIENCE: Oh. PROFESSOR: And if it swaps them again, what do you get? The original configuration. Right? If by swapping, if by p, you mean the thing that swaps those particles, then doing it twice is like not doing anything at all. You can define a different quantity which isn't this swapping operation which does this twice and gives you something else. That's perfectly reasonable. But I'm going to be interested in the operator, which is just swap. And if you do it twice, you get back to the identity. AUDIENCE: Oh. OK. PROFESSOR: Cool? OK. Yeah? AUDIENCE: So regarding the defaults, is what the operation is doing is changed particle number 1, particle number 2, [INAUDIBLE]?? PROFESSOR: Well-- AUDIENCE: What does it do with [INAUDIBLE]?? PROFESSOR: Right. So at any-- given any configuration, at any moment time, right, pick your wave function, pick your state. For example, two particles here. What the script p operator with swapping operator does, it just swaps them. So it swaps the position of one and the position of the other. AUDIENCE: [INAUDIBLE]. PROFESSOR: Right. Exactly. And that-- you do that, at some moment in time. You do that to a state. So in that experiment, what-- I mean, there's no answer the question, you know, does p swap them before or does it swap them after. It's up to you. You can apply the p operator anytime you want. Cool? OK. Yeah? AUDIENCE: So when you extracted the probabilities, obviously if you have a case where it's really far apart and the two particles end up [INAUDIBLE],, aren't they more likely that they have not moved the entire distance in between? PROFESSOR: Yeah, that sounds reasonable. AUDIENCE: It sounds like-- PROFESSOR: Yeah. This is disconcerting. AUDIENCE: --kind of like the wave function. [INAUDIBLE] PROFESSOR: That's exactly right. So that's an excellent observation. Let me rephrase that slightly. So here's the observation that she's making. It's exactly correct and it's where we're going to get in a few minutes. So the observation is this. Look, imagine I take two electrons, which for the moment we'll just call identical particles, so we take two identical particles. Put one in my right hand, one in my left hand. And I just hold them there and I wait for a while. A while later, is it the same electron in my right hand? AUDIENCE: Probably. PROFESSOR: I don't know. They're identical. I can't tell. They're completely identical. And so if you think about this like you do some like weak scattering-- if you did some weak scattering process between these where you pick them very far apart, very slowly moving along-- but they're very far away so the electrostatic interaction is very small and so they repel each other just a little bit. If this system is, in fact, identical and if the wave function is, let's say for the moment-- and we'll talk about whether this is correct or not-- if the wave function is invariant under swapping the particles-- let's just imagine that it's invariant and they're swapping the particles-- then there are two things that could have happened. The particles could have done this. Or there's also a contribution where they do this. Which kind of hurts. And in order for the system to be symmetric, you have to have both contributions. So let's come to that. But indeed, it's as if there's some additional interactions, or some additional correlations. And that's exactly what we want to study. So let's get to that. Very good observation. So what I'd like to do is make that precise. So there are two kinds of particles-- or three kinds of particles, I should say-- in the world from this point of view. The first kind of particle are distinguishable particles. Suppose I have two particles, one with a mass m and one with a mass 2000m. Say just to pick randomly a number. Right? Those are distinguishable because you can weigh them. So you can tell which one is the heavy one, which one is the light one. And you can tell. Cool? So there are distinguishable particles. And if we have distinguishable particles-- I'll call psi sub d for distinguishable-- then it's OK to have the following thing. Psi distinguishable first particle is-- the amplitude for the first particle would be an a and the amplitude for the second particle would be a b, could be chi of a, some function of a, and phi of b. This is not invariant under a goes to b. Because under a goes to b, it becomes chi of b, some function of b, phi of a, some different function of a. That's distinct. But it doesn't matter. They're distinguishable. So that's perfectly fine. It's not true p of ab is not equal to p of ba. But that's OK, because they're distinguishable. Everyone cool with that? AUDIENCE: So p of ba is i of p? PROFESSOR: Yeah. Exactly. So p of ab, this is by definition equal to norm squared of psi d-- and I should say d. D. Norm squared of psi d of ab squared, which is equal to norm squared of chi of a, phi of b squared. Whereas this guy would have been chi of b, phi of a, norm squared. And since those are just some stupid functions-- I haven't told you what they are, just some random functions-- then they're just different probability distributions. So on the other hand, if we have indistinguishable wave functions, then psi indistinguishable, we know that psi squared of a,b squared is equal to psi of ba norm squared. And this is not of that form. So we have two possibilities. I'll write this as psi plus minus. If I know one of the particles is in the state described by chi, and the other particle is in this state described by phi, this would be an example of a wave function with that property. However, it's not invariant under swapping a and b. So how could I make it invariant under swapping a and b? Well I could do the following. 1 over root 2, chi of a, phi of b. And if I want to make it invariant, I could add plus chi of b, phi of a. And now if I swap a and b, here this becomes chi of b, phi a, but that's exactly this term. And this becomes chi of a, phi b. That's this term. So just swap them. Yeah? But we don't need the wave function to be invariant under swapping. We just need it to be invariant under-- up to a sign. So the other option is to have a minus sign here. And this gives us that the swapping operation, p, acting on psi plus minus of a,b is equal to plus minus psi of a,b. And the plus is generally called the symmetric, and the minus, the anti-symmetric combination. OK. So distinguishable particles can just be in some random state, but there are constraints on what states, what combinations of states are allowed, for indistinguishable particles. If you can't tell the difference between two indistinguishable particles and you know one is in the state chi and the other in the state phi, this cannot be the wave function. It must be either chi phi plus chi phi in this fashion, or minus. Everyone cool with that? Yeah? AUDIENCE: I have a question about what we mean by p of ab. So normally, when we talk about probabilities we say that, yes, if you measure a system what's the probability that the metric value will equal that? PROFESSOR: Yes. AUDIENCE: But if we can't even determine anything from that type of system, what do we have a probability of? PROFESSOR: Good. So what this probability means is, what's the probability that if-- that upon observation in the system, I find the first particle to be at a and the second part to be at b. Right? And I can check that by saying like, look, I catch the first particle. I catch the second particle. Is this a? No. OK. Then that gives zero to the probability distribution. I do that a billion times and I build up statistics. And if I'm a fourth-- one out of four times, I'll find a particle, the first particle-- or I'll find a particle at a. One out of four times, I'll find a partial at b. And the probability that I find the first particle at a and the second particle at b is one tenth, say. OK. AUDIENCE: So we can never make that if they're identical, right? PROFESSOR: What you can't do if they're identical is you can't say which particle you caught at a. This is saying a particle at a or a particle at b, right? But it's-- but whether this is the same or not of probability that I find a particle at b-- the first particle a b and the second particle at a. If you can't tell the difference then they're just a particle at a and a particle at b. OK. OK. So what does this give us? So this gives us a couple of nice facts. So imagine-- that's an exciting sound. So this gives us a couple of nice facts with which we can find awesomeness in the world. The first is the following. If you have identical particles, then the energy can't depend on the order. If you have identical particles and they're truly identical, you swap them, then the energy will be the same. If it wasn't the same, then they're distinguishable by figuring out what the energy is. So in order that they're identical, it must be true that if you swap the particles, and then compute the energy, this should be the same as what you get if you first compute the energy and then swap them. Which is to say that the commutator of e with the swapping operator, p, is 0. OK? But what that tells you is that the expectation value of p doesn't change in time. In particular, if it's some initial state-- if you were initially p on psi is equal to plus psi at time 0, then psi-- then p at psi, p psi, is equal to plus psi for all future times. P psi of p is equal to plus psi of t. OK. So if you have two identical particles and the wave function is invariant under swapping them at some moment in time, then it will always be invariant under swapping them. It's a persistent property of particles that the wave function is invariant under swapping them. Yeah? Yeah. AUDIENCE: What about things that like, become indistinguishable. For example, you have atomic nuclei like, for uranium. And one of them is in like a heavier isotope. And during the time that you're like, holding them in your hands one of them decays, now they're the same isotope. PROFESSOR: Yeah. That's-- OK. AUDIENCE: [INAUDIBLE]. PROFESSOR: This tells you something. This is a very good question. So the question is, suppose I have an excited isotope of uranium that' s distinguishable from some other isotope of uranium. I wait for a while and then this thing decays down to the state-- it won't decay to the stabilized isotope of uranium, but whatever-- it decays down to-- we could imagine a universe in which it did. It decays down to a stable state. And-- I mean, uranium's never stable. But anyway, you get the idea. At which point they're indistinguishable. This sounds better. Because now the wave function should be invariant. But it started out not being invariant. What's the problem in this argument? The system has changed. In particular, something went flying out. So this is actually kind of a nice way to argue that there must have been something else. The wave function describes a full system. But if something leaves, then it's not the same system anymore and the wave functions isn't describing the same degrees of freedom. Something left. AUDIENCE: But suppose we keep track of that one, we still can't swap the two uraniums. PROFESSOR: Exactly. Then-- well, then there's some additional constraint, right? It must be invariant under swapping that. But it must-- but the wave function also knows about that extra bit that went flying off. And so the whole wave function has to be invariant under swapping the identical parts, but not invariant under swap-- the invariance is not just those two things. They're correlated with that thing that went flying away. So another way to think about this is imagine two different hydrogen atoms. Here I've got a hydrogen atom. It's an electron and a proton. Here's a deuterium atom. It's an electron bound to a proton and a neutron glued together, deuteron. So are the electrons identical? Yeah, they're totally identical. So is the wave function invariant-- does the wave function-- or the probability distribution have to be invariant under swapping the electrons? Yes, they're identical. So the probability distribution must be invariant under swapping the electrons. However, is an identical under swapping the hydrogen with the deuterium? AUDIENCE: No. PROFESSOR: No. So it's invariant under swapping the identical parts and not the non-identical parts. Cool? AUDIENCE: Yeah. PROFESSOR: OK. So this tells us that there are two kinds of particles. There are persistent-- or sorry, there are three kinds of particles. And these properties are persistent. The first kind of particle-- sets of particles-- are distinguishable particles. If you have two particles which are distinguishable then you're done. They're distinguishable. Nothing else to say. Two, you can have identical particles with the property that if you take p on psi a,b-- sorry, p on psi. If you swap the particles, this is equal to plus psi. And then you have-- so identical with plus. And you have three identical particles where if you swap the particles, you get a minus sign on psi. These particles are called-- we have a name for particles of this kind. Bosons. These are called fermions. My TA did a bad thing to me when I was taking quantum mechanics. And said, just imagine them as little tiny Fermis. So just take a picture of Fermi in your head and imagine little tiny-- and this is cruel, because I can't help it. Every time someone in a seminar is like, blah, blah, blah, Fermi. And I'm like, damn it. Little Fermi. It's really quite annoying. So now you have it too. Great. So what are the consequences of the fact that there are two kinds of particles in the universe? These fermions and bosons? This has a really lovely consequence. The first is, suppose we have two fermions. OK? Examples of fermions are electrons. Suppose we have a wave function for two fermions. The first might-- what's that probability amplitude that the first is at a and the second is t b? Well it's this psi of a,b. And the statement that it's a fermion is the statement that this is equal to minus psi of b, a. If we swap the positions of the two particles, we must pick up a minus sign. This tells us in particular that the probability amplitude for the first particle to be at a and the second particle to be at a is equal to minus itself. Because upon swapping the particles, we get minus psi of a,a. So the probability amplitude to find two fermions at the same place is equal to 0. Two fermions cannot occupy the same state. This was the Pauli exclusion principle, which we needed to get the periodic table. Pauli. Two. If we have-- so this is fermions-- if we have bosons, psi of a,b-- let me write this out. So psi fermion or boson-- so fermion is going to be with a minus and boson is going to be with a plus-- is equal to, suppose I have two particles 1 over root 2. And one particle is in the state chi and the other particle is in the state phi. But in order to be fermionic or bosonic, in order for this to be invariant under swapping a and b, we have to have a plus or minus chi of b, phi of a. And here we immediately see this Pauli principle at work. If I could take the fermionic example with the minus sign, then psi evaluated at a, a must be chi at a, phi at b, minus chi at b, phi at a. But if a is b, this is chi a, phi a, minus chi a, phi a. That's zero. But let's think about the bosonic case. If we have a bosonic field-- or, if we have-- sorry. If we have bosonic identical particles, then psi b at a with a-- for our fermion, it was zero. But psi b of the amplitude to be at two at the same place is equal to-- well, if b is a, then these two terms are identical and we have a plus. This is root 2 chi at a, phi at a. Which is greater than what you might have naively guessed, which would have been just chi a, phi a. For bosons, they really like being next to each other. They really like being in the same place. And this will eventually lead to lasers. So from this simple statistical property under swapping, picking up a minus sign, we get the Pauli principle, which gave us the periodic table. And is going to give us in the next lecture bands and solids in conductivity. From the same principle but with a plus, and the persistence of this sine, from the persistence of the statistics, from the fact that we have two identical bosons, we get that they like to be in the same spot. And we'll get lasers and Bose-Einstein condensates. And next time, we'll pick up with fermions in a periodic potential and we'll study solids and get to diamond.
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_7_More_on_Energy_Eigenstates.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: OK, so I want to start out by reviewing a few things and putting some machinery together. Unfortunately, this thing is sort of stuck. We're going to need a later, so I don't know. I'll put it up for now. So first just a bit of notation. This symbol, you should think of it like the dot product, or the inner product. It's just saying that bracket f g is the integral. It's a number that you get. So this is a number that you get from the function f and the function g by taking f, taking its complex conjugate, multiplying it by g, and then integrating overall positions. All right? So it's a way to get a number. And you should think about it as the analog for functions of the dot product for vectors. It's a way to get a number out of two vectors. And so, for example, with vectors we could do v dot w, and this is some number. And it has a nice property that v dot v, we can think it as v squared, it's something like a length. It's strictly positive, and it's something like the length of a vector. Similarly, if I take f and take its bracket with f, this is equal to the integral dx of f squared, and in particular, f could be complex, so f norm squared. This is strictly non-negative. It could vanish, but it's not negative at a point, hence the norm squared. So this will be zero if and only if what? f is 0, f is the 0 function, right. So the same way that if you take a vector, and you take its dot product with itself, take it the norm, it's 0 if an only if the vector is 0. So this beast satisfies a lot of the properties of a dot product. You should think about it as morally equivalent. We'll talk about that in more detail later. Second, basic postulate of quantum mechanics, to every observable is associated an operator, and it's an operator acting on the space of functions or on the space of wave functions. And to every operator corresponding to an observable in quantum mechanics are associated a special set of functions called the eigenfunctions, such that when the operator acts on that function, it gives you the same function back times a constant. What these functions mean, physically, is they are the wave functions describing configurations with a definite value of the corresponding observable. If I'm in an eigenfunction of position with eigenvalue x naught, awesome. Thank you, AV person, thank you. So if your system is described by a wave function which is an eigenfunction of the position operator with eigenvalue x naught, that means you can be confident that the system is in the configuration corresponding to having a definite position x naught. Right? It's not a superposition of different positions. It is at x naught. Similarly, momentum, momentum has eigenfunctions, and we know what these guys are. These are the exponentials, e to the iKX's. They're the eigenfunctions, and those are the wave functions describing states with definite value of the momentum, of the associated observable. Energy as an operator, energy is described by an operator, which has eigenfunctions which I'll call phi sub n, with energy as E sub n, those are the eigenvalues. And if I tell you that your wave function is the state phi sub 2, what that tells you is that the system has a definite energy, E sub 2, corresponding to that eigenvalue. Cool? And this is true for any physical observable. But these are sort of the basic ones that we'll keep focusing on, position, momentum, and energy, for the next while. Now a nice property about these eigenfunctions is that for different eigenvalues, the associated wave functions are different functions. And what I mean by saying they're different functions is that they're actually orthogonal functions in the sense of this dot product. If I have a state corresponding to be at x 0, definite position x 0, that means they're in eigenfunction of position with eigenvalue x 0, and I have another that corresponds to being at x1, an eigenfunction of the position operator or the eigenvalue x1, then these wave functions are orthogonal to each other. And we get 0 if x 0 is not equal to x1. Everyone cool with that? Now, meanwhile not only are they orthogonal but they're normalized in a particular way. The inner product gives me a delta function, which goes beep once, so that if I integrate against it I get a 1. Same thing with momentum. And you do this, this you're checking on the problem set. I don't remember if it was last one or this one. And for the energies, energy 1, if I know the system is in state energy 1, and let's say e sub n and e sub m, those are different states if n and m are not equal to each other. And this inner product is 0 if n and m are not equal to each other and 1 if they are. Their properly normalized. Everyone cool with that? Yeah. AUDIENCE: Is it possible that two eigenfunctions have the same eigenvalue? PROFESSOR: Absolutely. It is absolutely possible for two eigenfunctions to have the same eigenvalue. That is certainly possible. AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah, good. Thank you, this is a good technicality that I didn't want to get into, but I'll go and get into it. It's a very good question. So the question is, is it possible for two different eigenfunctions to have the same eigenvalue. Could there be two states with the same energy , different states, same energy? Yeah, that's absolutely possible. And we'll run into that. And there's nice physics encoded in it. But let's think about what that means. The subsequent question is well, if that's the case, are they really still orthogonal? And here's the crucial thing. The crucial thing is, let's say I take one function, I'll call the function phi 1, consider the function phi 1. And let it have energy E1, so that E acting on phi 1 is equal to E1 phi 1. And let there be another function, phi 2, such that the energy operator acting on phi 2 is also equal to E1 phi 2. These are said to be degenerate. Degenerate doesn't mean you go out and trash your car, degenerate that the energies are the same. So what does this tell me? This tells me a cool fact. If I take a wave function phi, and I will call this phi star, in honor of Shri Kulkarni, so I've got this phi star, which is a linear combination alpha phi 1 plus beta phi 2, a linear combination of them, a superposition of those two states. Is this also an energy eigenfunction? Yeah, because if I act on phi star with E, then it's linear, so E acting on phi star is E acting on alpha phi 1, alpha's a constant, doesn't care. Phi 1 gives me an E1. Similarly, E acting on phi 2 gives me an E1. So if I act with E on this guy, this is equal to, from both of these I get an overall factor of E1. So notice that we get the same vector back, times a constant, a common constant. So when we have degenerate eigenfunctions, we can take arbitrary linear combinations to them, get another degenerate eigenfunction. Cool? So this is like, imagine I have a vector, and I have another vector. And they share the property that they're both eigenfunctions of some operator. That means any linear combination of them is also, right? So there's a whole vector space, there's a whole space of possible functions that all have the same eigenvalue. So now you say, well, look, are these two orthogonal to each other? No. These two? No. But here's the thing. If you have a vector space, if you have a the space, you can always find orthogonal guys and a basis for that space, yes? So while it's not true that the eigenfunctions are always orthogonal, it is true-- we will not prove this, but we will discuss the proof of it later by pulling the mathematician out of the closet-- the proof will say that it is possible to find a set of eigenfunctions which are orthogonal in precisely this fashion, even if there are degeneracies. OK? That theorem is called the spectral theorem. And we'll discuss it later. So it is always possible to do so. But you must be alert that there may be degeneracies. There aren't always degeneracies. In fact, degeneracies are very special. Why should two numbers happen to be the same? Something has to be forcing them to be the same. That's going to be an important theme for us. But it certainly is possible. Good question. Other questions? Yeah. AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah, so using the triangular brackets-- so there's another notation for the same thing, which is f g, but this carries some slightly different weight. It mean something slightly-- you'll see this in books, and this means something very similar to this. But I'm not going to use this notation. It's called Dirac notation. We'll talk about it later in the semester, but we're not going to talk about it just yet. But when you see this, effectively it means the same thing as this. This is sort of like dialect. You know, it's like French and Quebecois. Other questions? My wife's Canadian. Other questions? OK. So given this fact, given the fact that we can associate observables to operators, operators come with special functions, the eigenfunctions, those eigenfunctions corresponding to have a definite value of the observable, and they're orthonormal. This tells us, and this is really the statement of the spectral theorem, that any function can be expanded in a basis of states with definite values of some observable. So for example, consider position. I claim that any wave function can be expanded as a superposition of states with definite position. So here's an arbitrary function, here's this set of states with definite position, the delta functions. And I can write any function as a superposition with some coefficients of states with definite position, integrating over all possible positions, x0. And this is also sort of trivially true, because what's this integral? Well, it's an integral, dx0 over all possible positions of this delta function. But we're evaluating at x, so this is 0 unless x is equal to x0. So I can just put in x instead of x0, and that gives me psi of x. Sort of tautological We can do the same thing for momentum eigenfunctions. I claim that any function can be expanded in a superposition of momentum eigenfunctions, where I sum over all possible values in the momentum with some weight. This psi tilde of K is just telling me how much amplitude there is at that wave number. Cool? But this is the Fourier theorem, it's a Fourier expansion. So purely mathematically, we know that this is true. But there's also the physical statement. Any state can be expressed as a superposition of states with definite momentum. There's a math in here, but there's also physics in it. Finally, this is less obvious from a mathematical point of view, because I haven't even told you what energy is, any wave function can be expanded in states with definite energy. So this is a state, my state En, with definite energy, with some coefficient summed over all possible values of the energy. Given any physical observable, any physical observable, momentum, position, angular momentum, whatever, given any physical observable, a given wave function can be expanded as some superposition of having definite values of that. Will it in general have definite values of the observable? Well a general state be an energy eigenfunction? No. But any state is a superposition of energy eigenfunctions. Will a random state have definite position? Certainly not. You could have this wave function. Superposition. Yeah. AUDIENCE: Why is the energy special such that you can make an arbitrary state with a countable number of energy eigenfunctions rather than having to do a continuous spectrum? PROFESSOR: Excellent question. So I'm going to phrase that slightly differently. It's an excellent question, and we'll come to that at the end of today's lecture. So the question is, those are integrals, that is a sum over discrete things. Why? Why is the possible values of the position continuous, possible values of momentum continuous, and possible values of energy discrete? The answer to this will become apparent over the course of your next few problem sets. You have to do some problems to get your fingers dirty to really understand this. But here's the statement, and we'll see the first version of this at the end of today's lecture. Sometimes the allowed energies of a system, the energy eigenvalues, are discrete. Sometimes they are continuous. They will be discrete when you have bound states, states that are trapped in some region and aren't allowed to get arbitrarily far away. They'll be continuous when you have states that can get arbitrarily far away. Sometimes the momentum will be allowed to be discrete values, sometimes it will be allowed to be continuous values. And we'll see exactly why subsequently. But the thing I want to emphasize is that I'm writing this to emphasize that it's possible that each of these can be discrete or continuous. The important thing is that once you pick your physical system, you ask what are the allowed values of position, what are the allowed values of momentum, and what are the allowed values of energy. And then you sum over all possible values. Now, in the examples we looked at yesterday, or last lecture, the energy could have been discrete, as in the case of the infinite well, or continuous, as in the case of the free particle. In the case of a continuous particle this would have been an integral. In the case of the system such as a free particle, where the energy could take any of a continuous number of possible values, this would be a continuous integral. To deal with that, I'm often going to use the notation, just shorthand, integral sum. Which I know is a horrible bastardization of all that's good and just, but on the other hand, emphasizes the fact that in some systems you will get continuous, in some systems discrete, and sometimes you'll have both continuous and discrete. For example, in hydrogen, in hydrogen we'll find that there are bound states where the electron is stuck to the hydrogen nucleus, to the proton. And there are discrete allowed energy levels for that configuration. However, once you ionize the hydrogen, the electron can add any energy you want. It's no longer bound. It can just get arbitrarily far away. And there are an uncountable infinity, a continuous set of possible states. So in that situation, we'll find that we have both the discrete and continuous series of possible states. Yeah. AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah, sure, if you work on a lattice. So for example, consider the following quantum system. I have an undergraduate. And that undergraduate has been placed in 1 of 12 boxes. OK? Now, what's the state of the undergraduate? I don't know. Is it a definite position state? It might be. But probably it's a superposition, an arbitrary superposition, right? Very impressive undergraduates at MIT. OK, other questions. Yeah. AUDIENCE: Do these three [INAUDIBLE] hold even if the probability changes over time? PROFESSOR: Excellent question. We'll come back to that. Very good question, leading question. OK, so we have this. The next thing is that energy eigenfunctions satisfy some very special properties. And in particular, energy eigenfunctions have the property from the Schrodinger equation i h bar d t on psi of x and t is equal to the energy operator acting on psi of x and t. This tells us that if we have psi x 0 time t 0 is equal to phi n of x, as we saw last time, then the wave function, psi at x at time t is equal to phi n of x. And it only changes by an overall phase, e to the minus i En t over h bar. And this ratio En upon h bar will often be written omega n is equal to En over h bar. This is just the Dupre relations. Everyone cool with that? So are energy eigenfunctions-- how to say. No wave function is more morally good than another. But some are particularly convenient. Energy eigenfunctions have the nice property that while they're not in a definite position and they don't necessarily have a definite momentum, they do evolve over time in a particularly simple way. And that and the superposition principle allow me to write the following. If I know that this is my wave function at psi at x at time 0, so let's say in all these cases, this is psi of x at time 0, how does this state evolve forward in time? It's kind of complicated. How does this description, how does psi tilde of k evolve forward in time? Again, kind of complicated. But when expressed in terms of the energy eigenstates, the answer to how it evolves forward in time is very simple, because I know that this is a superposition, a linear combination of states with definite energy. States with definite energy evolve with a phase. And the Schrodinger equation is linear, so solutions of the Schrodinger equation evolve to become solutions of the Schrodinger equation. So how does this state evolve forward in time? It evolves forward with a phase, e to the minus i omega n t. One for every different terms in this sum. Cool? So we are going to harp on energy functions, not because they're more moral, or more just, or more good, but because they're more convenient for solving the time evolution problem in quantum mechanics. So most of today is going to be about this expansion and qualitative features of energy eigenfunctions. Cool? OK. And just to close that out, I just want to remind you of a couple of examples that we did last time, just get them on board. So the first is a free particle. So for free particle, we have that our wave functions-- well, actually let me not write that down. Actually, let me skip over the free particle, because it's so trivial. Let me just talk about the infinite well. So the potential is infinite out here, and it's 0 inside the well, and it goes from 0 to L. This is just my choice of notation. And the energy operator, as usual, is p squared upon 2m plus u of x. You might say, where did I derive this, and the answer is I didn't derive this. I just wrote it down. It's like force in Newton's equations. You just declare some force and you ask, what system does is model. So here's my system. It has what looks like a classical kind of energy, except these are all operators. And the potential here is this guy, it's 0 between 0 and L, and it's infinite elsewhere. And as we saw last time, the solutions to the energy eigenvalue equation are particularly simple. Phi sub n of x is equal to root properly normalized 2 upon L sine of Kn x, where kn is equal to n plus 1 pi, where n is an integer upon L. And these were chosen to satisfy our boundary conditions, that the wave function must vanish here, hence the sine, and K was chosen so that it turned over and just hit 0 as we got to L. And that gave us that the allowed energies were discrete, because the En, which you can get by just plugging into the energy eigenvalue equation, was equal to h bar squared Kn squared upon 2m. So this tells us a nice thing. First off, in this system, if I take a particle and I throw it in here in some arbitrary state so that at time t equals zero the wave function x 0 is equal to sum over n phi n of x Cn. OK? Can I do this? Can I just pick some arbitrary function which is a superposition of energy eigenstates? Sure, because any function is. Any function can be described as a superposition of energy eigenfunctions. And if I use the energy eigenfunctions, it will automatically satisfy the boundary conditions. All good things will happen. So this is perfectly fine initial condition. What is the system at time t? Yeah, we just pick up the phases. And what phase is this guy? It's this, e to the minus i omega n t. And when I write omega n, let me be more explicit about that, that's En over h bar. So that's h bar Kn squared upon 2m t. Cool? So there is our solution for arbitrary initial conditions to the infinite square well problem in quantum mechanics. And you're going to study this in some detail on your problem set. But just to start with a little bit of intuition, let's look at the wave functions and the probability distributions for the lowest lying states. So for example, let's look at the wave function for the ground state, what I will call psi sub 0. And this is from 0 to L. And I put these bars here not because we're looking at the potential. I'm going to be plotting the real part of the wave function. But I put these walls here just to emphasize that that's where the walls are, at x equals 0 and x equals L. So what does it look like? Well, the first one is going to sine of Kn x. n is 0. Kn is going to be pi upon L. So that's again just this guy. Now, what's the probability distribution associated with psi 0? Where do you find the particle? So we know that it's just the norm squared of this wave function and the norm squared is here at 0, it's 0 and it rises linearly, because sine is linear for small values. That makes this quadratic, and a maximum, and then quadratic again. So there's our probability distribution. Now, here's a funny thing. Imagine I take a particle, classical particle, and I put it in a box. And you put it in a box, and you tell it, OK, it's got some energy. So classically it's got some momentum. So it's sort of bouncing back and forth and just bounces off the arbitrarily hard walls and moves around. Where are you most likely to find that particle? Where does it spend most of its time? It spends the same amount of time at any point. It's moving at constant velocity. It goes boo, boo, boo, boo, right? So what's the probability distribution for finding it at any point inside, classically? Constant. Classically, the probability distribution is constant. You're just as likely to find it near the wall as not near the wall. However, quantum mechanically, for the lowest lying state that is clearly not true. You're really likely to find it near the wall. What's up with that? So that's a question that I want to put in your head and have you think about. You're going to see a similar effect arising over and over. And we're going to see at the very end that that is directly related, the fact that this goes to 0, is directly related, and I'm not kidding, to the transparency of diamond. OK, I think it was pretty cool. They're expensive. It's also related to the transparency of cubic zirconium, which I guess is less impressive. So the first state, again, let's look at the real part of psi 1, the first excited state. Well, this is now a sine with one extra-- with a 2 here, 2 pi, so it goes through 0. So the probability distribution associated with psi 1, and I should say write this as a function of x, looks like, well, again, it's quadratic. But it has a 0 again in the middle. So it's going to look like-- oops, my bad art defeats me. OK, there we go. So now it's even worse. Not only is unlikely to be out here, it's also very unlikely to be found in the middle. In fact, there is 0 probability you'll find it in the middle. That's sort of surprising. But you can quickly guess what happens as you go to very high energies. The real part of psi n let's say 10,000, 10 to the 4, what is that going to look like? Well, this had no 0s, this had one 0, and every time you increase n by 1, you're just going to add one more 0 to the sign. That's an interesting and suggestive fact. So if it's size of 10,000, how many nodes are there going to be in the middle of the domain? 10,000. And the amplitude is going to be the same. I'm not to be able to do this, but you get the idea. And now if I construct the probability distribution, what's the probability distribution going to be? Probability of the 10,000th psi sub 10 to the 4 of x. Well, it's again going to be strictly positive. And if you are not able to make measurements on the scale of L upon 10,000, but just say like L over 3, because you have a thumb and you don't have an infinitely accurate meter, what do you see? You see effectively a constant probability distribution. And actually, I shouldn't draw it there. I should draw it through the half, because sine squared over 2 averages to one half, or, sorry, sine squared averages to one half over many periods. So what we see is that the classical probability distribution constant does arise when we look at very high energy states. Cool? But it is manifestly not a good description. The classical description is not a good description. Your intuition is crappy at low energies, near the ground state, where quantum effects are dominating, because indeed, classically there was no minimum energy. Quantum effects have to be dominating there. And here we see that even the probability distribution's radically different than our intuition. Yeah. AUDIENCE: [INAUDIBLE] PROFESSOR: Keep working on it. So I want you all to think about what-- you're not, I promise you, unless you've already seen some quantum mechanics, you're not going to be able to answer this question now. But I want you to have it as an uncomfortable little piece of sand in the back of your oyster mind-- no offense-- what is causing that 0? Why are we getting 0? And I'll give you a hint. In quantum mechanics, anytime something interesting happens it's because of superposition and interference. All right. So with all that said, so any questions now over this story about energy eigenfunctions expanding in a basis, et cetera, before we get moving? No, OK. In that case, get out your clickers. We're going to test your knowledge. Channel 41, for those of you who have to adjust it. [CHATTER] Wow. That's kind of worrying. Aha. OK, ready? OK, channel 41, and here we go. So go ahead and start now. Sorry, there was a little technical glitch there. So psi 1 and psi 2 are eigenstates. They're non-degenerate, meaning the energies are different. Is a superposition psi 1 plus psi 2 also an eigenstate? All right, four more seconds. All right. I want everyone to turn to the person next to you and discuss this. You've got about 30 seconds to discuss, or a minute. [CHATTER] All right. I want everyone, now that you've got an answer, click again, put in your current best guess. Oh, wait, sorry. For some reason I have to start over again. OK, now click. This is the best. I'm such a convert to clickers, this is just fantastic. So you guys went from, so roughly you all went from about 30, 60, 10, to what are we now? 8, 82, and 10. So it sounds like you guys are predicting answer b. And the answer is-- I like the suspense. There we go. B, good. So here's a quick question. So why? And the reason why is that if we have E on psi 1 plus psi 2, this is equal to E on psi 1 plus E on psi 2, operator, operator, operator, but this is equal to E 1 psi 1 E 2 psi 2, which if E1 and E2 are not equal, which is not equal to E times psi 1 plus psi 2. Right? Not equal to E anything times psi 1 plus psi 2. And it needs to be, in order to be an eigenfunction, an eigenfunction of the energy operator. Yeah. AUDIENCE: So I was thinking about this, if this was kind of a silly random case where one of the energies is 0. Does this only happen if you have something that's infinite? PROFESSOR: Yeah, that's a really good question. So first off, how do you measure an energy? Do you ever measure an energy? Do you ever measure a voltage, the actual value of the scalar potential, the electromagnetic scalar potential? No. You measure a difference. Do you ever measure the energy? No, you measure a difference in energy. So the absolute value of energy is sort of a silly thing. But we always talk about it as if it's not. We say, that's got energy 14. It's a little bit suspicious. So to answer your question, there's nothing hallowed about the number 0, although we will often refer to zero energy with a very specific meaning. What we really mean in that case is the value of the potential energy at infinity. So when I say energy, usually what I mean is relative to the value at infinity. So then let me ask your question again. Your question is it possible to have energy 0? Absolutely, and we'll see that. And it's actually going to be really interesting what's true of states with energy 0 in that sense. Second part of your question, though, is how does energy being 0 fit into this? Well, does that save us? Suppose one of the energies is 0. Then that says E on psi 1 plus psi 2 is equal to, let's say E2 is 0. Well, that term is gone. So there's just the one E1. Are we in energy eigenstate? No, because it's still not of the form E times psi 1 plus psi 2. So it doesn't save us, but it's an interesting question for the future. All right. Next question, four parts. So the question says x and p commute to i h bar. We've shown this. Is p x equal to i h bar, and is ip plus cx the same as cx plus ip? If you're really unsure you can ask the person next to you, but you don't have to. OK, so this is looking good. Everyone have an answer in? No? Five, four, three, two, one, OK, good. So the answer is C, which most of you got, but not everyone. A bunch of you put D. So let's talk through it. So remember what the definition of the commutator is. x with p by definition is equal to xp minus px. If we change the order here, px is equal to minus this, px minus xp. It's just the definition of the commutator. So on the other hand, if you add things, does 7 plus 6 equal 6 plus 7? Yeah. Well, of course 6 times 7 is 7 times 6. So that's not a terribly good analogy. Does the order of addition of operators matter? No. Yeah. Yeah, exactly. Exactly. So it's slightly sneaky. OK, next question. OK, this one has five. f and g are both wave functions. c is a constant. Then if we take the inner product c times f with g, this is equal to what? Three, two, one, OK. So the answer is-- so this one definitely discuss. Discuss with the person next to you. [CHATTER] All right. OK, go ahead and enter your guess again, or your answer again, let it not be a guess. OK, 10 seconds. Wow. OK, fantastic. That works like a champ. So what's the answer? Yes, complex conjugation. Don't screw that one up. It's very easy to forget, but it matters a lot. Cursor keeps disappearing. OK, next one. A wave function has been expressed as a sum of energy eigenfunctions. Here I'm calling them mu rather than phi, but same thing. Compared to the original wave function, the set of coefficients, given that we're using the energy basis, the set of coefficients contains more or less the same information, or it can't be determined. OK, five seconds. All right. And the answer is C, great. OK, next one. So right now we're normalizing. OK. All stationary states, or all energy eigenstates, have the form that spatial and time dependence is the spatial dependence, the energy eigenfunction, times a phase, so that the norm squared is time independent. Consider the sum of two non-degenerate energy eigenstates psi 1 and psi 2. Non-degenerate means they have different energy. Is the wave function stationary? Is the probability distribution time independent or is it time dependent? This one's not trivial. Oh, shoot. I forgot to get it started. Sorry. It's particularly non-trivial if you can't enter your answer. Right. So go ahead and enter your answer. Whoo, yeah. This one always kills people. No chatting just yet. Test yourself, not your neighbor. It's fine to look deep into your soul, but don't look deep into the soul of the person sitting next to you. All right. So at this point, chat with your neighbor. Let me just give you some presage. The parallel strategy's probably not so good, because about half of you got it right, and about half of you got it wrong. [CHATTER] All right. Let's vote again. And hold on, starting now. OK, vote again. You've got 10 seconds to enter a vote. Wow. OK, two seconds. Good. So the distribution on this one went from 30, 50, 20 initially, to now it is 10, 80, and 10. Amazingly, you guys got worse. The answer is C. And I want you to discuss with each other why it's C. [CHATTER] All right. OK. So let me talk you through it. So the wave function, we've said psi of x and t is equal to phi 1 at x, e to the minus i omega 1 t plus phi 2 of x e to the minus i omega 2 t. So great, we take the norm squared. What's the probability to find it at x at time t. The probability density is the norm squared of this guy, psi squared, which is equal to phi 1 complex conjugate e to the plus i omega 1 t plus phi 2 complex conjugate e to the plus i omega 2t times the thing itself phi 1 of x e to the minus i omega 1 t plus phi 2 of x e to the minus i omega 2t, right? So this has four terms. The first term is psi 1 norm squared. The phases cancel, right? You're going to see this happen a billion times in 804. The first term is going to be phi 1 norm squared. There's another term, which is phi 2 norm squared. Again the phases exactly cancel, even the minus i omega 2 t to the plus i omega 2 t. Plus phi 2 squared. But then there are two cross terms, the interference terms. Plus phi 1 complex conjugate phi 2 e to the i omega 1 t e to the plus i omega 1 t, i omega 1 t, and e to the minus i omega 2t, minus omega 2. So we have a cross-term which depends on the difference in frequencies. Frequencies are like energies modulo on h-bar, so it's a difference in energies. And then there's another term, which is the complex conjugate of this guy, phi 2 star times phi 1 phi 2 complex conjugate phi 1 and the phases are also the complex conjugate e to the minus i omega 1 minus omega 2 t of x of x of x of x. So is there time dependence in this, in principle? Absolutely, from the interference terms. Were we not in the superposition, we would not have interference terms. Time dependence comes from interference, when we expand in energy eigenfunctions. Cool? However, can these vanish? When? Sorry, say again? Great, so when omega 1 equals omega 2, what happens? Time dependence goes away. But omega 1 is e 1 over h bar, omega 2 is e 2 over h bar, and we started out by saying these are non-degenerate. So if they're non-degenerate, the energies are different, the frequencies are different, so that doesn't help us. How do we kill this time dependence? Yes. If the two functions aren't just orthogonal in a functional sense, but if we have the following. Suppose phi 1 is like this. It's 0 everywhere except for in some lump that's phi 1, and phi 2 is 0 everywhere except here. Then anywhere that phi 1 is non-zero, phi 2 is zero. And anywhere where phi 2 is non-zero, phi 1 is zero. So this can point-wise vanish. Do you expect this to happen generically? Does it happen for the energy eigenfunctions in the infinite square well? Sine waves? No. They have zero at isolated points, but they're non-zero generically. Yeah, so it doesn't work there. What about for the free particle? Well, those are just plain waves. Does that ever happen? No. OK, so this is an incredibly special case. We'll actually see it in one problem on a problem set later on. It's a very special case. So technically, the answer is C. And I want you guys to keep your minds open on these sorts of questions, when does a spatial dependence matter and when are there interference terms. Those are two different questions, and I want you to tease them apart. OK? Cool? Yeah? AUDIENCE: Is a valid way to think about this to think that you're fixing the initial frequency but then you have that group velocity is still time dependent. PROFESSOR: That's a very good way to think about it. That's exactly right. That's a very, very good question. Let me say that subtly differently, and tell me if this agrees with what you were just saying. So I can look at this wave function, and I already know that the overall phase of a wave function doesn't matter. That's what it is to say a stationary state is stationary. It's got an overall phase that's the only thing, norm squared it goes away. So I can write this as e to the minus i omega 1 t times phi 1 of x plus phi 2 of x e to the minus i omega 2 minus omega 1 t. Is that what you mean? So that's one way to do this. We could also do something else. We could do e to the minus i omega 1 plus omega 2 upon 2 t. And this is more, I think, what you were thinking of, a sort of average frequency and then a relative frequency, and then the change in the frequencies on these two terms. Absolutely. So you can organize this in many, many ways. But your question gets at a very important point, which is that the overall phase doesn't matter. But relative phases in a superposition do matter. So when does a phase matter in a wave function? It does not matter if it's an overall phase. But it does matter if it's a relative phase between terms in a superposition. Cool? Very good question. Other questions? If not, then I have some. So, consider a system which is in the state-- so I should give you five-- system is in a state which is a linear combination of n equals 1 and n equals 2 eigenstates. What's the probability that measurement will give us energy E1? And it's in this superposition. OK, five seconds. OK, fantastic. What's the answer? Yes, C, great. OK, everyone got that one. So one's a slightly more interesting question. Suppose I have an infinite well with width L. How does the energy, the ground state energy, compare to that of a system with a wider well? So L versus a larger L. OK, four seconds. OK, quickly discuss amongst yourselves, like 10 seconds. [CHATTER] All right. Now click again. Yeah. All right. Five seconds. One, two, three, four, five, great. OK, the answer is A. OK, great, because the energy of the infinite well goes like K squared. K goes like 1 over L. So the energy is, if we make it wider, the energy if we make it wider is going to be lower. And last couple of questions. OK, so t equals 0. Could the wave function for an electron in an infinite square well of width a, rather than L, be A sine squared of pi x upon a, where A is suitably chosen to be normalized? All right, you've got about five seconds left. And OK, we are at chance. We are at even odds, and the answer is not a superposition of A and B, so I encourage you to discuss with the people around you. [CHATTER] Great. What properties had it better satisfy in order to be a viable wave function? What properties should the wave function have so that it's reasonable? Yeah. Is it zero at the ends? Yeah. Good. Is it smooth? Yeah. Exactly. And so you can write it as a superposition. Excellent. So the answer is? Yeah. All right. Vote again. OK, I might have missed a few people. So go ahead and start. OK, five more seconds. All right. So we went from 50-50 to 77-23. That's pretty good. What's the answer? A. Why? Is this an energy eigenstate? No. Does that matter? No. What properties had this wave function better satisfy to be a reasonable wave function in this potential? Say again? It's got to vanish at the walls. It's got to satisfy the boundary conditions. What else must be true of this wave function? Normalizable. Is it normalizable? Yeah. What else? Continuous. It better not have any discontinuities. Is it continuous? Great. OK. Is there any reason that this is a stupid wave function? No. It's perfectly reasonable. It's not an energy eigenfunction, but-- Yeah, cool? Yeah. AUDIENCE: This is sort of like a math question. So to write that at a superposition, you have to write it like basically a Fourier sign series? Isn't the [INAUDIBLE] function even, though? PROFESSOR: On this domain, that and the sines are even. So this is actually odd, but we're only looking at it from 0 to L. So, I mean that half of it. The sines are odd, but we're only looking at the first peak. So you could just as well have written that as cosine of the midpoint plus the distance from the midpoint. Actually, let me say that again, because it's a much better question I just give it shrift for. So here's the question. The question is, look, so sine is an odd function, but sine squared is an even function. So how can you expand sine squared, an even function, in terms of sines, an odd function? But think about this physically. Here's sine squared in our domain, and here's sine. Now what do you mean by even? Usually by even we mean reflection around zero. But I could just as well have said reflection around the origin. This potential is symmetric. And the energy eigenfunctions are symmetric about the origin. They're not symmetric about reflection around this point. But they are symmetric about reflection around this point. That's a particularly natural place to call it 0. So I was calling them sine because I was calling this 0, but I could have called it cosine if I called this 0, for the same Kx. And indeed, can we expand this sine squared function in terms of a basis of these sines on the domain 0 to L? Absolutely. Very good question. And lastly, last clicker question. Oops. Whatever. OK. At t equals 0, a particle is described by the wave function we just saw. Which of the following is true about the wave function at subsequent times? 5 seconds. Whew. Oh, OK. In the last few seconds we had an explosive burst for A, B, and C. So our current distribution is 8, 16, 10, and 67, sounds like 67 is popular. Discuss quickly, very quickly, with the person next to you. [CHATTER] OK, and vote again. OK, five seconds. Get your last vote in. All right. And the answer is D. Yay. So let's think about the logic. Let's go through the logic here. So as was pointed out by a student up here earlier, the wave function sine squared of pi x can be expanded in terms of the energy eigenfunction. Any reasonable function can be expanded in terms of a superposition of definite energy states of energy eigenfunctions. So that means we can write psi at some time as a superposition Cn sine of n pi x upon a e to the minus i e n t upon h bar, since those are, in fact, the eigenfunctions. So we can do that. Now, when we look at the time evolution, we know that each term in that superposition evolves with a phase. The overall wave function does not evolve with a phase. It is not an energy eigenstate. There are going to be interference terms due to the fact that it's a superposition. So its probability distribution is not time-independent. It is a superposition. And so the wave function doesn't rotate by an overall phase. However, we can solve the Schrodinger equation, as we did before. The wave function is expanded at time 0 as the energy eigenfunctions times some set of coefficients. And the time evolution corresponds to adding two each independent term in the superposition the appropriate phase for that energy eigenstate. Cool? All right. So the answer is D. And that's it for the clicker questions. OK, so any questions on the clicker questions so far? OK, those are going to be posted on the web site so you can go over them. And now back to energy eigenfunctions. So what I want to talk about now is the qualitative behavior of energy eigenfunctions. Suppose I know I have an energy eigenfunction. What can I say generally about its structure? So let me ask the question, qualitative behavior. So suppose someone hands you a potential U of x. Someone hands you some potential, U of x, and says, look, I've got this potential. Maybe I'll draw it for you. It's got some wiggles, and then a big wiggle, and then it's got a big wiggle, and then-- do I want to do that? Yeah, let's do that. Then a big wiggle, and something like this. And someone shows you this potential. And they say, look, what are the energy eigenfunctions? Well, OK, free particle was easy. The infinite square well was easy. We could solve that analytically. The next involved solving a differential equation. So what differential equation is this going to lead us to? Well, we know that the energy eigenvalue equation is minus h bar squared upon 2 m phi prime prime of x plus U of x phi x, so that's the energy operator acting on phi, is equal to, saying that it's an energy eigenfunction, phi sub E, says that it's equal to the energy operator acting on this eigenfunction is just a constant E phi sub E of x. And I'm going to work at moment in time, so we're going to drop all the t dependence for the moment. So this is the differential equation we need to solve where U of x is this god-awful function. Do you think it's very likely that you're going to be able to solve this analytically? Probably not. However, some basic ideas will help you get an intuition for what the wave function should look like. And I cannot overstate the importance of being able to eyeball a system and guess the qualitative features of its wave functions, because that intuition, that ability to estimate, is going to contain an awful lot of physics. So let's try to extract it. So in order to do so, I want to start by massaging this equation into a form which is particularly convenient. So in particular, I'm going to write this equation as phi sub E prime prime. So what I'm going to do is I'm going to take this term, I'm going to notice this has two derivatives, this has no derivatives, this has no derivatives. And I'm going to move this term over here and combine these terms into E minus U of x, and I'm going to divide each side by 2m upon h bar squared with a minus sign, giving me that phi prime prime of E of x upon phi E of x dividing through by this phi E is equal to minus 2m over h bar squared. And let's just get our signs right. We've got the minus from here, so this is going to be E minus U of x. So you might look at that and think, well, why is that any better than what I've just written down. But what is the second derivative of function? It's telling you not its slope, but it's telling you how the slope changes. It's telling about the curvature of the function. And what this is telling me is something very, very useful. So for example, let's look at the function. Let's assume that the function is real, although we know in general it's not. Let's assume that the function is real for simplicity. So we're going to plot the real part of phi in the vertical axis. And this is x. Suppose the real part of phi is positive at some point. Phi prime prime, if it's positive, tells us that not only is the slope positive, but it's increasing. Or it doesn't tell us anything about the slope, but it tells us that whatever the slope, it's increasing. If it's negative, the slope is increasing as we increase x. If it's positive, it's increasing as we increase x. So it's telling us that the wave function looks like this, locally, something like that. If phi is negative, if phi is negative, then if this quantity is positive, then phi prime prime has to be negative. But negative is curving down. So if this quantity, which I will call the curvature, if this quantity is positive, it curves away from the axis. So this is phi prime prime over phi greater than 0. If this quantity is positive, the function curves away from the axis. Cool? If this quantity is negative, phi prime prime upon phi less than 0, exactly the opposite. This has to be negative. If phi is positive, then phi prime prime has to be negative. It has to be curving down. And similarly, if phi is negative, then phi prime prime has to be positive, and it has to curve up. So if this quantity is positive, if the curvature is positive, it curves away from the axis. If the curvature is negative, if this quantity is negative, it curves towards the axis. So what does that tell you about solutions when the curvature is positive or negative? It tells you the following. It tells you that, imagine we have a function where phi prime prime over phi is constant. And in particular, let's let phi prime prime over phi be a constant, which is positive. And I'll call that positive constant kappa squared. And to emphasize that it's positive, I'm going to call it kappa squared. It's a positive thing. It's a real number squared. What does the solution look like? Well, this quantity is positive. It's always going to be curving away. So we have solutions that look like this or solutions that look like this. Can it ever be 0? Yeah, sure, it could be an inflection point. So for example, here the curvature is positive, but at this point the curvature has to switch to be like this. What functions are of this form? Let me give you another hint. Here's one. Is this curvature positive? Yes. What about this one? Yup. Those are all positive curvature. And these are exponentials. And the solution to this differential equation is e to the plus kappa x or e to the minus kappa x. And an arbitrary solution of this equation is a superposition A e to the kappa x plus B e to the minus kappa x. Everyone cool with that? When this quantity is positive, we get growing and collapsing exponentials. Yeah? On the other hand, if phi prime prime over phi is a negative number, i.e. minus what I'll call k squared, then the curvature has to be negative. And what functions have everywhere negative curvature? Sinusoidals. Cool? And the general solution is A e to the i K x plus B e to the minus i K x. So that differential equation, also known as sine and cosine. Cool? So putting that together with our original function, let's bring this up. So we want to think about the wave functions here. But in order to think about the energy eigenstates, we need to decide on an energy. We need to pick an energy, because you can't find the solution without figuring the energy. But notice something nice here. So suppose the energy is e. And let me just draw E. This is a constant. The energy is this. So this is the value of E. Here we're drawing potential. But this is the value of the energy, which is a constant. It's just a number. If you had a classical particle moving in this potential, what would happen? It would roll around. So for example, let's say you gave it this energy by putting it here. And think of this as a gravitational potential. You put it here, you let go, and it falls down. And it'll keep rolling until it gets up here to the classical turning point. And at that point, its kinetic energy must be 0, because its potential energy is its total energy, at which point it will turn around and fall back. Yes? If you take your ball, and you put it here, and you let it roll, does it ever get here, to this position? No, because it doesn't have enough energy. Classically, this is a forbidden position. So given an energy and given a potential, we can break the system up into classically allowed zones and classically forbidden zones. Cool? Now, in a classically allowed zone, the energy is greater than the potential. And in a classically forbidden zone, the energy is less than the potential. Everyone cool with that? But this tells us something really nice. If the energy is greater than the potential, what do you know about the curvature? Yeah. If we're in a classically allowed zone, so the energy is greater than the potential, then this quantity is positive, there's a minus sign here, so this is negative. So the curvature is negative. Remember, curvature is negative means that we curve towards the axis. So in a classically allowed region, the wave function should be sinusoidal. What about in the classically forbidden regions? In the classically forbidden regions, the energy is less than the potential. That means in magnitude this is less than this, this is a negative number, minus sign, the curvature is going to be minus times a minus is a positive, so the curvature's positive. So the solutions are either growing exponentials or shrinking exponentials or superpositions of them. Everyone cool with that? So let's think about a simple example. Let's work through this in a simple example. And let me give you a little bit more board space. Simple example would be a potential that looks like this. And let's just suppose that we want to find an energy eigenfunction with energy that's E. Well, this is a classically allowed zone, and these are the classically forbidden regions. Now I want to ask, what does the wave function look like? And I don't want to draw it on top of the energy diagram, because wave function is not an energy. Wave function is a different quantity, because it's got different axes and I want it drawn on a different plot. So but as a function of x-- so just to get the positions straight, these are the bounds of the classically allowed and forbidden regions. What do we expect? Well, we expect that it's going to be sinusoidal in here. We expect that it's going to be exponential growing or converging out here, exp. But one last important thing is that not only is the curvature negative in here in these classically allowed regions, but the magnitude of the curvature, how rapidly it's turning over, how big that second derivative is, depends on the difference between the energy and the potential. The greater the difference, the more rapid the curvature, the more rapid the turning over and fluctuation. If the differences between the potential and the true energy, the total energy, is small, then the curvature is very small. So the derivative changes very gradually. What does that tell us? That tells us that in here the wave function is oscillating rapidly, because the curvature, the difference between the energy and the potential is large, and so the wave function is oscillating rapidly. As we get out towards the classical turning points, the wave function will be oscillating less rapidly. The slope will be changing more gradually. And as a consequence, two things happen. Let me actually draw this slightly differently. So as a consequence two things happen. One is the wavelength gets longer, because the curvature is smaller. And the second is the amplitude gets larger, because it keeps on having a positive slope for longer and longer, and it takes longer to curve back down. So here we have rapid oscillations. And then the oscillations get longer and longer wavelength, until we get out to the classical turning point. And at this point, what happens? Yeah, it's got to be [INAUDIBLE].. Now, here we have some sine, and some superpositions of sine and cosines, exponentials. And in particular, it arrives here with some slope and with some value. We know this side we've got to get exponentials. And so this sum of sines and cosines at this point must match the sum of exponentials. How must it do so? What must be true of the wave function at this point? Can it be discontinuous? Can its derivative be discontinuous? No. So the value and the derivative must be continuous. So that tells us precisely which linear combination of positive growing and shrinking exponentials we get. So we'll get some linear combination, which may do this for awhile. But since it's got some contribution of positive exponential, it'll just grow exponentially off to infinity. And as the energy gets further and further away from the potential, now in their negative sine, what happens to the rate of growth? It gets more and more rapid. So this just diverges more and more rapidly. Similarly, out here we have to match the slope. And we know that the curvature has to be now positive, so it has to do this. So two questions. First off, is this sketch of the wave function a reasonable sketch, given what we know about curvature and this potential of a wave function with that energy? Are there ways in which it's a bad estimate? AUDIENCE: [INAUDIBLE] PROFESSOR: OK, excellent. AUDIENCE: On the right side, could it have crossed zero? PROFESSOR: Absolutely, it could have crossed zero. So I may have drawn this badly. It turned out it was a little subtle. It's not obvious. Maybe it actually punched all the way through zero, and then it diverged down negative. That's absolutely positive. So that was one of the quibbles you could have. Another quibble you could have is that it looks like I have constant wavelength in here. But the potential's actually changing. And what you should chalk this up to, if you'll pardon the pun, is my artistic skills are limited. So this is always going to be sort of inescapable when you qualitatively draw something. On a test, I'm not going to bag you points on things like that. That's what I want to emphasize. But the second thing, is there something bad about this wave function? Yes, you've already named it. What's bad about this wave function? It's badly non-normalizable. It diverges off to infinity out here and out here. What does that tell you? It's not physical. Good. What else does it tell you about this system? Sorry? Excellent. Is this an allowable energy? No. If the wave function has this energy, it is impossible to make it continuous, assuming that I drew it correctly, and have it converge. Is this wave function allowable? No, because it does not satisfy our boundary conditions. Our boundary conditions are that the wave function must vanish out here and it must vanish out here at infinity in order to be normalizable. Here we failed. Now, you can imagine that-- so let's decrease the energy a little bit. If we decrease the energy, our trial energy just a little tiny bit, what happens? Well, that's going to decrease the curvature in here. We decrease, we bring the energy in just a little tiny bit. That means this is a little bit smaller. The potential stays the same. So the curvature in the allowed region is just a little tiny bit smaller. And meanwhile, the allowed region has got just a little bit thinner. And what that will do is the curvature's a little less, the region's a little less, so now we have-- Sorry, I get excited. And if we tweak the energy, what's going to happen? Well, it's going to arrive here a little bit sooner. And let's imagine something like this. And if we chose the energy just right, we would get it to match to a linear combination of collapsing and growing exponentials, where the contribution from the growing exponential in this direction vanishes. There's precisely one value of the energy that lets me do that with this number of wiggles. And so then it goes through and does its thing. And we need it to happen on both sides. Now if I take that solution, so that it achieves convergence out here, and it achieves convergence out here, and I take that energy and I increase it by epsilon, by just the tiniest little bit, what will happen to this wave function? It'll diverge. It will no longer be normalizable. When you have classically forbidden regions, are the allowed energies continuous or discrete? And that answers a question from earlier in the class. And it also is going to be the beginning of the answer to the question, why is the spectrum of hydrogen discrete. See you next time.
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_1_Introduction_to_Superposition.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ALLAN ADAMS: Hi everyone. Welcome to 804 for spring 2013. This is the fourth, and presumably final time that I will be teaching this class. So I'm pretty excited about it. So my name is Allan Adams. I'll be lecturing the course. I'm an assistant professor in Course 8. I study string theory and its applications to gravity, quantum gravity, and condensed matter physics. Quantum mechanics, this is a course in quantam mechanics. Quantam mechanics Is my daily language. Quantum mechanics is my old friend. I met quantum mechanics 20 years ago. I just realized that last night. It was kind of depressing. So, old friend. It's also my most powerful tool. So I'm pretty psyched about it. Our recitation instructors are Barton Zwiebach, yea! And Matt Evans-- yea! Matt's new to the department, so welcome him. Hi. So he just started his faculty position, which is pretty awesome. And our TA is Paolo Glorioso. Paolo, are you here? Yea! There you go. OK, so he's the person to send all complaints to. So just out of curiosity, how many of you all are Course 8? Awesome. How many of you all are, I don't know, 18? Solid. 6? Excellent. 9? No one? This is the first year we haven't had anyone Course 9. That's a shame. Last year one of the best students was a Course 9 student. So two practical things to know. The first thing is everything that we put out will be on the Stellar website. Lecture notes, homeworks, exams, everything is going to be done through Stellar, including your grades. The second thing is that as you may notice there are rather more lights than usual. I'm wearing a mic. And there are these signs up. We're going to be videotaping this course for the lectures for OCW. And if you're happy with that, cool. If not, just sit on the sides and you won't appear anywhere on video. Sadly, I can't do that. But you're welcome to if you like. But hopefully that should not play a meaningful role in any of the lectures. So the goal of 804 is for you to learn quantum mechanics. And by learn quantum mechanics, I don't mean to learn how to do calculations, although that's an important and critical thing. I mean learn some intuition. I want you to develop some intuition for quantum phenomena. Now, quantam mechanics is not hard. It has a reputation for being a hard topic. It is not a super hard topic. So in particular, everyone in this room, I'm totally positive, can learn quantum mechanics. It does require concerted effort. It's not a trivial topic. And in order to really develop a good intuition, the essential thing is to solve problems. So the way you develop a new intuition is by solving problems and by dealing with new situations, new context, new regimes, which is what we're going to do in 804. It's essential that you work hard on the problem sets. So your job is to devote yourself to the problem sets. My job is to convince you at the end of every lecture that the most interesting thing you could possibly do when you leave is the problem set. So you decide who has the harder job. So the workload is not so bad. So we have problem sets due, they're due in the physics box in the usual places, by lecture, by 11 AM sharp on Tuesdays every week. Late work, no, not so much. But we will drop one problem set to make up for unanticipated events. We'll return the graded problem sets a week later in recitation. Should be easy. I strongly, strongly encourage you to collaborate with other students on your problem sets. You will learn more, they will learn more, it will be more efficient. Work together. However, write your problem sets yourself. That's the best way for you to develop and test your understanding. There will be two midterms, dates to be announced, and one final. I guess we could have multiple, but that would be a little exciting. We're going to use clickers, and clickers will be required. We're not going to take attendance, but they will give a small contribution to your overall grade. And we'll use them most importantly for non-graded but just participation concept questions and the occasional in class quiz to probe your knowledge. This is mostly so that you have a real time measure of your own conceptual understanding of the material. This has been enormously valuable. And something I want to say just right off is that the way I've organized this class is not so much based on the classes I was taught. It's based to the degree possible on empirical lessons about what works in teaching, what actually makes you learn better. And clickers are an excellent example of that. So this is mostly a standard lecture course, but there will be clickers used. So by next week I need you all to have clickers, and I need you to register them on the TSG website. I haven't chosen a specific textbook. And this is discussed on the Stellar web page. There are a set of textbooks, four textbooks that I strongly recommend, and a set of others that are nice references. The reason for this is twofold. First off, there are two languages that are canonically used for quantum mechanics. One is called wave mechanics, and the language, the mathematical language is partial differential equations. The other is a matrix mechanics. They have big names. And the language there is linear algebra. And different books emphasize different aspects and use different languages. And they also try to aim at different problems. Some books are aimed towards people who are interested in materials science, some books that are aimed towards people interested in philosophy. And depending on what you want, get the book that's suited to you. And every week I'll be providing with your problem sets readings from each of the recommended texts. So what I really encourage you to do is find a group of people to work with every week, and make sure that you've got all the books covered between you. This'll give you as much access to the texts as possible without forcing you to buy four books, which I would discourage you from doing. So finally I guess the last thing to say is if this stuff were totally trivial, you wouldn't need to be here. So ask questions. If you're confused about something, lots of other people in the class are also going to be confused. And if I'm not answering your question without you asking, then no one's getting the point, right? So ask questions. Don't hesitate to interrupt. Just raise your hand, and I will do my best to call on you. And this is true for both in lecture, also go to office hours and recitations. Ask questions. I promise, there's no such thing as a terrible question. Someone else will also be confused. So it's a very valuable to me and everyone else. So before I get going on the actual physics content of the class, are there any other practical questions? Yeah. AUDIENCE: You said there was a lateness policy. ALLAN ADAMS: Lateness policy. No late work is accepted whatsoever. So the deal is given that every once in a while, you know, you'll be walking to school and your leg is going to fall off, or a dog's going to jump out and eat your person standing next to you, whatever. Things happen. So we will drop your lowest problem set score without any questions. At the end of the semester, we'll just dropped your lowest score. And if you turn them all in, great, whatever your lowest score was, fine. If you missed one, then gone. On the other hand, if you know next week, I'm going to be attacked by a rabid squirrel, it's going to be horrible, I don't want to have to worry about my problem set. Could we work this out? So if you know ahead of time, come to us. But you need to do that well ahead of time. The night before doesn't count. OK? Yeah. AUDIENCE: Will we be able to watch the videos? ALLAN ADAMS: You know, that's an excellent question. I don't know. I don't think so. I think it's going to happen at the end of the semester. Yeah. OK. So no, you'll be able to watch them later on the OCW website. Other questions. Yeah. AUDIENCE: Are there any other videos that you'd recommend, just like other courses on YouTube? ALLAN ADAMS: Oh. That's an interesting question. I don't off the top of my head, but if you send me an email, I'll pursue it. Because I do know several other lecture series that I like very much, but I don't know if they're available on YouTube or publicly. So send me an email and I'll check. Yeah. AUDIENCE: So how about the reading assignments? ALLAN ADAMS: Reading assignments on the problem set every week will be listed. There will be equivalent reading from every textbook. And if there is something missing, like if no textbook covers something, I'll post a separate reading. Every once in a while, I'll post auxiliary readings, and they'll be available on the Stellar website. So for example, in your problem set, first one was posted, will be available immediately after lecture on the Stellar website. There are three papers that it refers to, or two, and they are posted on the Stellar website and linked from the problem set. Others? OK. So the first lecture. The content of the physics of the first lecture is relatively standalone. It's going to be an introduction to a basic idea then is going to haunt, plague, and charm us through the rest of the semester. The logic of this lecture is based on a very beautiful discussion in the first few chapters of a book by David Albert called Quantum Mechanics and Experience. It's a book for philosophers. But the first few chapters, a really lovely introduction at a non-technical level. And I encourage you to take a look at them, because they're very lovely. But it's to be sure straight up physics. Ready? I love this stuff. today I want to describe to you a particular set of experiments. Now, to my mind, these are the most unsettling experiments ever done. These experiments involve electrons. They have been performed, and the results as I will describe them are true. I'm going to focus on two properties of electrons. I will call them color and hardness. And these are not the technical names. We'll learn the technical names for these properties later on in the semester. But to avoid distracting you by preconceived notions of what these things mean, I'm going to use ambiguous labels, color and hardness. And the empirical fact is that every electron, every electron that's ever been observed is either black or white and no other color. We've never seen a blue electron. There are no green electrons. No one has ever found a fluorescent electron. They're either black, or they are white. It is a binary property. Secondly, their hardness is either hard or soft. They're never squishy. No one's ever found one that dribbles. They are either hard, or they are soft. Binary properties. OK? Now, what I mean by this is that it is possible to build a device which measures the color and the hardness. In particular, it is possible to build a box, which I will call a color box, that measures the color. And the way it works is this. It has three apertures, an in port and two out ports, one which sends out black electrons and one which sends out white electrons. And the utility of this box is that the color can be inferred from the position. If you find the particle, the electron over here, it is a white electron. If you find the electron here, it is a black electron. Cool? Similarly, we can build a hardness box, which again has three apertures, an in port. And hard electrons come out this port, and soft electrons come out this port. Now, if you want, you're free to imagine that these boxes are built by putting a monkey inside. And you send in an electron, and the monkey, you know, with the ears, looks at the electron, and says it's a hard electron, it sends it out one way, or it's a soft electron, it sends it out the other. The workings inside do not matter. And in particular, later in the semester I will describe in considerable detail the workings inside this apparatus. And here's something I want to emphasize to you. It can be built in principle using monkeys, hyper intelligent monkeys that can see electrons. It could also be built using magnets and silver atoms. It could be done with neutrons. It could be done with all sorts of different technologies. And they all give precisely the same results as I'm about to describe. They all give precisely the same results. So it does not matter what's inside. But if you want a little idea, you could imagine putting a monkey inside, a hyper intelligent monkey. I know, it sounds good. So a key property of these hardness boxes and color boxes is that they are repeatable. And here's what I mean by that. If I send in an electron, and I find that it comes out of a color box black, and then I send it in again, then if I send it into another color box, it comes out black again. So in diagrams, if I send in some random electron to a color box, and I discover that it comes out, let's say, the white aperture. And so here's dot dot dot, and I take the ones that come out the white aperture, and I send them into a color box again. Then with 100% confidence, 100% of the time, the electron coming out of the white port incident on the color box will come out the white aperture again. And 0% of the time will it come out the black aperture. So this is a persistent property. You notice that it's white. You measure it again, it's still white. Do a little bit later, it's still white. OK? It's a persistent property. Ditto the hardness. If I send in a bunch of electrons in to a hardness box, here is an important thing. Well, send them into a hardness box, and I take out the ones that come out soft. And I send them again into a hardness box, and they come out soft. They will come out soft with 100% confidence, 100% of the time. Never do they come out the hard aperture. Any questions at this point? So here's a natural question. Might the color and the hardness of an electron be related? And more precisely, might they be correlated? Might knowing the color infer something about the hardness? So for example, so being male and being a bachelor are correlated properties, because if you're male, you don't know if you're a bachelor or not, but if you're a bachelor, you're male. That's the definition of the word. So is it possible that color and hardness are similarly correlated? So, I don't know, there are lots of good examples, like wearing a red shirt and beaming down to the surface and making it back to the Enterprise later after the away team returns. Correlated, right? Negatively, but correlated. So the question is, suppose, e.g., suppose we know that an electron is white. Does that determine the hardness? So we can answer this question by using our boxes. So here's what I'm going to do. I'm going to take some random set of electrons. That's not random. Random. And I'm going to send them in to a color box. And I'm going to take the electrons that come out the white aperture. And here's a useful fact. When I say random, here's operationally what I mean. I take some piece of material, I scrape it, I pull off some electrons, and they're totally randomly chosen from the material. And I send them in. If I send a random pile of electrons into a color box, useful thing to know, they come out about half and half. It's just some random assortment. Some of them are white, some of them come out black. Suppose I send some random collection of electrons into a color box. And I take those which come out the white aperture. And I want to know, does white determine hardness. So I can do that, check, by then sending these white electrons into a hardness box and seeing what comes out. Hard, soft. And what we find is that 50% of those electrons incident on the hardness box come out hard, and 50% come out soft. OK? And ditto if we reverse this. If we take hardness, and take, for example, a soft electron and send it into a color box, we again get 50-50. So if you take a white electron, you send it into a hardness box, you're at even odds, you're at chance as to whether it's going to come out hard or soft. And similarly, if you send a soft electron into a color box, even odds it's going to come out black or white. So knowing the hardness does not give you any information about the color, and knowing the color does not give you any information about the hardness. cool? These are independent facts, independent properties. They're not correlated in this sense, in precisely this operational sense. Cool? Questions? OK. So measuring the color give zero predictive power for the hardness, and measuring the hardness gives zero predictive power for the color. And from that, I will say that these properties are correlated. So H, hardness, and color are in this sense uncorrelated. So using these properties of the color and hardness boxes, I want to run a few more experiment's. I want to probe these properties of color and hardness a little more. And in particular, knowing these results allows us to make predictions, to predict the results for set a very simple experiments. Now, what we're going to do for the next bit is we're going to run some simple experiments. And we're going to make predictions. And then those simple experiments are going to lead us to more complicated experiments. But let's make sure we understand the simple ones first. So for example, let's take this last experiment, color and hardness, and let's add a color box. One more monkey. So color in, and we take those that come out the white aperture. And we send them into a hardness box. Hard, soft. And we take those electrons which come out the soft aperture. And now let's send these again into a color box. So it's easy to see what to predict. Black, white. So you can imagine a monkey inside this, going, aha. You look at it, you inspect, it comes out white. Here you look at it and inspect, it comes out soft. And you send it into the color box, and what do you expect to happen? Well, let's think about the logic here. Anything reaching the hardness box must have been measured to be white. And we just did the experiment that if you send a white electron into a hardness box, 50% of the time it comes out a hard aperture and 50% of the time it comes out the soft aperture. So now we take that 50% of electrons that comes out the soft aperture, which had previously been observed to be white and soft. And then we send them into a color box, and what happens? Well, since colors are repeatable, the natural expectation is that, of course, it comes out white. So our prediction, our natural prediction here is that of those electrons that are incident on this color box, 100% should come out white, and 0% should come out black. That seem like a reasonable-- let's just make sure that we're all agreeing. So let's vote. How many people think this is probably correct? OK, good. How many people think this probably wrong? OK, good. That's reassuring. Except you're all wrong. Right? In fact, what happens is half of these electrons exit white, 50%. And 50% percent exit black. So let's think about what's going on here. This is really kind of troubling. We've said already that knowing the color doesn't predict the hardness. And yet, this electron, which was previously measured to be white, now when subsequently measured sometimes it comes out white, sometimes it comes out black, 50-50% of the time. So that's surprising. What that tells you is you can't think of the electron as a little ball that has black and soft written on it, right? You can't, because apparently that black and soft isn't a persistent thing, although it's persistent in the sense that once it's black, it stays black. So what's going on here? Now, I should emphasize that the same thing happens if I had changed this to taking the black electrons and throwing in a hardness and picking soft and then measuring the color, or if I had used the hard electrons. Any of those combinations, any of these ports would have given the same results, 50-50. Is not persistent in this sense. Apparently the presence of the hardness box tampers with the color somehow. So it's not quite as trivial is that hyper intelligent monkey. Something else is going on here. So this is suspicious. So here's the first natural move. The first natural move is, oh, look, surely there's some additional property of the electron that we just haven't measured yet that determines whether it comes out the second color box black or white. There's got be some property that determines this. And so people have spent a tremendous amount of time and energy looking at these initial electrons and looking with great care to see whether there's any sort of feature of these incident electrons which determines which port they come out of. And the shocker is no one's ever found such a property. No one has ever found a property which determines which port it comes out of. As far as we can tell, it is completely random. Those that flip and those that don't are indistinguishable at beginning. And let me just emphasize, if anyone found such a-- it's not like we're not looking, right? If anyone found such a property, fame, notoriety, subverting quantum mechanics, Nobel Prize. People have looked. And there is none that anyone's been able to find. And as we'll see later on, using Bell's inequality, we can more or less nail that such things don't exist, such a fact doesn't exist. But this tells us something really disturbing. This tells us, and this is the first real shocker, that there is something intrinsically unpredictable, non-deterministic, and random about physical processes that we observe in a laboratory. There's no way to determine a priori whether it will come out black or white from the second box. Probability in this experiment, it's forced upon us by observations. OK, well, there's another way to come at this. You could say, look, you ran this experiment, that's fine. But look, I've met the guy who built these boxes, and look, he's just some guy, right? And he just didn't do a very good job. The boxes are just badly built. So here's the way to defeat that argument. No, we've built these things out of different materials, using different technologies, using electrons, using neutrons, using bucky-balls, C60, seriously, it's been done. We've done this experiment, and this property does not change. It is persistent. And the thing that's most upsetting to me is that not only do we get the same results independent of what objects we use to run the experiment, we cannot change the probability away from 50-50 at all. Within experimental tolerances, we cannot change, no matter how we build the boxes, we cannot change the probability by part in 100. 50-50. And to anyone who grew up with determinism from Newton, this should hurt. This should feel wrong. But it's a property of the real world. And our job is going to be to deal with it. Rather, your job is going to be to deal with it, because I went through this already. So here's a curious consequence-- oh, any questions before I cruise? OK. So here's a curious consequence of this series of experiments. Here's something you can't do. Are you guys old enough for you can't do this on television? This is so sad. OK, so here's something you can't do. We cannot build, it is impossible to build, a reliable color and hardness box. We've built a box that tells you what color it is. We've built a box that tells you what hardness it is. But you cannot build a meaningful box that tells you what color and hardness an electron is. So in particular, what would this magical box be? It would have four ports. And its ports would say, well, one is white and hard, and one is white and soft, one is black and hard, and one is black and soft. So you can imagine how you might try to build a color and hardness box. So for example, here's something you might imagine. Take your incident electrons, and first send them into a color box. And take those white electrons, and send them into a hardness box. And take those electrons, and this is going to be white and hard, and this is going to be white and soft. And similarly, send these black electrons into the hardness box, and here's hard and black, and here's soft and back. Everybody cool with that? So this seems to do the thing I wanted. It measures both the hardness and the color. What's the problem with it? AUDIENCE: [INAUDIBLE] ALLAN ADAMS: Yeah, exactly. So the color is not persistent. So you tell me this is a soft and black electron, right? That's what you told me. Here's the box. But if I put a color box here, that's the experiment we just ran. And what happens? Does this come out black? No, this is a crappy source of black electrons. It's 50/50 black and white. So this box can't be built. And the reason, and I want to emphasize this, the reason we cannot build this box is not because our experiments are crude. And it's not because I can't build things, although that's true. I was banned from a lab one day after joining it, actually. So I really can't build, but other people can. And that's not why. We can't because of something much more fundamental, something deeper, something in principle, which is encoded in this awesome experiment. This can be done. It does not mean anything, as a consequence. It does not mean anything to say this electron is white and hard, because if you tell me it's white and hard, and I measure the white, well, I know if it's hard, it's going to come out 50-50. It does not mean anything. So this is an important idea. This is an idea which is enshrined in physics with a term which comes with capital letters, the Uncertainty Principle. And the Uncertainty Principle says basically that, look, there's some observable, measurable properties of a system which are incompatible with each other in precisely this way, incompatible with each other in the sense not that you can't know, because you can't know whether it's hard and soft simultaneously, deeper. It is not hard and white simultaneously. It cannot be. It does not mean anything to say it is hard and white simultaneously. That is uncertainty. And again, uncertainty is an idea we're going to come back to over and over in the class. But every time you think about it, this should be the first place you start for the next few weeks. Yeah. Questions. No questions? OK. So at this point, it's really tempting to think yeah, OK, this is just about the hardness and the color of electrons. It's just a weird thing about electrons. It's not a weird thing about the rest of the world. The rest of the world's completely reasonable. And no, that's absolutely wrong. Every object in the world has the same properties. If you take bucky-balls, and you send them through the analogous experiment-- and I will show you the data, I think tomorrow, but soon, I will show you the data. When you take bucky-balls and run it through a similar experiment, you get the same effect. Now, bucky-balls are huge, right, 60 carbon atoms. But, OK, OK, at that point, you're saying, dude, come on, huge, 60 carbon atoms. So there is a pendulum, depending on how you define building, in this building, a pendulum which is used, in principle which is used to improve detectors to detect gravitational waves. There's a pendulum with a, I think it's 20 kilo mirror. And that pendulum exhibits the same sort of effects here. We can see these quantum mechanical effects in those mirrors. And this is in breathtakingly awesome experiments done by Nergis Malvalvala, whose name I can never pronounce, but who is totally awesome. She's an amazing physicist. And she can get these kind of quantum effects out of a 20 kilo mirror. So before you say something silly, like, oh, it's just electrons, it's 20 kilo mirrors. And if I could put you on a pendulum that accurate, it would be you. OK? These are properties of everything around you. The miracle is not that electrons behave oddly. The miracle is that when you take 10 to the 27 electrons, they behave like cheese. That's the miracle. This is the underlying correct thing. OK, so this is so far so good. But let's go deeper. Let's push it. And to push it, I want to design for you a slightly more elaborate apparatus, a slightly more elaborate experimental apparatus. And for this, I want you to consider the following device. I'm going to need to introduce a couple of new features for you. Here's a hardness box. And it has an in port. And the hardness box has a hard aperture, and it has a soft aperture. And now, in addition to this hardness box, I'm going to introduce two elements. First, mirrors. And what these mirrors do is they take the incident electrons and, nothing else, they change the direction of motion, change the direction of motion. And here's what I mean by doing nothing else. If I take one of these mirrors, and I take, for example, a color box. And I take the white electrons that come out, and I bounce it off the mirror, and then I send these into a color box, then they come out white 100% of the time. It does not change the observable color. Cool? All it does is change the direction. Similarly, with the hardness box, it doesn't change the hardness. It just changes the direction of motion. And every experiment we've ever done on these, guys, changes in no way whatsoever the color or the hardness by subsequent measurement. Cool? Just changes the direction of motion. And then I'm going to add another mirror. It's actually a slightly fancy set of mirrors. All they do is they join these beams together into a single beam. And again, this doesn't change the color. You send in a white electron, you get out, and you measure the color on the other side, you get a white electron. You send in a black electron from here, and you measure the color, you get a black electron again out. Cool? So here's my apparatus. And I'm going to put this inside a big box. And I want to run some experiments with this apparatus. Everyone cool with the basic design? Any questions before I cruise on? This part's fun. So what I want to do now is I want to run some simple experiments before we get to fancy stuff. And the simple experiments are just going to warm you up. They're going to prepare you to make some predictions and some calculations. And eventually we'd like to lead back to this guy. So the first experiment, I'm going to send in white electrons. Whoops. Im. I'm going to send in white electrons. And I'm going to measure at the end, and in particular at the output, the hardness. So I'm going to send in white electrons. And I'm going to measure the hardness. So this is my apparatus. I'm going to measure the hardness at the output. And what I mean by measure the hardness is I throw these electrons into a hardness box and see what comes out. So this is experiment 1. And let me draw this, let me biggen the diagram. So you send white into-- so the mechanism is a hardness box. Mirror, mirror, mirrors, and now we're measuring the hardness out. And the question I want to ask is how many electrons come out the hard aperture, and how many electrons come out the soft aperture of this final hardness box. So I'd like to know what fraction come out hard, and what fraction come out soft. I send an initial white electron, for example I took a color box and took the white output, send them into the hardness box, mirror, mirror, hard, hard, soft. And what fraction come out hard, and what fraction come out soft. So just think about it for a minute. And when you have a prediction in your head, raise your hand. All right, good. Walk me through your prediction. AUDIENCE: I think it should be 50-50. ALLAN ADAMS: 50-50. How come? AUDIENCE: [INAUDIBLE] color doesn't have any bearing on hardness. [INAUDIBLE] ALLAN ADAMS: Awesome. So let me say that again. So we've done the experiment, you send a white electron into the hardness box, and we know that it's non-predictive, 50-50. So if you take a white electron and you send it into the hardness box, 50% of the time it will come out the hard aperture, and 50% of the time it will come out the soft aperture. Now if you take the one that comes out the hard aperture, then you send it up here or send it up here, we know that these mirrors do nothing to the hardness of the electron except change the direction of motion. We've already done that experiment. So you measure the hardness at the output, what do you get? Hard, because it came out hard, mirror, mirror, hardness, hard. But it only came out hard 50% of the time because we sent in initially white electron. Yeah? What about the other 50%? Well, the other 50% of the time, it comes out the soft aperture and follows what I'll call the soft path to the mirror, mirror, hardness. And with soft, mirror, mirror, hardness, you know it comes out soft. 50% of the time it comes out this way, and then it will come out hard. 50% it follows the soft path, and then it will come out soft. Was this the logic? Good. How many people agree with this? Solid. How many people disagree? No abstention. OK. So here's a prediction. Oh, yep. AUDIENCE: Just a question. Could you justify that prediction without talking about oh, well, half the electrons were initially measured to be hard, and half were initially measured to be soft, by just saying, well, we have a hardness box, and then we joined these electrons together again, so we don't know anything about it. So it's just like sending white electrons into one hardness box instead of two. ALLAN ADAMS: Yeah, that's a really tempting argument, isn't it? So let's see. We're going to see in a few minutes whether that kind of an argument is reliable or not. But so far we've been given two different arguments that lead to the same prediction, 50-50. Yeah? Question. AUDIENCE: Are the electrons interacting between themselves? Like when you get them to where-- ALLAN ADAMS: Yeah. This is a very good question. So here's a question look you're sending a bunch of electrons into this apparatus. But if I take-- look, I took 802. You take two electrons and you put them close to each other, what do they do? Pyewww. Right? They interact with each other through a potential, right? So yeah, we're being a little bold here, throwing a bunch of electrons in and saying, oh, they're independent. So I'm going to do one better. I will send them in one at a time. One electron through the apparatus. And then I will wait for six weeks. [LAUGHTER] See, you guys laugh, you think that's funny. But there's a famous story about a guy who did a similar experiment with photons, French guy. And, I mean, the French, they know what they're doing. So he wanted to do the same experiment with photons. But the problem is if you take a laser and you shined it into your apparatus, there there are like, 10 to the 18 photons in there at any given moment. And the photons, who knows what they're doing with each other, right? So I want to send in one photon, but the problem is, it's very hard to get a single photon, very hard. So what he did, I kid you not, he took an opaque barrier, I don't remember what it was, it was some sort of film on top of glass, I think it was some sort of oil-tar film. Barton, do you remember what he used? So he takes a film, and it has this opaque property, such that the photons that are incident upon it get absorbed. Once in a blue moon a photon manages to make its way through. Literally, like once every couple of days, or a couple of hours, I think. So it's going to take a long time to get any sort of statistics. But he this advantage, that once every couple of hours or whatever a photon makes its way through. That means inside the apparatus, if it takes a pico-second to cross, triumph, right? That's the week I was talking about. So he does this experiment. But as you can tell, you start the experiment, you press go, and then you wait for six months. Side note on this guy, liked boats, really liked yachts. So he had six months to wait before doing a beautiful experiment and having the results. So what did he do? Went on a world tour in his yacht. Comes back, collects the data, and declares victory, because indeed, he saw the effect he wanted. So I was not kidding. We really do wait. So I will take your challenge. And single electron, throw it in, let it go through the apparatus, takes mere moments. Wait for a week, send in another electron. No electrons are interacting with each other. Just a single electron at a time going through this apparatus. Other complaints? AUDIENCE: More stories? ALLAN ADAMS: Sorry? AUDIENCE: More stories? ALLAN ADAMS: Oh, you'll get them. I have a hard time resisting. So here's a prediction, 50-50. We now have two arguments for this. So again, let's vote after the second argument. 50-50, how many people? You sure? Positive? How many people don't think so? Very small dust. OK. It's correct. Yea. So, good. I like messing with you guys. So remember, we're going to go through a few experiments first where it's going to be very easy to predict the results. We've got four experiments like this to do. And then we'll go on to the interesting examples. But we need to go through them so we know what happens, so we can make an empirical argument rather than an in principle argument. So there's the first experiment. Now, I want to run the second experiment. And the second experiment, same as the first, a little bit louder, a little bit worse. Sorry. The second experiment, we're going to send in hard electrons, and we're going to measure color at out. So again, let's look at the apparatus. We send in hard electrons. And our apparatus is hardness box with a hard and a soft aperture. And now we're going to measure the color at the output. Color, what have I been doing? And now I want to know what fraction come out black, and what fraction come out white. We're using lots of monkeys in this process. OK, so this is not rocket science. Rocket science isn't that complicated. Neuroscience is much harder. This is not neuroscience. So let's figure out what this is. Predictions. So again, think about your prediction your head, come to a conclusion, raise your hand when you have an idea. And just because you don't raise your hand doesn't mean I won't call on you. AUDIENCE: 50-50 black and white. ALLAN ADAMS: 50-50 black and white. I like it. Tell me why. AUDIENCE: It's gone through a hardness box, which scrambled the color, and therefore has to be [INAUDIBLE] ALLAN ADAMS: Great. So the statement, I'm going to say that slightly more slowly. That was an excellent argument. We have a hard electron. We know that hardness boxes are persistent. If you send a hard electron in, it comes out hard. So every electron incident upon our apparatus will transit across the hard trajectory. It will bounce, it will bounce, but it is still hard, because we've already done that experiment. The mirrors do nothing to the hardness. So we send a hard electron into the color box, and what comes out? Well, we've done that experiment, too. Hard into color, 50-50. So the prediction is 50-50. This is your prediction. Is that correct? Awesome. OK, let us vote. How many people think this is correct? Gusto, I like it. How many people think it's not? All right. Yay, this is correct. Third experiment, slightly more complicated. But we have to go through these to get to the good stuff, so humor me for a moment. Third, let's send in white electrons, and then measure the color at the output port. So now we send in white electrons, same beast. And our apparatus is a hardness box with a hard path and a soft path. Do-do-do, mirror, do-do-do, mirror, box, join together into our out. And now we send those out electrons into a color box. And our color box, black and white. And now the question is how many come out black, and how many come out white. Again, think through the logic, follow the electrons, come up with a prediction. Raise your hand when you have a prediction. AUDIENCE: Well, earlier we showed that [INAUDIBLE] so it'll take those paths equally-- ALLAN ADAMS: With equal probability. Good. AUDIENCE: Yeah. And then it'll go back into the color box. But earlier when we did the same thing without the weird path-changing, it came out 50-50 still. So I would say still 50-50. ALLAN ADAMS: Great. So let me say that again, out loud. And tell me if this is an accurate extension of what you said. I'm just going to use more words. But it's, I think, the same logic. We have a white electron, initially white electron. We send it into a hardness box. When we send a white electron into a hardness box, we know what happens. 50% of the time it comes out hard, the hard aperture, 50% of the time it comes out the soft aperture. Consider those electrons that came out the hard aperture. Those electrons that came out the hard aperture will then transit across the system, preserving their hardness by virtue of the fact that these mirrors preserve hardness, and end up at a color box. When they end at the color box, when that electron, the single electron in the system ends at this color box, then we know that a hard electron entering a color box comes out black or white 50% of the time. We've done that experiment, too. So for those 50% that came out hard, we get 50/50. Now consider the other 50%. The other half of the time, the single electron in the system will come out the soft aperture. It will then proceed along the soft trajectory, bounce, bounce, not changing its hardness, and is then a soft electron incident on the color box. But we've also done that experiment, and we get 50-50 out, black and white. So those electrons that came out hard come out 50-50, and those electrons that come out soft come out 50/50. And the logic then leads to 50-50, twice, 50-50. Was that an accurate statement? Good. It's a pretty reasonable extension. OK, let's vote. How many people agree with this one? OK, and how many people disagree? Yeah, OK. So vast majority agree. And the answer is no, this is wrong. In fact, all of these, 100% come out white and 0 come out black. Never ever does an electron come out the black aperture. I would like to quote what a student just said, because it's actually the next line in my notes, which is what the hell is going on? So let's the series of follow up experiments to tease out what's going on here. So something very strange, let's just all agree, something very strange just happened. We sent a single electron in. And that single electron comes out the hardness box, well, it either came out the hard aperture or the soft aperture. And if it came out the hard, we know what happens, if it came out the soft, we know what happens. And it's not 50-50. So we need to improve the situation. Hold on a sec. Hold on one sec. Well, OK, go ahead. AUDIENCE: Yeah, it's just a question about the setup. So with the second hardness box, are we collecting both the soft and hard outputs? ALLAN ADAMS: The second, you mean the first hardness box? AUDIENCE: The one-- are we getting-- no, the-- ALLAN ADAMS: Which one, sorry? This guy? Oh, that's a mirror, not a hardness box. Oh, thanks for asking. Yeah, sorry. I wish I had a better notation for this, but I don't. There's a classic-- well, I'm not going to go into it. Remember that thing where I can't stop myself from telling stories? So all this does, it's just a set of mirrors. It's a set of fancy mirrors. And all it does is it takes an electron coming this way or an electron coming this way, and both of them get sent out in the same direction. It's like a beam joiner, right? It's like a y junction. That's all it is. So if you will, imagine the box is a box, and you take, I don't know, Professor Zwiebach, and you put him inside. And every time an electron comes up this way, he throws it out that way, and every time it comes in this way, he throws it out that way. And he'd be really ticked at you for putting him in a box, but he'd do the job well. Yeah. AUDIENCE: And this also works if you go one electron at a time? ALLAN ADAMS: This works if you go one electron at a time, this works if you go 14 electrons at a time, it works. It works reliably. Yeah. AUDIENCE: Just, maybe [INAUDIBLE] but what's the difference between this experiment and that one? ALLAN ADAMS: Yeah, I know. Right? Right? So the question was, what's the difference between this experiment and the last one. Yeah, good question. So we're going to have to answer that. Yeah. AUDIENCE: Well, you're mixing again the hardness. So it's like as you weren't measuring it at all, right? ALLAN ADAMS: Apparently it's a lot we weren't measuring it, right? Because we send in the white electron, and at the end we get out that it's still white. So somehow this is like not doing anything. But how does that work? So that's an excellent observation. And I'm going to build you now a couple of experiments that tease out what's going on. And you're not going to like the answer. Yeah. AUDIENCE: How were the white electrons generated in this experiment? ALLAN ADAMS: The white electrons were generated in the following way. I take a random source of electrons, I rub a cat against a balloon and I charge up the balloon. And so I take those random electrons, and I send them into a color box. And we have previously observed that if you take random electrons and throw them into a color box and pull out the electrons that come out the white aperture, if you then send them into a color box again, they're still white. So that's how I've generated them. I could have done it by rubbing the cat against glass, or rubbing it against me, right, just stroke the cat. Any randomly selected set of electrons sent into a color box, and then from which you take the white electrons. AUDIENCE: So how is it different from the experiment up there? ALLAN ADAMS: Yeah. Uh-huh. Exactly. Yeah. AUDIENCE: Is the difference that you never actually know whether the electron's hard or soft? ALLAN ADAMS: That's a really good question. So here's something I'm going to be very careful not to say in this class to the degree possible. I'm not going to use the word to know. AUDIENCE: Well, to measure. [INAUDIBLE] ALLAN ADAMS: Good. Measure is a very slippery word, too. I've used it here because I couldn't really get away with not using it. But we'll talk about that in some detail later on in the course. For the moment, I want to emphasize that it's tempting but dangerous at this point to talk about whether you know or don't know, or whether someone knows or doesn't know, for example, the monkey inside knows or doesn't know. So let's try to avoid that, and focus on just operational questions of what are the things that go in, what are the things that come out, and with what probabilities. And the reason that's so useful is that it's something that you can just do. There's no ambiguity about whether you've caught a white electron in a particular spot. Now in particular, the reason these boxes are such a powerful tool is that you don't measure the electron, you measure the position of the electron. You get hit by the electron or you don't. And by using these boxes we can infer from their position the color or the hardness. And that's the reason these boxes are so useful. So we're inferring from the position, which is easy to measure, you get beaned or you don't, we're inferring the property that we're interested in. It's a really good question, though. Keep it in the back of your mind. And we'll talk about it on and off for the rest of the semester. Yeah. AUDIENCE: So what happens if you have this setup, and you just take away the bottom right mirror? ALLAN ADAMS: Perfect question. This leads me into the next experiment. So here's the modification. But thank you, that's a great question. Here's the modification of this experiment. So let's rig up a small-- hold on, I want to go through the next series of experiments, and then I'll come back to questions. And these are great questions. So I want to rig up a small movable wall, a small movable barrier. And here's what this movable barrier will do. If I put the barrier in, so this would be in the soft path, when I put the barrier in the soft path, it absorbs all electrons incident upon it and impedes them from proceeding. So you put a barrier in here, put a barrier in the soft path, no electrons continue through. An electron incident cannot continue through. When I say that the barrier is out, what I mean is it's not in the way. I've moved it out of the way. Cool? So I want to run the same experiment. And I want to run this experiment using the barriers to tease out how the electrons transit through our apparatus. So experiment four. Let's send in a white electron again. I want to do the same experiment we just did. And color at out, but now with the wall in the soft path. Wall in soft. So that's this experiment. So we send in white electrons, and at the output we measure the color as before. And the question is what fraction come out black, and what fraction come out white. So again, everyone think through it for a second. Just take a second. And this one's a little sneaky. So feel free to discuss it with the person sitting next to you. [CHATTER] ALLAN ADAMS: All right. All right, now that everyone has had a quick second to think through this one, let me just talk through what I'd expect from the point of these experiments. And then we'll talk about whether this is reasonable. So the first thing I expect is that, look, if I send in a white electron and I put it into a hardness pass, I know that 50% of the time it goes out hard, and 50% of the time it goes out soft. If it goes out the soft aperture, it's going to get eaten by the barrier, right? It's going to get eaten by the barrier. So first thing I predict is that the output should be down by 50%. However, here's an important bit of physics. And this comes to the idea of locality. I didn't tell you this, but these armlinks in the experiment I did, 3,000 kilometers long. 3,000 kilometers long. That's too minor. 10 million kilometers long. Really long. Very long. Now, imagine an electron that enters this, an initially white electron. If we had the barriers out, if the barrier was out, what do we get? 100% white, right? We just did this experiment, to our surprise. So if we did this, we get 100%. And that means an electron, any electron, going along the soft path comes out white. Any electron going along the hard path goes out white. They all come out white. So now, imagine I do this. Imagine we put a barrier in here 2 million miles away from this path. How does a hard electron along this path know that I put the barrier there? And I'm going to make it even more sneaky for you. I'm going to insert the barrier along the path after I launched the electron into the apparatus. And when I send in the electron, I will not know at that moment, nor will the electron know, because, you know, they're not very smart, whether the barrier is in place. And this is going to be millions of miles away from this guy. So an electron out here can't know. It hasn't been there. It just hasn't been there. It can't know. But we know that when we ran this apparatus without the barrier in there, they came out 100% white. But it can't possibly know whether the barrier's in there or not, right? It's over here. So what this tells us is that we should expect the output to be down by 50%. But all the electrons that do make it through must come out white, because they didn't know that there was a barrier there. They didn't go along that path. Yeah. AUDIENCE: Not trying to be wise, but why are you using the word know? ALLAN ADAMS: Oh, sorry, thank you. Thank you, thank you, thank you, that was a slip of the tongue. I was making fun of the electron. So in that particular case, I was not referring to my or your knowledge. I was referring to the electron's tragically impoverished knowledge. Yeah. AUDIENCE: But if they come out one at a time white, then wouldn't we know then with certainty that that electron is both hard and white, which is like a violation? ALLAN ADAMS: Well, here's the more troubling thing. Imagine it didn't come out 100% white. Then the electron would have demonstrably not go along the soft path. It would have demonstrably gone through the hard path, because that's the only path available to it. And yet, it would still have known that millions of miles away, there's a barrier on a path it didn't take. So which one's more upsetting to you? And personally, I find this one the less upsetting of the two. So the prediction is our output should down by 50%, because a half of them get eaten. But they should all come out white, because those that didn't get eaten can't possibly know that there was a barrier here, millions of miles away. So we run this experiment. And here's the experimental result. In fact, the experimental result is yes, the output is down by 50%. But no, not 100% white, 50% white. 50% white. The barrier, if we put the barrier in the hardness path. If we put the barrier in the hardness path, still down by 50%, and it's at odds, 50-50. How could the electron know? I'm making fun of it. Yeah. AUDIENCE: So I guess my question is before we ask how it knows that there's a block in one of the paths, how does it know, before, over there, that there were two paths, and combine again? ALLAN ADAMS: Excellent. Exactly. So actually, this problem was there already in the experiment we did. All we've done here is tease out something that was existing in the experiment, something that was disturbing. The presence of those mirrors, and the option of taking two paths, somehow changed the way the electron behaved. How is that possible? And here, we're seeing that very sharply. Thank you for that excellent observation. Yeah. AUDIENCE: What if you replaced the two mirrors with color boxes, so that both color boxes [INAUDIBLE] ALLAN ADAMS: Yeah. So the question is basically, let's take this experiment, and let's make it even more intricate by, for example, replacing these mirrors by color boxes. So here's the thing I want to emphasize. I strongly encourage you to think through that example. And in particular, think through that example, come to my office hours, and ask me about it. So that's going to be setting a different experiment. And different experiments are going to have different results. So we're going to have to deal with that on a case by case basis. It's an interesting example, but it's going to take us a bit afar from where we are right now. But after we get to the punchline from this, come to my office hours and ask me exactly that question. Yeah. AUDIENCE: So we had a color box, we put in white electrons and we got 50-50, like random. How do you know the boxes work? ALLAN ADAMS: How do I know the boxes work? These are the same boxes we used from the beginning. We tested them over and over. AUDIENCE: How did you first check that it was working? [INAUDIBLE] ALLAN ADAMS: How to say-- there's no other way to build a box that does the properties that we want, which is that you send in color and it comes out color again, and the mirrors behave this way. Any box that does those first set of things, which is what I will call a color box, does this, too. There's no other way to do it. I don't mean just because like, no one's tested-- AUDIENCE: Because you can't actually check it, you can't actually [INAUDIBLE] you know which one is white. ALLAN ADAMS: Oh, sure, you can. You take the electron that came out of the color box. That's what we mean by saying it's white. AUDIENCE: [INAUDIBLE] ALLAN ADAMS: But that's what it means to say the electron is white. It's like, how do you know that my name is Allan? You say, Allan, and I go, what? Right? But you're like, look that's not a test of whether I'm Allan. It's like, well, what is the test? That's how you test. What's your name? I'm Allan. Oh, great, that's your name. So that's what I mean by white. Now you might quibble that that's a stupid thing to call an electron. And I grant you that. But it is nonetheless a property that I can empirically engage. OK, so I've been told that I never ask questions from the people on the right. Yeah. AUDIENCE: Is it important whether the experimenter knows if the wall is there or not? ALLAN ADAMS: No. This experiment has been done again by some French guys. The French, look, dude. So there's this guy, Alain Aspect, ahh, great experimentalist, great physicist. And he's done lots of beautiful experiments on exactly this topic. And send me an email, and I'll post some example papers and reviews by him-- and he's a great writer-- on the web page. So just send me an email to remind me of that. OK, so we're lowish on time, so let me move on. So what I want to do now is I want to take the lesson of this experiment and the observation that was made a minute ago, that in fact the same problem was present when we ran this experiment and go 100%. We should have been freaked out already. And I want to think through what that's telling us about the electron, the single electron, as it transits the apparatus. The thing is, at this point we're in real trouble. And here's the reason. Consider a single electron inside the apparatus. And I want to think about the electron inside the apparatus while all walls are out. So it's this experiment. Consider the single electron. We know, with total confidence, with complete reliability, that every electron will exit this color box out the white aperture. We've done this experiment. We know it will come out white. Yes? Here's my question. Which route did it take? AUDIENCE: Spoiler. ALLAN ADAMS: Not a spoiler. Which route did it take? AUDIENCE: Why do we care what route? ALLAN ADAMS: I'm asking you the question. That's why you care. I'm the professor here. What is this? Come on. Which route did it take? OK, let's think through the possibilities. Grapple with this question in your belly. Let's think through the possibilities. First off, did it take the hardness path? So as it transits through, the single electron transiting through this apparatus, did it take the hard path or did it take the soft? These are millions of miles long, millions of miles apart. This is not a ridiculous question. Did it go millions of miles in that direction, or millions of miles in that direction? Did it take the hardness path? Ladies and gentlemen, did it take the hard path? AUDIENCE: Yes. ALLAN ADAMS: Well, we ran this experiment by putting a wall in the soft path. And if we put a wall in the soft path, then we know it took the hard path, because no other electrons come out except those that went through the hard path. Correct? On the other hand, if it went through the hard path, it would come out 50% of the time white and 50% of the time black. But in fact, in this apparatus it comes out always 100% white. It cannot have taken the hard path. No. Did it take the soft path? Same argument, different side, right? No. Well, this is not looking good. Well, look, this was suggested. Maybe it took both. Maybe electrons are sneaky little devils that split in two, and part of it goes one way and part of it goes the other. Maybe it took both paths. So this is easy. We can test this one. And here is how I'm going to test this one. Oh, sorry. Actually, I'm not going to do that yet. So we can test this one. So if it took both paths, here's what you should be able to do. You should be able to put a detector along each path, and you'd be able to follow, if you've got half an electron on one side and half an electron on the other, or maybe two electrons, one on each side and one on the other. So this is the thing that you'd predict if you said it went both. So here's what we'll do. We will take detectors. We will put one along the hard path and one along the soft path. We will run the experiment and then observe whether, and ask whether, we see two electrons, we see half and half, what do we see. The answer is you always, always see one electron on one of the paths. You never see half an electron. You never see a squishy electron. You see one electron on one path, period. It did not take both. You never see an electron split in two, divided, confused. No. Well, it didn't take the hard path, didn't take the soft path, it didn't take both. There's one option left. Neither. Well, I say neither. But what about neither? And that's easy. Let's put a barrier in both paths. And then what happens? Nothing comes out. So no. So now, to repeat an earlier prescient remark from one of the students, what the hell? So here's the world we're facing. I want you to think about this. Take this seriously. Here's the world we're facing. And when I say, here's the world we're facing, I don't mean just these experiments. I mean the world around you, 20 kilo mirrors, bucky-balls, here is what they do. When you send them through an apparatus like this, every single object that goes through this apparatus does not take the hard path, it does not take the soft path, it doesn't take both, and it does not take neither. And that pretty much exhausts the set of logical possibilities. So what are electrons doing when they're inside the apparatus? How do you describe that electron inside the apparatus? You can't say it's on one path, you can't say it's on the other, it's not on both, and it's not on neither. What is it doing halfway through this experiment? So if our experiments are accurate, and to the best of our ability to determine, they are, and if our arguments are correct, and that's on me, then they're doing something, these electrons are doing something we've just never thought of before, something we've never dreamt of before, something for which we don't really have good words in the English language. Apparently, empirically, electrons have a way of moving, electrons have a way of being which is unlike anything that we're used to thinking about. And so do molecules. And so do bacteria. So does chalk. It's just harder to detect in those objects. So physicists have a name for this new mode of being. And we call it superposition. Now, at the moment, superposition is code for I have no idea what's going on. Usage of the word superposition would go something like this. An initially white electron inside this apparatus with the walls out is neither hard, nor soft, nor both, nor neither. It is, in fact, in a superposition of being hard and of being soft. This is why we can't meaningfully say this electron is some color and some hardness. Not because our boxes are crude, and not because we're ignorant, though our boxes are crude and we are ignorant. It's deeper. Having a definite color means not having a definite hardness, but rather being in a superposition of being hard and being soft. Every electron exits a hardness box either hard or soft. But not every electron is hard or soft. It can also be a superposition of being hard or being soft. The probability that we subsequently measure it to be hard or soft depends on precisely what superposition it is. For example, we know that if an electron is in the superposition corresponding to being white then there are even odds of it being subsequently measured be hard or to be soft. So to build a better definition of superposition than I have no idea what's going on is going to require a new language. And that language is quantum mechanics. And the underpinnings of this language are the topic of the course. And developing a better understanding of this idea of superposition is what you have to do over the next three months. Now, if all of this troubles your intuition, well, that shouldn't be too surprising. Your intuition was developed by throwing spears, and running from tigers, and catching toast as it jumps out of the toaster, all of which involves things so big and with so much energy that quantum effects are negligible. As a friend of mine likes to say, you don't need to know quantum mechanics to make chicken soup. However, when we work in very different regimes, when we work with atoms, when we work with molecules, when we work in the regime of very low energies and very small objects, your intuition is just not a reasonable guide. It's not that the electrons-- and I cannot emphasize this strongly enough-- it is not that the electrons are weird. The electrons do what electrons do. This is what they do. And it violates your intuition, but it's true. The thing that's surprising is that lots of electrons behave like this. Lots of electrons behave like cheese and chalk. And that's the goal of 804, to step beyond your daily experience and your familiar intuition and to develop an intuition for this idea of superposition. And we'll start in the next lecture. I'll see you on Thursday.
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_20_Periodic_Lattices_Part_1.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So today we begin our study of solids and in particular of conductivity in solids and periodic potentials, and that diagram will mean something in a few weeks. So before I get started, questions on everything previous? Yeah. AUDIENCE: So when you're talking about fermions and bosons [INAUDIBLE]. Statistics. So we said That you had to write the statement particular way for fermions with a minus sign particular. PROFESSOR: Yes. AUDIENCE: Well, we were assuming that you had to write the statement a product statement of the two. Will it be that you can't do that sometimes because they can't be separated. PROFESSOR: Absolutely. That's a wonderful question. We'll come to that in the second next to the last lecture where we'll talk about something called entanglement, which is really what you're talking about. But here's the crucial thing, for fermions, so what I wrote last time, I said suppose we have two particles that are identical one is in the state we know one is in the state chi of x and the other is in the state described by phi of x. I'm going to call the position of the first vertical x and the position of the second particle y. And I can write the two states without property, well, three. I could fill many, but two in particular of the following from, chi of x phi of y plus or minus chi of y phi of x. And the point of this linear combination is that when they exchange positions, the x and y So here this is the amplitude for the first particle will be an x, and the second particle to be a y. If I switch the sign or the positions to psi plus minus of yx or if the first particle is at y, the second particle is at x, this is equal to just swap these two terms, plus or minus psi plus minus of x,y. So this wave function has been contracted to ensure that under the exchange of the position of the two particles, the wave function stays the same up to a sign. This plus sign is associated with systems we call bosons, and the minus sign is associated with systems we call fermions. Now as I've discussed previously, this is a persistent property of a system. If a system is bosonic, at some moment in time, it will always be bosonic. If it's fermionic in some moment in time, it will always be fermionic, and that is a consequence of the fact that the exchange operator that swaps the two particles commutes with the energy eigenfunction if they're identical particles. And so if it commutes with the energy operator, then its value is preserved under time evolution with that energy operator. So now the question that was asked is, if this is our definition of bosons and fermions, is it true that you can always write the wave function in this form of differences of products of two states? Well, the answer is both yes and no. The answer is yes for sufficiently-simple systems. The answer is no when you have many, many particles, and we can't always write it as a simple product of two states. We can't always write as the sum of two terms. What you will discover if you studied multi-particle systems in 805 and 806 is that there's a nice way of organizing this anti symmetrization property, and it doesn't necessarily have two terms, but it may have many, many terms. With that said, this is not the defining property. This is really the defining property, whether your a boson or a fermion. OK? Cool. Yeah. AUDIENCE: So I had a question about fermions. [INAUDIBLE] the explanation being makes perfect sense about why two things can't be in the same state. PROFESSOR: Yeah. AUDIENCE: But Used in 804-4, during recitation one of the instructors said, well actually, this takes like really hard quantum field theory to actually prove this. What does that actually mean? PROFESSOR: OK. Good. So the question is, look, there's this legendary statement that this fact that the Pauli exclusion principle is incredibly hard and requires the magical details of quantum field theory to explain, and that some surprising. So your recitation lecturer said this to you. AUDIENCE: Yes. PROFESSOR: I'm sure I love this person dearly, but I disagree with him in an important way. So there are two things are confusing. Two things that require explanation I should say. One, we can explain at the level of 804, and the other does require machinery, but it's not crazy complicated machinery. So let me tell you what we can explain with 804, and then let me sketch for you how the more sophisticated argument goes. So this fact, the basic fact that wave functions can be symmetric or anti-symmetric, and when you have identical particles, they must be either symmetric or anti-symmetric. This follows from the statement that we have identical particles. If you have identical particles, it must be true that under the exchange, they are either even or odd. They're either symmetric or anti-symmetric under exchange. So if we have an exchange of identical particles, we need to have identical symmetric or identical anti-symmetric. And now it's just an empirical question. What particles in the world are identical bosons and what particles are identical fermions. Identical fermions? Well, how could you tell the difference? You can tell that identical fermions do not want to be the same state, and identical bosons do as we saw last time, as an extra factor of two for two particles that are actually n for n particles. So bosons want to be in the same state. Fermions want to not be in the same state, so if we just look at the world and say, which particles are well modeled by assuming we have bosons, which particles are well modeled by assuming we have fermions. And answer turns out to be the following, any particle that has intrinsic angular momentum, which is a half integer, any particle which spin like the electron, which has a limit to 1/2 and we'll study that in more detail later, any particle that has half-integer intrinsic angular momentum empirically turns out to be a fermion. Take two electrons, take their wave function, and swap the positions of the two electrons, the way function picks up the minus sign. Take two neutrons, also spin 1/2. Swap them and you pick up a minus sign. It's a fermion. Take two particles of light, two photons, swap them and you get a plus sign, and these guys have integer spin. Angular momentum of light is 1, and similarly, if we could, for the Higgs boson, if you take two Higgs bosons and swap them, you'll discover that they're bosons and as a consequence there's a plus sign under exchange. So that is not surprising, it's just an observable property of the world. The thing that's shocking is why that's true, right? Why is it that things that have half-integer angular momentum, which sounds like an independent property from whether they're bosonic or whether identical bosons are identified, why is it that all things with half-integer angular momentum or spin are fermions, and all things with integer angular momentum like light or the Higgs boson or me we're for bosonic, we're identical with an even sign, and that requires quantum field theory. But it's not insanely difficult. It requires the bearest minima of quantum field theory. So anyone who wants to understand, come to my office hours and I happily explain that. It takes about 15 minutes just to give you the basics of quantum field theory, just the bare basics. It's pretty straightforward. But it's a beautiful thing about relativistic quantum mechanics, so I'm going to turn this all around. So let's look at the history. The history of this was people looked at atomic spectra and said, this is bizarre. It almost fits what we get from central potential except it does buy this factor of two integer indices. Ah-ha. There must be an extra quantum number and on top of that, so the spin, the two possible states of the electron. And secondly, it must be that electrons can't be in the same state. So it sounds like that's two hypotheses. Now, if you take those two hypotheses, everything else follows. Fine. What Dirac discovered when he studied the relativistic version of the Schrodinger equation, which really is quantum field theory, but when he studied the relativistic version of electrons, quantum mechanical relativistic version, he discovered that these two things are linked. It is cool. I grant you. So you he discovered that these two things are linked, and he identified what's referred to as the spin statistics theorem. If you know the spin, you determine the statistics. So exactly how that works, that does require a little bit of effort-- that's the 15 minutes-- but you can dispense with that entirely and not be at all shocked by anything if you just accept two principles instead of one, first, that electrons are anti-fermionic, so they don't want to be the same state, and secondly, they have half-integer spin. AUDIENCE: Given that the only thing that's really measurable is [? the square ?] or non-square of the wave function and the way the function itself is never completely actually, measurable. Is there a mathematical reason why exchange can't transform a wave function just by arbitrary phase? It has to be plus or minus 1? PROFESSOR: Well, the reason it has to be plus or minus 1 for two particles, so you said a couple of things. Let me answer the first part. So what I will call the first part of your question is, why did it have to be plus or minus 1. Why couldn't it have been an arbitrary phase? And the reason is that if you do this exchange operation twice for two particles, you get back the same thing. And that's what I mean by saying I have identical particles. Because I could quibble with this. You could say, well look, you could define some different notion of exchange, or under exchange, I pick up an extra phase. I swap them and then I swap them and I swap them again, I get a phase. But here's the problem with that. The problem with that is, and this can be dealt with, but an interesting question is if it can only be dealt with in two dimensions or it can be dealt with in general number of dimensions. This is a story called antions, which Frank Wilczek here has really pounded hard on. But here's the basic question. So does the wave function have a single value? I'd like to think it does. The wave function really should have a single value. If it has multiple values, then it's ambiguous what the value of the wave function is there. That's not good. That mean you have to specify more than just the wave function, you have to specify which of the values the wave function takes in a particular point in the value of the wave function. So if under double exchange we've returned to the original configuration, you pick up a phase, something is screwed. You must be able to tell whether or not you've exchanged twice, and here's why you can tell. Because while it's true that you can't tell the overall phase, imagine if I take my system and I put in a splitter and I have a beam splitter I could think of it as a two-split experiment, one component of that wave function I will double swap and the other component I will not. And then I will interfere them. I will combine them back together, do an interference experiment. So you measure that relative phase. So that relative phase definitely matters. Do you just see that point? So it's true that they overall phase they can't measure, but by doing a superposition and only exchanging one of the superposition pairs, by physically separating those, then you can see that in interference experiments again. So you can still deal with this, but it requires being able to know whether or not you can exchange, so there has to be some way of telling that you sort of entwined these guys, and that's something you could do in two dimensions. It turns out to be difficult to do in general number of dimensions. Well, it's an act of research topic. If you want to a more detailed answer, ask me later. OK. One more question. Yeah. AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah. it has precisely one observable property, which is this sign, this it's eigenvalue, value, and you can tell because the wave function either vanishes when you take the two points together or it doesn't. So that's it. AUDIENCE: So it's not like the momentum [INAUDIBLE]. So it's only basically just [INAUDIBLE]. PROFESSOR: Yeah. It's less interesting than position or momentum. It has two eigenvalues rather than many. But it's no less interesting than, for example, spin. It contains information that you can learn about system. OK. One more. AUDIENCE: I was wondering why you can write this fermion as two electrons I guess when it seems to me [INAUDIBLE]. PROFESSOR: That's a fantastic question. OK. Good. I was going to gloss over this. That's a fantastic question. So here's the question. Look, I have in my box here, I have a hydrogen atom, and I've quantized the cooling potential, and I know what the energy eigenvalues are for the cooling potential, and I put the electron in one of those states. And I say the wave function describing my system as sign is equal to phi [? nlm ?] for some particular [? nlm. ?] That's what we've been doing. But there are a lot of electrons in the world. In fact, there are electrons in the walls. There are electrons in your nose. There are electrons everywhere, and they are fermion. So the wave function describing the entire system must be invariant up to a sign under the exchange of any two electrons. If I take this electron and I swap it with an electron in Matt's ear-- Hi, Matt-- then the wave function had better pick up a minus sign. So what business do I ever have writing a single wave function for a single electron. Is that your question? AUDIENCE: Yes. PROFESSOR: That's an excellent question. So let me answer that. So suppose I have two electrons, and I have the wave function for two electrons x and y. I mean, I write this as if it's in one dimension. This is some position, some position, but this generally is just an arbitrary number of dimensions, and one is in the state chi of x, and the other is in the state phi of y. But I know that I can't have this be the state because it must be invariant under swapping x and y of two minus signs. So I need minus chi at y phi at x, and let's normalize this, so 1 over 2. So that's our electron. Now, at this point, you might really worry, because say chi is the electron is the wave function bound to my atom, and phi is the wave function for the electron bound to an atom in Matt's ear. So we've just in writing this. Why can we do that? Well, let's think about this. What does the combination of terms give us? What do they tell us is that they're interference term. If I ask them what's the probability to find an electron at x and an electron at y, this is equal to, well, it's this whole quantity norm squared sine norm squared, which is equal to 1/2, and then there's a term where this gets term squared, this term squared, and then two cross terms. So the first term gives us the chi at x norm squared phi at y norm squared. The second term gives us plus chi at y norm squared phi at x norm squared plus twice the real part of I'm going to write this subscript chi sub x phi y, complex conjugate, complex conjugate chi y phi x. Everybody agree? That's just the norm squared of everything. AUDIENCE: Shouldn't that be minus? PROFESSOR: Oh, shoot. Sorry. Thank you. Minus. Yes, minus. Importantly minus. Now, let's be a little more explicit about this. Saying that chi is the state localized at the hydrogen, then that's as if the hydrogen is here. If the proton is here, then the wave function for chi has a norm squared that looks like this. And if I say that phi is the state corresponding to being localized, at Matt's ear, then here is the ear, and here is the wave function for it, so that's phi. So let's then ask, what's the probability that I find the first electron here and the second electron here? Well, if this is x, and this is y, what's the probability that I find a particle there and a particle there? Well, what's this term? Well, it's chi of x, which is not so small-- that's just some number-- times phi of y, some value norm squared, so this is some number. What about this one? What's chi of y? Well, chi of y, this is all sharply-localized wave function over here chi, and it's 0 out here. The probability that an electron bound to the hydrogen is found far away is exponentially small as we've seen from the wave functions of hydrogen. It's bound. So this is the negligibility small as is phi of x. Phi of x is negligibility do small. The wave function is localized around the ear, but x is way over there by the hydrogen. It's exponentially small. This term is also exponentially small. Now in here, what about this? Chi of x phi of y, that's good. Those are both fine, but chi of y phi of x phi of x, these are both negligibility small, so these terms go away. So this is just equal to chi of x norm squared phi of y norm squared. And this is what you would get approximately, and that approximately is fantastically good as they get far apart. This chi of x is just the probability that the particle that x given that it was in the wave function bound to the hydrogen, and ditto given that it was bound to the ear. Cool? So the important thing is when fermions are separated and the wave functions are localized, you can treat them independently. When the fermions are not separated or when their wave functions are not localized, you can't treat them independently and you have to include all the terms. That cool? OK. Yeah. AUDIENCE: That's times 1/2, right? PROFESSOR: Yeah, that is times 1/2. What did I do? I'm screwing something up with the normalization. Ask me that after lecture. It's a good question. I'm screwing something up. I'm not going to get the factor of 2 straightened out right now. OK. Does that answer your question? OK. Good. This is a really deep-- it's important that this is true. I'm going to stop questions because I need to pick up with what we need to do. So this is a really important question because if it weren't true that you could independently deal with these electron, then everything we did up until now in quantum mechanics would have been totally useless for any identical particles. And since as far as we can tell, all fundamental particles are identical, this would have been very bad. So with that, let's move on to the study of solids. So I want to pick up where your problem set left off. The last part of your problem set, which I hope everyone did, the left part of your problem set involved looking at this simulation from the PhET people of a particle in a series of n wells. And I want to look at the result, at least the results that I got for that simulation. So let's see, here are the data points that I got-- shall I make that larger? Is that impossible to see? Those are the data points that I pulled off of PhET just the same way you did. You look, you point, you move the cursor, and you pull off the data points. So those are approximately then. And this is for 1, 2, 3, 4, 5, 6, 7, 8, 10 wells, and here's the plot of them as a function of the index which state they are. So as a function, this is the first state. This is the second state, third, fourth, et cetera. Here are the energies vertically. And hopefully, you all got to plot some more of this and they seem to become bunched together, and the energy of the ground state without the potential was this. So I'm not going to add in a parabola corresponding to a free particle with the associated wavelengths. Remember I asked you to say take that state that looks most like a momentum. eigenstate with a definite wavelength and calculate the energy of a free particle with the corresponding wavelength. So this parabola-- I should make this smaller-- is a parabola describing a free particle which agrees with those free particle energies at the three points where you're supposed to compute them. So there's the parabola. And so what I've done here is I've grouped them together into the bands of states. So vertically we have energy. On the horizontal side, we have n but remember that n is also corresponding kind to a momentum because each state wiggles and especially the states at the top of each band have a reasonably well-defined momenta, and so they look a lot like momentum eigenstates. So I'm going to interpret this horizontal direction, it's something like a momentum. And at the top of the states, it certainly makes sense, but then what it actually is is just the number the level, and what we see is that the energy is a function of the level is close-- near the top anyway-- to the energy of a free particle, but the actual allowed energies are bound. So which energies correspond to allowed states, and which energies have no allowed states associate with them is encoded in this diagram. Are we cool with that? In this shaded region, there are states with an energy in that region, at least approximately, and in this unshaded region, there aren't any. And in each band, there are n energy eigenstates, and each band corresponds to one of the bound states of the potential. OK. Everyone cool with that? So does this look more or less like what y'all got? AUDIENCE: Yes. PROFESSOR: Good. OK. Questions about this before I move on? It's an important one. And let me just remind you of a couple of facts about this PhET simulation so here's what we see if we have the single well. We have three states 1, 0, two nodes, two zeroes, and they satisfy the node theorem. If we have many states, then if we look at the bottom state in each well. So there's the bottom state and the bottom band. It has no zeroes. It satisfies the node theorem, and the top one by comparison looks extremely similar. It looks extremely similar to the ground state, which is the orange one, the lowest energy state. It is a little higher energy. You can see that because the curvature is greater. If you look at the top and bottom of each wave function, you see that the curvature is greater for the yellow guy, which is the guy at the top of the band, and the yellow guy, meanwhile, looks like a wave with a definite wavelength. But a general state in here doesn't have any simple symmetry properties. It's not periodic. It's not approximately periodic. Well, that one's kind of approximately periodic. But the top guy and the bottom guy look approximately periodic. They have some nice symmetry properties. But in general, they're just not translationally invariant. They're not symmetric. They don't have any simple, nice structure. They are just some horrible, ugly things. The bottom though and the top guy always have some nice symmetry property. Anyway to keep that in mind. And the last thing I want you to note from all of this is that as we make the potential barriers stronger, two things happen. First off, bands become very narrow, the gaps between them become very large, and secondly, the wave functions take now an even more sort of complicated and messy, which is more obvious that the wave functions are not periodic. This thing it looks almost periodic, but is periodic with some funny period, and it's not a single period by any stretch of the imagination. OK. So any questions about the wave functions in here? Yeah. AUDIENCE: In this one, there is an overall [INAUDIBLE] period. [INAUDIBLE] PROFESSOR: Excellent. OK. So let me give you a little bit of intuition for that. So suppose I have a big box. What's the ground state? What does the ground state look like? Yeah. Good. Everyone's doing this. That's good. So the ground state looks like this, and the first excited state is going to do something like this, and so on. Cool. And what's the third excited state going to look like? I guess I shouldn't do that. The third excited state is going to do something like this. Cool? Now, meanwhile, imagine I take my periodic well and I add to it a bunch of delta function scatterers with some strength. When the strength is zero, I just recover my original well. When the strength is non-zero though, we know that what a delta function is going to do is, it's going to make a little kink. It's going to induce a little condition on the first derivative at the delta function. So what's going to this ground state when we turn a little tiny bit of that delta function? Yeah. What we're going to get is, we're going to get something that kinks at each of the delta functions. But it still has to have those boundary conditions. Everybody cool with that? And as we make the delta function stronger and stronger, the effect of this is going to eventually be to give us something that looks like-- yeah? Now that should look a lot like that. Everyone cool with that? Meanwhile, what's going to happen to the second guy? Well the second guy, same thing. The effect on the second state is going to be-- cool? I'm not sure of I need the sound effects. So let's magnify and let's look at the second state here. Ooh, it's going to be hard to deal with that tiny energy splitting. Come on finger, you can do this. There we go. There is two. Ah-ha. That looks a lot like this. Yes, that second state looks an awful lot like this. And let's go back to that funny state that we were looking at before. Let' see if I can get it. Should be the third guy here. So there's a second guy in the third band. Again, you see an envelope and then the fluctuation from being in the third state and in the second state in this band as well. So on the other hand, what's the scale of the energy splitting due to a well of it this width compared to the energy splitting between states due to a well of this width? A wide well means that its states are close together. An ENDOR well means the states are far apart. So what state do we have? We have the state that was the ground state of the whole box with a correction due to the delta function versus states which are, for example, excited states in the box. The splitting between this state and this state is going to be much larger than the splitting between this state and this state, just because this is a tiny little well and so the link scale is much shorter, the energy is much greater. That's the intuition you should have from these guys. So when you see that there's some approximate structure, let's see let's make this nice and separated again. And Oh, yeah. these are kind of difficult to control when they're so close together. The trouble with touch pads. There we go. OK. There we go. That was the guy we were looking at before. And so you see that there is an overall envelope which has three maxima, and then there's the n inside each well. It's got two zeroes inside each well. And it's that envelope which is coming from the fact that you're in a box and the two zeroes inside each well, which is coming from this. All the states here in this band are of the form two zeroes inside the well. All the states here are the form two zeroes is inside each well in this band except they're being modulated by an overall sine wave due to the fact that they're in a box. And so they're very closely split states with a different overall modulation but with each state inside each band corresponding to either the ground state for the individual well or the first excited state in the individual well or the second excited state. So each band corresponds to which excited state you are inside the well, and which state you are within the band is your modulation of the overall wave from being inside a box. That make sense? AUDIENCE: I guess so, but why does the overall modulation have a smaller effect than the-- PROFESSOR: Great. If I have a well that's this wide, what are the ground state energies? And let's say the width is L. What are the energies in this square well, an infinite well with L? Yeah, but what are the eigenvalues? AUDIENCE: [INAUDIBLE]. PROFESSOR: E sub n. AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah. Exactly. So they go like, I'm just going to write proportional to, n squared over L squared, because it's a sine wave with period n pi upon L. So the k is n upon L with constants, and that means that the energy again, constant h squared k upon 2n is k squared, which is n square over L squared. But the important thing is that the splitting between subsequent energy levels is controlled by the width. It goes like 1 over L squared. So if you have a very wide well, the splittings are very small, the ground state. On the other hand, if you have a thin, narrow well, then the splittings also go like L squared, but L is very small here so the splittings are gigantic. When we have a superposition of a big box inside of which we have a bunch of barriers, then the splitting due to being excited in the individual wells goes like 1 over the little distance squared. Let's call this a squared. So these go like 1 over a squared, delta a delta E, goes like 1 over a squared, and from here it goes like if this is L, delta E goes like 1 over L squared. So the combined effect is that you get splittings due to both. These splittings are tiny, and these splittings are really large because this is a much smaller distance, and this is a much larger distance. Cool? So the only question is how big in amplitude are these-- how effective are these barriers. If they were strict delta functions, then this is all we would get. And if there were no delta functions, we would just get precisely these. And when we have some barrier, we get a combination of the two, and that's what we're seeing here with the splitting. So what I'm doing when I'm tuning the separation here I'm controlling effectively how strong is that barrier. And so as we make the barriers stronger-- there we go-- then the splittings are controlled by the individual wells, an as we make the barrier very inefficient, then the splittings are controlled by the overall box. Cool? Excellent. Other questions about the simulation? Yeah. AUDIENCE: How close are the actual [? wavelength ?] potentials or the combinations of the original wavelength? PROFESSOR: Excellent question. So you can answer that immediately from this. So the question is this, look, if we had arbitrarily separated wells, if we had something that looked like-- so they're very, very far separated. Then the ground state would be effectively degenerate, because the wave function or the ground state would be the completely symmetric combination of these guys-- you guys actually studied this in our problem set-- then there's also a combination where you have this, this, this. You could also take this constant this. There are many things you could do. I'm sorry, this, many combinations. The point is since each wave function for each well effectively drops off to zero inside, we can just linear combinations of these, and there's no penalty for using the true ground seats in here because the potential is so high, that the true solution has an exponential tail, but that exponential tail is ridiculously small if the barrier is big. So in the limit that the barriers are gigantic, the true energy eigenstates are just arbitrary linear combinations of these guys, of the individual eigenstates of each individual well. Agreed? However, when the barriers a large but finite, then the true energetic states have little tiny exponential tails, and the curvature of that little tiny exponential tail will determine exactly what the energy is. So a state that curves a little bit versus a state that curves more will have slightly different energies. So when the barriers are very, very large, the linear combinations of the individual well eigenstates should be very good approximations, but not exact. And as the barriers become less and less effective, they should become less and less exact, and we can see that right here. So let's take the ground state. The bottom of the band is the completely symmetric combination of the ground seat of each well. Everyone see that? It's just well, well, well, well. It's a completely symmetric combination. At the top of the band is the completely anti symmetric combination, and let's put them for comparison. The top of the band is the orange one, and the bottom of the band is the yellow one. So the bottom of the band is the completely symmetric, and the top of the band is alternating combination of the ground state in each well. And all the other states inside here-- well, which are extremely difficult to-- Let's see. There we go. These states are also linear combinations of the ground state in each well with different coefficients in front of them. There are the different coefficients, and they are almost degenerate. But if we go to higher energy states where the barrier is less effective-- because they have high energy and the ratio between the barrier height and their energy is small-- then you see that these states are not particularly well approximated by linear combinations. And the energies correspondingly are not degenerating. AUDIENCE: So you said linear combinations of the [INAUDIBLE]. PROFESSOR: That gives you a better approximation, but for the same reason. It's a good approximation, but it's not exact. But it becomes excellent as the barriers become infinite, even when they also go over to the infinites very well. Very good question. Yeah. AUDIENCE: [INAUDIBLE]. PROFESSOR: Very good question. We'll come to that in little bit. We'll come to that shortly. So I'm done at the moment with the PhET simulations. I'm motivated by all this, and effectively by that question, I'm going to ask the following thing. Look, real materials like metal in my laptop, the real material is actually a periodic potential. It's built out of a crystal of metal atoms that are bound together, and each atom is some positively-charged beast, to which some positively-charged nucleus, to which is bound an electron. And if I want to understand properties of solids, like the fact that metals conduct electricity, a basic fact I'd like to explain, if I want to explain the fact that metals conduct electricity, but plastic doesn't, which we will be able to explain, and diamond doesn't, which is cool, if we want to explain that property, we probably ought to study the physics of electrons in periodic systems where the potential is, atom, I don't want to be stuck to you. Atom, I don't want to be stuck to you. Atom I don't want to be stuck to you. So it's a periodic well a potential. So in order to study the physics of solids, I need to understand first the physics of electrons in periodic potentials. And these PhET simulations were a first start at that. We did it n's wells. I want to now think about what happens if I take not n, but an infinite number of wells. What if I really strictly periodic lattice on the line? What do we expect to happen? Well, looking at the results of these PhET simulations, as you did this for different numbers, n, what you found was the same sort of band structure. You just find for more and more wells, you find more and more states inside each band. In fact, you find n states within each band. Both the top and bottom of the band quickly asymptote to fixed values. As we take at large, what do you expect to happen? Well, this was a problem on your problem set. What did you predict? What should happen when you take n large? AUDIENCE: [? It ?] depends on the experiment. PROFESSOR: Yeah. How many states should there be in each bend? AUDIENCE: N. PROFESSOR: N. So there are arbitrarily many. Exactly. So there should be arbitrarily many states in each band, but they all have to fit within this energy band, the width, which is asymptoting to a constant. So there has to be a continuum of states, and they're not going to be free particle states because the system has a potential. What is the shape going to be? Well, a reasonable guess is that the shape is going to be something that fills out this curve. It's going to be a little bit different from the free particle state. So let's find out. Let's just solve the problem of an electron in a periodic potential, and look, the periodic potential is extremely similar to things we've already solved. So I want to walk you through just a basic argument. So periodic potentials. Instead of n wells, consider a system of an infinite number of wells, each one identical and just for symmetry purposes. And let the width here be L, so the period of the lattice is L. And what I mean by that is saying, so this is V of x seeing that it's periodic is the statement as that V of x plus L is equal to V of x. Potential doesn't change if I shift it by a lattice vector, Now there's a nice way to say this which is that, if I take the potential, v of x, and I translate it by L, I get the same potential back V of x. And more to the point, if I take the translation if I think of these as operators expressions, if I take TL and act on Vx and then the potential of Vx f of x, then this is equal to V of x TL f of x. So if I think of these as operators instead of just little functions operator, operator, then if I take the translator operator and I translate V of x f of x, I'm going to get V of x plus L f of x plus L. And if I do the right-hand side I get translator of L that's f of x plus L but just times V of x. But if V of x is equal to V of x plus L-- this is V of x plus L. So if I translate by L says take this thing and translate it, plus L f of x plus L. And on the right-hand side, we have V of x, not V of x plus L translate by L f of x, f of x plus L And these are equal because V of x plus L is equal to V of x. Yeah. So what that tells you is as operators translate by L and V of x commute. Equals 0. Everyone happy with that? So you should just be able to immediately see this, the potential is periodic with period L if by translated by L, nothing should happen. So the potential respects T of L it commutes with it. So this tells you a neat thing. Remember that translate by L is equal to E to the i PL upon H bar, so in particular, TL commutes with P because this is just a polynomial in P dL P, and in particular with P squared is equal to 0, and that just follows from the definition. So what that tells us is that if we take the system with our periodic potential, periodic V of x plus L is equal to V of x, then the energy, which is P squared upon 2 m of V of x commutes with TL. So in this system, with a periodic potential, is momentum conserved? What must be true for momentum to be conserved? So momentum conservation come from a symmetry principle. AUDIENCE: Translation invariance. PROFESSOR: Translation invariance. In order for momentum be conserved, the system must be translationally invariant. Is the system translationally invariant? No. If I shift it by a little bit, it's not invariant. It is, however, invariant under a certain subset of translations, which is finite shifts by L. Yeah. So it's invariant under shifts by L, which is a subset of translations, and the legacy of that is the fact that the energy commutes with translations by L. The energy does not commute with P, Because P acting on V of x, it takes the derivative and gives you the prime of x. It is not the same thing. But we have a subset of translations under which the system is invariant, and that's good. That's less powerful than being a free particle, but it's more powerful than not knowing anything. In particular, it tells us that we can find energy eigenfunctions, which are simultaneously eigenfunctions of the energy, and, I'll call it Q for a moment, they have an eigenvalue under TL. So we can find eigenfunctions which are simultaneously eigenfunctions the energy operator you might E of phi E is equal to E phi Eq and which are eigenfunctions under translate by L on phi Eq is equal to something times phi Eq. So we're going to find simultaneous eigenfunctions. Everyone cool with that? Yeah. AUDIENCE: Do you know if the momentum operator is still the momentum operator? PROFESSOR: Sorry. Say it again. AUDIENCE: The momentum operator is still in remission, right? PROFESSOR: Yeah. The momentum operator is still the momentum operator. [INAUDIBLE]. AUDIENCE: Do you know if [INAUDIBLE]? PROFESSOR: Excellent. Excellent. Good Yes. OK. So if T of L is unitary, note, T of L is unitary. And you actually showed this on a problem set before, and let me remind you a couple of facts about it. The first is that T of L is unitary, and that says that TL dagger is equal to the identity, right? But what is TL dagger? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah. TL dagger, well, P is remission, so we just pick up a minus sign, so TL dagger is equal to e to the minus i PL over H bar, and we can do think of this two ways. First thing you can notice that it's clearly the inverse of that guy, by definition of the exponential, but the other is, TL is the thing that translates you by L, and this operator, e to the minus i PL, well, I could put my H wit h the and that's just translation by minus L. So this is translated by L and then translated by minus L. And of course, if you translate by L and then you untranslate by L, you haven't done anything with the identity. What are the eigenvalues? What are the form of the eigenvalues of a unitary operator? AUDIENCE: [INAUDIBLE]. PROFESSOR: Well, that's true. The eigenfunctions are orthonormal, but the eigenvalues, are they real numbers? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah. So the eigenvalues of a unitary operator are of the form, well, imagine we're in a moment eigenstate, for a momentum eigenstate, then P is a real number. And then the eigenvalue of TL is e to the i, the P over H bar, that's a real number. So is e to the i a real number? That's a pure phase. It's e to i a real number. So the eigenvalues are pure phases, and this let's just do a nice thing. We can note that here the translate by L of the phi, if these are common eigenfunctions of E and of TL, well, TL is unitary, its eigenvalue must be a pure phase, e to the i a real number. So let's call that real number alpha, e to the i alpha. So maybe just label this by alpha for the moment, and you'll see why I want to change this to q later, but for the moment let's just call it alpha. Everyone cool with that? So you actually show the following thing on a problem set, but I'm going to re-drive it for you now. We can say something more about the form of an eigenfunction of the translation operator. This is quite nice. So suppose we have a function phi sub alpha such that TL and i is equal to e to the i alpha phi alpha. I want to know what is the form. What can I say about the function phi f of x? What can I say about the form of this function? We can actually say something really useful for this. This is going to turn out to be necessary for us. The first thing I'm going to do is, I'm going to just make the following observation. Define the function u of x, which is equal to e to the minus i qx phi sub alpha. So if phi sub alpha is an eigenfunction of TL with an eigenvalue e to the alpha, and I'm just defining this function u to be e to minus i qx times phi sub alpha. Cool? It's just a definition. I'm going to construct some stupid u. Note the following thing. T sub L on u, if we translate u of x, this is u of x plus L. Translating this, I get e to the i qx goes to e to the minus i qx plus L, which is just the e to the minus i qx times e to the minus i qL e to the minus i qx plus L phi alpha of x plus L. But this is equal to from here e to the minus i qL e to the minus i qx. And from phi of x plus a, I know that if I translate by a to get phi of x plus L, I just pick up a phase, e to the phi alpha, so e to the phi alpha phi alpha of x. But this is equal to e to the i alpha minus qL, putting these two terms together, and this is nothing but u of x. Yeah. So another following cool thing. So here I define some function u with some stupid value q, which I just pulled out of thin air, and I'm just defining this function. What I've observed is that if I translate this function by L, it picks up an overall phase times its original value where the phase depends on alpha and on q. So then I'm going to do the following thing. This becomes really simple if I pick a particular value of q, for a special value of q, q is equal to alpha divided by L, then this is equal to e to the i alpha minus qL. That's going to be equal to e to the i 0, which is just 1, so I have the phase, so this u of x. So that means if I translate by L on u and I pick q is equal to alpha over L, I get back u, so u is periodic. Everyone see that? Question? So as a result, I can always write my wave function phi sub e and we'll now call this q or q is related to alpha as q is equal to alpha over L, every energy eigenstate can find a basis of energy eigenstates with a definite eigenvalue under the energy, a definite eigenvalue under T sub L, and I can write them in the form, since u is equal to e to the minus i qx phi, then phi is equal to e to the i qx u e to the i qx u of x where q, where the eigenvalue, e to the i phi alpha, is equal to e to the i qL and u of x is periodic. Everyone cool with that? It's not totally obvious how much this is helping us here, but what we've done is, we've extracted, we've observe that there is some lingering symmetry in the system, and I've used that symmetry to deduce the form of the energy eigenfunctions as best as possible. So I haven't completely determined the energy eigenfunctions, I've just determined that the energy eigenfunctions are of the form, a phase, e to the i qx, so it varies as we change x, times a periodic function. So the wave function is periodic up to an overall phase, not a constant phase, a position-dependent phase. Does everybody agree with that? Questions about that? So a couple things to note about this q, the first is, the eigenvalue under TL e to the i alpha, this is the eigenvalue under L also equal to e to the i qL, but if I take q to q plus 2 pi over L, then nothing changes to the eigenvalue, because the 2 pi over L times L is just 2 pi, e to the 2 pi is 1, so we don't change the eigenvalue. So different ways of q only correspond to different eigenvalues of translate by L of a translation by one lattice vector, one lattice basing. They only correspond to the different eigenvalues of TL if they don't differ by 2 pi over L. If different values of q differ by 2 pi over L, then they really mean the same thing because we're just talking about translate by one period. So q is equivalent to q plus 2 pi over L. So that's just the first thing to know. So what have we done so far? What we've done so far is nothing whatsoever except extract, take advantage of the translational symmetry that's left, the remaining lingering little bit of translational symmetry to constrain the for of the energetic functions. What I want to do now is observe some consequences that follows immediately from this. So there are some very nice things to follow just from this. We can learn something great about the system without knowing anything else, without knowing anything about the detailed structure of the potential. At this point, I've made no assumptions about the potential other than the fact that it's periodic. So the thing I'm about to tell you were to be true for any periodic potential. They follow only from the structure. So let's see what they are. So the first one is that the wave function sine of x itself with a definite value of e and a definite value of q is not periodic by L. Because under a shift by L, u is periodic, but this part picks up a phase e to the i qL. That's what it is to say that an eigenfunction, the translation operator. It's not periodic by L unless q is equal to 0. So for q equals 0, there is precisely one wave function which is periodic by L. Because if q is equal to 0, then this phase is 0. So only if is equal to 0 is the wave function periodic. Cool? So this should look familiar because back in the band structure, these guys are not periodic. So these states in the middle of the band, they're horrible. They're not periodic. They're some horrible things, but that top guy in the band is periodic, and it turns out that if we had made the system infinitely large, it would become exactly periodic. The fact that it has an envelope on it like this is just the fact that we have an finite number of wells, we're in a box. If we got rid of that box and we made many, many wells, we would find it perfectly periodic. Yeah. AUDIENCE: Could q not be something like 2 pi over L? PROFESSOR: Yeah, but 2 pi over L would be equal to, it's equivalent to q equals 0, because it corresponds to the same eigenvalue. So q's that differ by 2 pi over L, I'm going to call the same thing. AUDIENCE: OK. Got it. PROFESSOR: Good. Other questions? So the wave function is not periodic except for one special case, and we already have a guess as to which special case. It looks like it's the state on the top of the energy band, which is approximately periodic. Two, while the wave function is not periodic, the probability that you find the particle at x plus L is equal to the probability that you find the particle at x, and this immediately falls in this form, because when you take the norm squared, the phis cancels out at every point and we're left with the periodic u of x. So the probability is equal to norm squared of u of x, which is equal to norm squared of u of x plus L because u is periodic. So probability distribution is periodic. On the one hand, this is reassuring, because if you look at the potential, the potential is perfectly periodic, and it would be really weird if you could tell by some probability, by some measure, like how likely you would find an electron here as supposed to over here, potential is the same, the probability should be the same find the electron in each spot. The problem with that logic though, is that these are wells and usually we think that when we have wells, we have bound states, and those bound states are localized. But the probability has to be invariant under translations, so it cannot be that the electron wave functions are delocalized. The electrons wave functions must have equal overall amplitude to be in any given well. It must be invariant under translations by a lattice spacing. So that's really weird. We have a whole bunch of quantum wells, and yet there are no localized states. All the states are extended just like free particle states are extended. It had to be this way by periodicity, but it should surprise you a little bit. Anyway it surprises me. I shouldn't tell you what to be surprised by. Yeah. AUDIENCE: To normalize probability, what do you do? PROFESSOR: Yeah. Excellent. OK. So this is for the same reasons that free particle wave functions aren't normalizable, these aren't going to be normalized either. So what are we going to have to do? AUDIENCE: Build wave packets. PROFESSOR: Yeah. We've going to have to build wave packets. In order to really meaningfully talk about that stuff, we're going to have to built wave packets. We need wave packets to make normalizable states. But the localized wave packets are necessarily not going to be energy eigenstates just like for a free particle. And so there's going to be dispersion, and the whole story that you saw three particles we're going to see again for the particles in the periodic potential. Cool? So next thing. Therefore, all phi sub Eq extended. This is going to have some really cool consequences, and understanding this fact in detail is going to explain to us the difference between connectivity in the middle and not conductivity in plastic. Although there's a surprising hook in there. Naively, it would have gone the other way around. So the last thing to say three, is that phi sub to Eq, so this is the state with definite energy and definite eigenvalue under translation E to the i qL Yeah. And you showed on a problem set that if you take a wave function, you multiply it by E to the i qx, what have you done to the expectation value of the momentum? AUDIENCE: You've increased it by q. PROFESSOR: You've increased it by q, right? This is the boost it by q operation. So q incidentally has units of a wave number, so times h bar has units of a momentum. q is like a momentum. Is q a momentum? Well, we can answer that question by asking, is this an eigenfunction of P? Is it an eigenfunction a momentum operator? And the answer is definitely not. That's an eigenfunction of a momentum operator. I take a derivative, I get q times i and I multiply by h bar, I get h bar q, momentum, but when I have this extra periodic function, this thing together is no longer an eigenfunction of q, because u is some periodic function, which is necessarily not a momentum eigenfunction. So this guy is not a momentum eigenfunction. You can see that by just taking the derivative. I get one term that picks up a q from this, but I get another term that gets a derivative with respect to u, and that's not proportional with the constant to u e to the i qx unless u is in the form e to the i kx. But if u is in the form e to the i kx, this is just a free particle wave function, which it can't be if we have a periodic potential. So finally q is not a momentum eigenstate. And correspondingly, q and more important h bar q is not a momentum, I'll say the momentum in the sense that it's not the eigenvalue of the momentum operator. However, it has-- semi colon-- it has units of momentum. And not only does it have units of momentum, it's the thing that tells you how the state transforms under translations. It translates by an e to the i qL under translations by L. And there is usually what momentum is. it's the constant that tells you how the state transfers under momentum. So it's kind of like a momentum, but it's not the eigenvalue of the momentum operator. We have a name for it. We call it the crystal momentum, and we'll understand what it does for us. Part of the goal for the next two lectures is going to be to understand exactly what this crystal moment is. So for the moment, I'm just giving it a name sort of like the beginning we give super position a name, and we'll exploit its properties basically now. So these are the things that follow just from periodicity. I didn't know anything about the potential. I just used periodicity. So before we move on to talking about a specific potential, you all have questions about the general structure of periodic systems like this. Yeah. AUDIENCE: What's the difference between the psi and the [INAUDIBLE]? PROFESSOR: Oh, excellent. I shouldn't have used psi, I should have used phi. Thanks. I'll use psi only when we talk about general super positions and wave packets. Other questions? All right. So with all that, let's talk about a specific potential. So far we've extracted about as much as of the physics out of the system that we can just from periodicity. In order to make more progress, in order to talk about, for example, what are the allowed energy eigenvalues and for that matter, how does that allowed energy eigenvalue depend on q, the crystal momentum, in order to answer that, we have to talk about a specific system. So let's go ahead and talk about a specific system. I'm going to pick my favorite. Now the problem set, you're going to do something really cool about it, which I'll explain in a moment. So you'll do a general case in the problem set, but for the moment, let's work with a simple example. And the simplest example is going to be a periodic potential with the simplest possible barriers in the potential. What's the simplest barrier? Delta function. Yes. So the potential is going to be for this example V of x is equal to sum from n is equal to minus infinity of h bar squared over 2mL g naught delta of x minus nL. So what is this? So this is just some overall constant out front. Normally we call this V naught, but I've made g naught dimensionless by pulling out an h bar squared over 2nL. L is the spacing between delta barriers. So the potential looks like this. Here is 0. We have a delta function barrier. We have a delta function barrier L. We have a delta function barrier at 2L, dot, dot, dot, minus L, dot, dot, dot. So there's our potential. And I want to know what are the energy eigenfunctions for a single particle in this potential. And again, g naught is the dimensionless strength of the potential. So how do we solve this problem? Well, we've done this so many times. What we see is that in between each barrier, the particle is just free, so we know the form of the wave function between each barrier, it's just e to the i kx plus 2 minus e to the minus i kx, where k is defined from the energy h bar squared k squared upon 2n is e. At the barrier, we have to satisfy appropriate matching conditions. So I could put a and b here and c and d and e and f. And under each one of these imposed boundary conditions, and this is going to be an infinite number of coefficients with an infinite number of boundary conditions. And that sounds like it's going to take some time. But we can use something really nice. We already know that the wave function is periodic. So suppose between 0 and L, let's say less than x, less than L the wave function phi takes the form, and we know that phi of eq takes the form e to the i qx, and now some periodic function, which would write as a e to the i kx-- actually, I'm not going to write that. So in between 0 and L, it's a free particle, and it's equal to A e to the i kx plus B e to the minus akx, where k is defined purely from the energy h bar squared k squared upon 2m is the energy. So this is what we mean by k. And in between 2L it will take the similar form, but what we can do is, we can define, so this is sine of x, between L less than x less than 2L phi eq is equal to-- well, I could define it with new constants, but I know what it has to be because whatever the value is here it's the same as the value here up to a shift by L the translation to phase e to the i qL. So this is equal between L and 2L, it's equal to e to the i qL and periodicity condition, the same thing, A plus B, with the same coefficients. Everyone cool with that? So I can now translate, so let's think about what the boundary conditions are going to have to be. At each of the delta functions, the delta function boundary condition is going to say that the slope here minus the slope here is something proportional to the amplitude here. And the slope here is going to be minus something proportional to the amplitude. But the slope here is the same as the slope here up to an overall phase from the translation. And the slope here is the same as the slope here up to a phase coming from the translation. So we can now turn this into instead of this, the condition between the slopes here and here, I can turn that into a condition between the slopes here and here. Wow. This is horrible. Here are my two delta functions. The slope here and here, the boundary condition I put at 0, I can translate this into the slope here up to an overall phase e to the i qL. So I can turn the boundary condition here into a boundary condition between these two guys inside one domain. And so I can turn this into a problem that just involved A and B and that's it. Everyone see that? So let's do it. So questions on the basic strategy at that point? So let's do it. So what we need is we got the general form of the wave function, and I'm going to erase this because we don't need it. Got the general form of the wave function, and then we need to impose the boundary conditions of the delta function. And in particular, let's just impose them at 0 because imposing them at 0 is going to be equivalent to imposing them everywhere else by periodicity if I use the periodicity of the wave function. So the boundary conditions at the delta functions. So the first is that phi at 0 plus is equal to phi at 0 minus, so this is a statement that the wave function is continuous. But phi at 0 minus is equal to-- well, that's this point. But this point is the same as this point except the wave function picks up a phase e to the i qL. This is equal to phi at L minus, right here, times e to the i qL. And we have to be careful because the statement is at the wave function here, is the wave function here time e to the i qL. So the wave function here times the wave function here times e to the i qL minus. And this leads to phi zero plus-- the value right here-- is A plus B because it's just this wave function evaluated at x equal 0, and this guy at L is going to be this evaluated at x is equal to L times e to the minus i qL. So this is equal to A e to the i k minus qL plus B e to the minus i k plus L, So there's my first condition. Second boundary condition is the derivative condition, and the derivative condition is that phi prime at 0 plus minus phi prime at 0 minus is equal to g naught upon L phi at 0. And this gives me that ik A minus B. So that's this guy, and now the derivative of the second guy picked up our extra phase minus ik A e to the ik minus q L minus B e to the minus ik plus qL-- this is all in the notes so I'm not going to worry too much about it-- is equal to, on the right-hand side is g naught upon L A plus B. So now note the following properties of these equations. These equations determine an equation, so from the first, I'll call this 1, and I'll call this 2. So 1 implies that A is equal to B times something. It's just a linear equation. So if we pull the side over, put this over here, A is equal to B times something. Everybody agree with that? Some horrible expression of k and q times. And 2 also gives an expression form A is equal to B times a horrible expression in terms of k and q but a different one. Call this expression 1, I'll call this expression 2. So as usual, so this is two different expressions for A given B and this only makes sense it's only a solution if these two horrible expressions are equal to each other. And so this is what we've done many times before, and if you've set these two expressions equal to each other and do a little bit of trigonometry, you get the following relation. Cosine of qL is equal to cosine of kL plus g naught upon 2kL sine kL. Where what k means is nothing other than the square root of 2ne upon h bar. k is defined from that relation, and what q means is q is the eigenfunction or e to the i qL anyway, is the eigenfunction of the wave function under translation by L. So k is really playing the role of the energy. Everyone cool with that? I just skipped the algebra, but if you do the horrible algebra from setting these two expressions equal to each other, then you get this. Yeah. AUDIENCE: Is q some like undetermined thing as of right now, or do we know-- PROFESSOR: Excellent. What values of q are allowed by this expression? Is exactly the right question to ask here. What values of q's are allowed for our wave function? Is there any condition on what the value q is so far? No. It could have been absolutely any number. For any number q, we can find a eigenfunction to translate by L, e to the i qL as the eigenvalue, any value of q. However, q is equivalent to q plus 2 pi upon L. If you skipped by 2 pi over L, then it's not really a different eigenvalue. It's the same eigenvalue. So q is defined. It's any continuous number between let's say minus pi over L and pi over L for simplicity, make it nice and small. i could have said 0 and 2 pi. It doesn't make any difference, minus pi and pi over L. So q is a free parameter. So there exists a state. Another way to say this is that there exist states and in particular eigenstates for any value of q, for any q, and any e, so for any value of q and e which satisfy this equation corresponds to a state. There exists states for any q E satisfying this equation. Cool? This is just like the finite well. In the case of the finite well, we go through exactly the same analysis the only difference is that we don't have the qL. What happens in the finite well case we impose a boundary condition off in infinity. So the thing has normalized, and we put another boundary condition in infinity, and what we ended up getting is something of the form roughly 1 is equal to something like this. It's actually not exactly that, but here we have kL, and if you combine these two together, you get something kL plus a phase shift over kL and then a 1. If you multiply, you get kL is equal to cosine of kL plus a phase shift, which is exactly the form for the delta potential. Good. So this is similar. However, here we have an extra parameter q, which is free to vary. q could be anything between minus pi over L and pi over L So what does cosine of qL turn into? Well, it's anything qL can take q is 2 pi upon L or pi upon L to minus pi over L. So qL can go between pi and minus pi, so cosine of qL varies between cosine of pi, which is 1 and cosine of minus pi, which is minus 1. So any value of cosine of qL between 1 minus 1 is a valid value of qL On the other hand, if this is equal to 7, is there a q such that this is equal to 7? No. So what we need to do now is, we need to solve this equation, but it's obviously a god awful transcendental equation. So how do we solve horrible transcendental equations? AUDIENCE: Graphically. PROFESSOR: Graphically. So let's solve it graphically. So to solve it graphically, that lets plot the left-hand side and the right-hand side as a function of qL So in this direction, I'm going to plot cosine of qL. That's the left side. And cosine of qL we know is going to vary between plus 1 and minus 1. Because q between q is equal to pi over L, and q is equal to minus pi over L. And in the horizontal direction, I'm going to plot kL. And when kL is 0, remember that the definition of kL is that E, the energy eigenvalue is equal to h bar squared k squared upon 2m. And what we're looking for are points, are any common solution to this equation where there's a value of q and a value of E of k such that these two expressions are equal to each other. Yeah. AUDIENCE: [INAUDIBLE]. PROFESSOR: Sorry? AUDIENCE: [INAUDIBLE]. PROFESSOR: Oh, sorry. Thank you. This is qL is equal to 0 and qL is equal to pi. Thank you. Yes. Are there types of other questions? So kL equals 0 is going to be equal to 0, k equals 0, and this is going to be a function of kL. I'm going to plot it as a function of kL because that's what shows up. But remember that k is nothing but code for E. And so what we're looking for a common solutions of that equation. Now let's first just plot. So if this is 0, let's just plot cosine of kL, the right-hand side cosine of kL plus g naught over 2kL sine kL. And let's do this first for the free particle. Oh, I wish I had colored chalk. For the free particle, free is going to be g naught equals 0. There is a potential and the potential is 0 times the delta function, which is not so much delta function. So for free particle when g is equal to 0, that second term vanishes, and we have cosine of qL is equal to cosine of kL. Kale, if you fry it, it's crispy. So what is that going to be? Well, this has an obvious solution which is that q is equal to k, but the problem is, q is only defined up to 2 pi over L, so mod 2 pi over L. Well, let's see what this looks like in here. So for the free particle what does this tell us? So for the free particle, let me just write this as therefore, q is equal to L or q is equal to k, and therefore E is equal to h bar squared is equal to q squared. So any q or E are allowed such that q squared is equal E times 2 upon h bar squared. So that's just the usual condition. It's the free particle has energy h bar squared k squared upon 2n, and we just chose to call it k equals q. So what we're doing is instead of organizing things as momentum eigenstates of the free particle or organizing the free particle energy eigenstates in terms of transient by L eigenstates. Perfectly valid. It commutes with the energy operator. What does this look like? Well, cosine of kL is cosine of qL, when k is equal to 0, q is equal to 0. So we get this. And the points, when the points where qL is equal to pi or minus 1, these are the points where we get the qL is equal to n pi or equivalently plus or minus kL is equal to n pi plus or minus n pi. So this is n is 1, n is 2, n is 3. Everyone cool with that? And these guys are 2 pi times n or 2n plus 1 2n pi. And I really want this to be odd, so n odd. And this is n pi kL so n pi for n even. Everyone cool with that? So that's what it would look like for a free particle. What happens if we don't have a free particle? What happens if we have the interacting potential? So now let's have g naught equal to 1 or 0. And again, here is plus 1. Here is minus 1. And we're plotting as a function of kL. And now it changes, so let g naught be a small, positive number. If g naught is a small, positive number, then near k goes to 0. Sine goes like it's argument, and so that means kL over kL we get cosine of 0, which is 1, plus g naught over 2. So at 0, the value of the right-hand side is greater than 1. Is there an energy eigenvalue with energy equal to 0? No. We have to wait until cosine gets sufficiently small and sine is increasing, so what does this curve do? It looks like this. It's again periodic with the appropriate period, but the amplitude of the deviation from the free particle is falling off as we go to higher and higher values. So we've taken this point and pushed it out. So what does this tell us? Well, remember that there is an energy eigenfunction with a given q and remember this is cosine of qL, there is an energy eigenvalue. There exists an energy eigenstate with an e and TL eigenstate for any E and q such that there exists a solution. And so what does that mean? Well, is there an energy eigenvalue with this value of E or this value of k? No. Right? Because here's the right-hand side and the left-hand side is any value in here. There's a q for which any left-hand side is any value between 1 and minus 1. So there's no value of q you can pick so the left-hand side is equal to the right-hand side for the value of k. And similarly here, here, here, here, here, specifically for this value of k or this value of the energy, E minimum, there's one value of q. This one. q is equal to 0 where there is an energy eigenstate. So there's a minimum energy in the system. What about for this value of energy? Yeah. There's exactly one solution there, and actually if you're careful, it's 2 because it's the cosine of qL. q could have been positive or negative. So we have a solution here with this value of energy and with this value of cosine qL. Any q that gives us this value of cosine qL is going to work, and that's one of two values. And similarly for each of these guys, there's a state for every value of energy in this region until we get here, and when we get here, this is E maximum. If we go to any higher energy, any higher k, then there's no solution for any value of q and any value of E. So what we get are these bands, continuous bands for any energy separated by gaps and then again continuous bands for any energy and bands for any energy. So all of these shaded in regions, any energy in this region corresponds to a state and similarly here. So here's the upshot of all this. We're going to study the details of this in detail next time. When we have a periodic potential, every energy eigenfunction is extended through the entire material. Every energy eigenfunction is extended across the entire lattice. None of them are localized. And that's like a free particle. However, not every energy is an allowed energy. Only some energies are allowed. Some energies do not correspond to allowed energy eigenfunctions eigenvalues, and some energies do. And they come in continuous bands of allowed energies and continuous gaps of disallowed energies. And it's going to turn out to be exactly this structure of bands and gaps, or band gaps, that is going to give us the structure of conductivity in metals and explain to us why we don't have conductivity in plastic. We'll pick up on that next time.
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_5_Operators_and_the_Schrödinger_Equation.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Today we're going to just continue what Allan Adams was doing. He's away on a trip in Europe, so he asked me to give their lecture today. And we'll just follow what he told me to do. He was sort of sad to give me this lecture, because it's one of the most interesting ones. This is the one where you get to see the Schrodinger equation. But anyway, had to be a way, so we'll do it here. He also told me to take off my shoes, but I won't do that. So let's go ahead. So what do we have so far. It's going to be a list of items that we have. And what have we learned? We know that particles, or systems, are governed by wave functions, described by wave functions. Wave functions. And those are psi of x at this moment. And these are complex numbers, belong to the complex numbers. And they're continuous and normalizable. Their derivatives need not be continuous, but the wave function has to be continuous. It should be also normalizable, and those are two important properties of it, continuous and normalizable. Second there's a probability associated with this thing. And this probability is described by this special p of x. And given that x is a continuous variable, you can say well, what is the probability that the particle is just at this point would be zero in general. You have to ask, typically, what's the probability that I find it in a range. It's any continuous variable that postulated to be given by the square of the wave function and the x. Third there's superposition of allowed states. So particles can be in superpositions. So a wave function that depends of x may quickly or generally be given as a superposition of two wave functions. And this is seen in many ways. You have these boxes. A particle was a superposition. A top, and side, and hard, and soft. And photon superposition of linearly polarized here or that way. That's always possible to explain. Now in addition to this, to motivate the next one we talk about relations between operators. There was an abstraction going on in this course in the previous lectures in which the idea of the momentum of a particle became that of an operator on the wave function. So as an aside, operators momentum, we have the momentum of a particle has been associated with an operator h bar over i [INAUDIBLE] x. Now two things here. My taste, I like to put h bar over i. Allan likes to put I h bar. That's exactly the same thing, but no minus sign is a good thing in my opinion. So I avoid them when possible. Now there's is d dx here is a partial derivative, and there seems to be no need for partial derivatives here. Why partial derivatives? I only see functions of x. Anybody? Why? Yes. AUDIENCE: The complete wave function also depends on time, doesn't it? PROFESSOR: Complete wave function depends on time as well. Yes, exactly. That's where we're going to get to today. This is the description of this particle at some instant. So within [INAUDIBLE] time, the time here is implicit. It could be at some time, now, later, some time, but that's all we know. So in physics you're allowed to ask the question, well, if I know the wave function of this time and that seems to be what I need to describe the physics, what will it be later? And later time will come in, so therefore we'll stick to this partial d dx in here. All right, so how do we use that? We think of this operator as acting on the wave functions to give you roughly the momentum of the particle. And we've made it in such a way that when we talk about the expectation value of the momentum, the expected value of the momentum of the particle, we compute the following quantity. We compute the integral from minus infinity to infinity dx. We put the conjugate of the wave function here. And we put the operator, h bar over i d dx acting on the wave function here. And that's supposed to be sort of like saying that this evaluates the momentum of the wave function. Why is that so? It is because if you're trying to say, oh any wave function, a general wave function need not be state in which the particle has definite momentum. So I kind of just say the momentum of a particle is the value of this operator on the function. Because if I act with this operator on the function, on the wave function, it might not return me, the wave function. In fact in general, we've seen that, for special wave functions, wave functions of the form psi, a number, e to the ikx. Then p, let's think p hat as the operator on psi. Would be h bar over i d dx on psi. Gives you what? Well, this h over I, the h remains there. When you differentiate with respect to x, the ik goes down, and the i cancels, so you get hk times the same wave function. And for this, we think that this wave function is a wave function with momentum hk. Because if you act with the operator p on that wave function, it returns for you hk. So we think of this as has p equal hk, h bar k. So the thing that we want to do now is to make this a little more general. This is just talking about momentum, but in quantum mechanics we're going to have all kinds of operators. So we need to be more general. So this is going to be, as Allan calls it, a math interlude. Based on the following question, what this is an operator? And then the physics question, what do measurable things have to do with our operators? So about operators aren't measurable things, quantities. Now your view of operators is going to evolve. It's going to evolve in this course, It's going to evolve in 805. It probably will continue to evolve. So we need to think of what operators are. And there is a simple way of thinking of operators that is going to be the basis of much of the intuition. It's a mathematical way of thinking of operators, and what we'll sometimes use it as a crutch. And the idea is that basically operators are things that act on objects and scramble them. So whenever you have an operator, you really have to figure out what are the objects you're talking about. And then the operator is some instruction on how to scramble it, scramble this object. So for example, you have an operator. You must see what it does to any object here. The operator acts on the object and gives you another object. So in a picture you have all kinds of objects sets, a set of objects. And the operators are things. You can have a list of operators. And they come here and move those objects, scramble them, do something to them. And we should distinguish them, because the objects are not operators, and the operators are not the objects. So what is the simplest example in mathematics of this thing is vectors and matrices. Simplest example. Objects are vectors. Operators are matrices. And how does that work? Well, you have a two by two, the case where you have a set in which you have vectors with two components. So example, a vector that has two components v1 v2. And the matrices, this is the object. And the operator is a matrix. a 11, a 12, a 21, a 22 as an operator. An m on a vector is a vector. If you are with a matrix on a vector, this 2 by 2 matrix on this common vector, you get another vector. So that's this simplest example of operators acting on objects. In our case, we're going to talk about a more-- we're going to have to begin, in quantum mechanics we're required to begin with a more sophisticated one in which the objects are going to be-- objects are going to be functions. In fact, I will write complex functions of x. So let's see it, the list of operators. And what do the operators do? The operators act on the functions. So what is an operator? It's a rule on how to take any function, and you must give a rule on how to obtain from that function another function. So let's start with an examples. It's probably the easiest thing to do. So an operator acts on the functions. So an operator for any function f of x will give you another function. Function of x. And here's operator o, we put a hat sometimes for operators. So the simplest operator, the operator one. We always, mathematicians, love to begin with trivial examples. Illustrate anything almost, and just kind of confuse you many times. But actually it's good to get them of the way. So what is the operator one? One possibility it takes any function and gives you the number one. Do you think that's it? Who thinks that's it? Nobody? Very good. that definitely is not a good thing to do to give you the number one. So this operator does the following. I will write it like that. The operator one takes f of x and gives you what? AUDIENCE: f of x. PROFESSOR: f of x. Correct. Good. So it's a very simple operator, but it's an operator. It's like what matrix? The identity matrix. Very good. There could be a zero operator that gives you nothing and would be the zero matrix. So let's write the more interesting operator. The operator would d dx. That's interesting. The derivative can be thought of as an operator because if you start with f of x, it gives you another function, d dx of f of x. And that's a rule to get from one function to another. Therefore that's an operator, qualifies as an operator. Another operator that typically can confuse you is the operator x. x an operator? What does that mean? Well, you just have to define it. At this moment, it could mean many things. But you will see that [INAUDIBLE] is the only thing that probably makes some sense. So what is the operator x? Well, it's the operator that, given f of x, gives you the function x times f of x. That's a reasonable thing to do. It's multiplying by x. It changes the function. You could define the operator x squared that multiplies by x squared. And the only reasonable thing is to multiply it by. You could divide by it, and you may need to divide by it as well. And you could define the operator 1 over x gives you the function times 1 over x. We will need that sometime, but not now. Let's see another set of operators where we give a name. It doesn't have a name because it's not all that's useful in fact. But it's good to illustrate things. They operator s sub q for squared. S q for the first two letters of the word square. That takes f of x into f of x squared. That's another function. You could define more functions like that. The operator p 42. That's another silly operator. Well certainly a lot more silly than this one. That's not too bad. But the p 42 takes f of x And gives you the number 42 times the constant function. So that's a function of x. It's trivial function of x. Now enough examples. So you get the idea. Operators act on functions and give you functions. And we just need to define them, and then we know what we're talking about. Yes? AUDIENCE: Is the Dirac delta and operator? PROFESSOR: The Dirac delta, well, you can think of it as an operator. So it all depends how you define things. So how could I do find the Dirac delta function to be an operator? So delta of x minus a. Can I call it the operator delta hat of a? Well, I would have to tell you what it does in functions. And probably I would say delta had on a on a function of x is equal to delta of x minus a times the function of x. And I'd say it's an operator. Now the question is, really, is it a useful operator? And sometimes it will be useful in fact. This is a more general case of another operator that maybe I could define. o sub h of x is the operator that takes f of x into h of x times f of x. So that would be another operator. Now there are operators that are particularly nice, and there are the so-called linear operators. So what is a linear operator? It's one that respects superposition. So linear operator respects superposition. So o hat is linear. o hat is a linear operator. If o hat on a times f of x plus b times g of x does what you would imagine it should do, it that's on the first, and then it acts on the second. Acting on the first, the number goes out and doesn't do anything, say, on the number. It's linear. It's part of that idea. And it gives you o on f of x plus b times o on g of x. So your operator may be linear, or it may not be linear. And we have to just guess them. And you would imagine that we can decide that, of the list of operators that we have, let's see, one d dx-- how much? Which one? Sq, p 42, and o sub h of x. Let's see. Let's vote on each one whether it's linear or not. A shouting match whether I hear a stronger yes or no. OK? One is a linear operator, yes? AUDIENCE: Yes. PROFESSOR: No? Yes. All right. d dx linear. Yes? AUDIENCE: Yes. PROFESSOR: Good. That's strong enough. Don't need to hear the other one. x hat. Linear operator? Yes or no? AUDIENCE: Yes. PROFESSOR: Yes, good. Squaring, linear operator? AUDIENCE: No. PROFESSOR: No. No way it could be a linear operator. It just doesn't happen. If you have sq on f plus g, it would be f plus g squared, which is f squared plus g squared plus, unfortunately too, fg. And this thing ruins it, because this is sq of f plus sq of g. It's even worse than that. You put sq on af, by linearity it should be a times the operator. But when you square a times f, you get a squared f squared. So you don't even need two functions to see that it's not real. So definitely no. How about p 42? AUDIENCE: No. PROFESSOR: No, of course not. Because if you add two functions, it still gives you 42. It doesn't get you 84, so no. How about oh of x? AUDIENCE: Yes. PROFESSOR: Yes, it does that. If you act with this operator on a sum of functions distributive law, it works. So this is linear. Good. Linear operators are important to us because we have some superposition of allowed states. So if this is a state and this is a state, this is also good state. So if we want superposition to work well with our theory, we want linear operator. So that's good. So we have those linear operators. And now operators have another thing that makes them something special. It is the idea that there's simpler object they can act on. We don't assume you've studied linear algebra in this course, so whatever I'm going to say, take it as motivation to learn some linear algebra at some stage. You will be a little more linear algebra in 805. But at this moment, just basic idea. So whenever you have matrices, one thing that people do is to see if there are special vectors. Any arbitrary vector, when you act within the matrix, is going to just jump and go somewhere else, point in another direction. But there are some special vectors that do act-- if you have a given matrix m, there are some funny vectors sometimes that acted by n remain the same direction. They may grow a little or become smaller, but they remain the same direction. These are called eigenvectors. And that constant of proportionality, proportional to the action of the operator on the vector, is called the eigenvalue. So these things have generalizations for our operators. So operators can have special functions, eigenfunctions. What are these eigenfunctions? So let's consider that operator a hat. It's some operator. I don't know which one of these, be we're going to talk about linear operator. So linear operators have eigenfunctions. A hat. So a hat. There may be functions that, when you act with the operator, you sort of get the same function up to possibly a constant a. So you may get a times the function. And that's a pretty unusual function, because, on most functions, any given operator is going to make a mess out of the function. But sometimes it does that. So to label them better with respect to operator, I would put a subscript a, which means that there's some special function that has a parameter a, for which this operator gives you a times that special function. And that special function is called-- this is the eigenfunction and this is the eigenvalue. And that the eigenvalue is a number. It's a complex number c there over there. So these are special things. They don't necessarily happen all the time to exist, but sometimes they do, and then they're pretty useful. And we have one example of them that is quite nice. For the operator a equal p, we have eigenfunctions e to the ikx with eigenvalue hk. So this is the connection to this whole thing. We wanted to make clear for you that what you saw here, that this operator acting on this function gives you something times this function is a general fact about the operators. Operators have eigenfunctions. So eigenfunction e of x with eigenvalue hk, because indeed p hat on this e to ikx, as you see this h bar k times e to the ikx. So here you have p hat is the a. This is the function labeled a would be like k. Here is something like k again. And here is this thing. But the main thing operator on the function number times the function is an eigenfunction. Yes? AUDIENCE: For a given operator, is the eigenvalue [INAUDIBLE]?? PROFESSOR: Well, for a given operator good question. a is a list of values. So there may be many, many, many eigenfunctions. Many cases infinitely many eigenfunctions. In fact, here I can put for k any number I want, and I get a different function. So a belongs to c and may take many, or even infinite, values. If you remember for nice matrices, n by n matrix may be a nice n by n matrix because n eigenvectors and eigenvalues are sometimes hard to generate, sometimes eigenvalues have the same numbers and things like that. OK. Linearity is this some of two eigenvectors and eigenvector. Yes? No? AUDIENCE: No. PROFESSOR: No, no. Correct. That's not necessarily true. If you have two eigenvectors, they have different eigenvalues. So things don't work out well necessarily. So an eigenvector plus another eigenvector is not an eigenvector. So you have here, for example, A f1 equals a1f1. And A f2 equal a2f2, then a on f1 plus f2 would be a1f1 plus a2f2, and that's not equal to something times f1 plus f2. It would have to be something times f1 plus f2 to be an eigenvector. So this is not necessarily an eigenvector. And it doesn't help to put a constant in front of here. Nothing helps. There's no way to construct an eigenvector from two eigenvectors by adding or subtracting. The size of the eigenvector is not fixed either. If f is an eigenvector, then 3 times f is also an eigenvector. And we call it the same eigenvector. Nobody would call it a different eigenvector. It's really the same. OK, so how does that relate to physics? Well, we've seen it here already. that one operator that we've learned to work with is the momentum operator. It has those eigenfunctions. So back to physics. We have other operators. Therefore we have the P operator. That's good. We have the X operator. That's nice. It's multiplication by x. And why do we use it? Because sometimes you have the energy operator. And what is the energy operator? The energy operator is just the energy that you've always known, but think of it as an operator. So how do we do that? Well, what is the energy of a particle we've written p squared over 2m plus v of x. Well, that was the energy of a particle, the momentum squared over 2m plus v of x. So the energy operator is hat here, hat there. And now it becomes an interesting object. This energy operator will be called E hat. It acts and functions. It is not a number. The energy is a number, but the energy operator is not a number. Far from a number in fact. The energy operator is minus h squared over 2m. d second the x squared. Why that? Well, because p was h bar over i d dx as an operator. So this sort of arrow here, it sort of the introduction. But after a while you just say P hat is h bar over a i d dx. End of story. It's not like double arrow. It's just what it is. That operator. That's what we call it. So when we square it, the i squares, the minus h squares, and d dx and d dx applied twice is the second derivative. And here we get v of X hat, which is your good potential, whatever potential you're interested in, in which, whenever you see an x, you put an X hat. And now this is an operator. So you see this is not a number, not the function. It's just an operator. The operator has this sort of operator v of x. Now what is this v of x here as an operator? This is v of x as an operator is just multiplication by the function v of x. You see, you have here that the operator x is f of x like that. I could have written the operator X hat to the n. What would it be? Well, if I add to the function, this is a lot of X hats acting on the function. Well, let the first one out. You let x times f of x. The second, that's another x, another x. So this is just x to the n times f of x. So lots of X hats. X hats To the 100th on a function is just X to the 100th times a function. So v of x on a function is just v of the number x on a function. It's just like this operator, the O in which you multiply by a function. So please I hope this is completely clear what this means as an operator. You take the wave function, take two derivatives, and add the product of the wave function times v of plane x. So I'll write it here maybe. So important. E hat and psi of x is therefore minus h squared over 2m the [INAUDIBLE] the x squared of psi of x plus just plain v of x times psi of x. That's what it does. That's an operator. And for these operators in general. Math interlude, is it over? Not quite. Wow. No, yes. Allan said at this moment it's over, when you introduce it here. I'll say something more here, but it's going to be over now. Our three continues here then with four. Four, to each observable we have an associated operator. So for momentum, we have the operator P hat. And for position we have the operator X hat. And for energy we have the operator E hat. And these are examples of operators. Example operator A hat could be any of those. And there may be more observables depending on the system you're working. If you have particles in a line, there's not too many more observables at this moment. If you have a particle in general, you can have angular momentum. That's an interesting observable. It can be others. So for any of those, our definition is just like with it for momentum. The expectation value of the operator is computed by doing what you did for momentum. You act with the operator on the wave function here and multiply by the compass conjugate function. And integrate just like you did for momentum. This is going to be the value that you expect. After many trials on this wave function, you would expect the measured value of this exhibit a distribution which its expectation value, the mean, is given by this. Now there are other definitions. One definition that already has been mentioned is that the uncertainty of the operator on the state psi, the uncertainty, is computed by taking the square root of the expectation value of A squared minus the expectation value of A, as a number, squared. Now the expectation value of A squared, just simply here instead of A you put A squared, so you've got A squared here. That unless the function is very special, it's very different whole is bigger than the expectation value of A squared. So this is a number, and it's called the uncertainty. And that's the uncertainty of the uncertainty principle. So for operators, we need to have another observation that comes from matrices that is going to be crucial for us is the observation that operators don't necessarily commute. And we'll do the most important example of that. So we'll try to see in the operators associated with momentum and position commute. And what we mean by commute or don't communicate? Whether the order of multiplication matters. Now we talked about matrices at the beginning, and we said matrices act on vectors to give you vectors. So do they commute? Well, matrices don't commute. The order matters for matrices multiplication. So these operators we're inventing here for physics, the order does matter as well. So commutation. So let's try to see if we compute the operator p and x hat. Is it equal to the operator x hat times p? This is a very good question. These are two operators that we've defined. And we just want to know if the order matters or if it doesn't matter. So how can I check it? I cannot just check it like this, because operators are only clear what they do is when they act on functions. So the only thing that I can do is test if this thing acting on functions give the same. So I'm going act with this on the function f of x. And I'm going to have act with this on the function f of x. Now what do I mean by acting with p times x hat on the function f of x. This is by definition you act first with the operator that is next to the f and then with the other. So this is p hat on the function x hat times f of x. So here I would have, this is x hat on the function p hat f of x. See, if you have a series of matrices, m1, m2, m3 acting on a vector, what do you mean? Act with this on the vector, then with this on the vector, then with this. That's multiplication. So we're doing that. So let's evaluate. What is x operator on f of x? This is p hat on x times f of x. That's what the x operator in the function is. Here, what this x hat? And now I have this, so I have here h over i d dx of f. Let's go one more step here. This is h over i d ddx now of this function, x f of x. And here I have just the x function times this function. So h over i x df dx. Well, are these the same? No, because this d dx here is not only acting on f like here. It's acting on the x. So this gives you two terms. One extra term on the d dx acts on the x. And then one term that is equal to this. So you don't get the same. So you get from here h over i f of x, when you [INAUDIBLE] the x plus h over i x the df dx. So you don't get the same. So when I subtract them, so when I do xp minus px acting on the function f of x, what do I get? Well, I put them in this order, x before the p. Doesn't matter which one you take, but many people like this. Well, these terms cancel and I get minus this thing. So I get minus h over i f of x, or i h bar f of x. Wow. You got something very strange. The x times p minus p times x gives you a number-- an imaginary number, even worse-- times f of x. So from this we write the following. We say look, operators are defined by the action and function, but for any function, the only effect of xp minus px, which we call the commutator of x with p. This definition, the bracket of two things, of A B. Is defined to be A B minus B A. It's called the commutator. x p is an operator acting on f of x, gives you i h bar times f of x. So our kind of silly operator that does nothing has appeared here. Because I could now say that x hat with p is equal to i h bar times the unit operator. Apart from the Schrodinger equation, this is probably the most important equation in quantum mechanics. It's the fact that x and b are incompatible operators as you will see later. They don't commute. Their order matters. What's going to mean is that when you measure one, you have difficulties measuring the other. They interfere. They cannot be measured simultaneously. All those things are encapsulated in a very lovely mathematical formula, which says that this is the way these operators work. Any questions? Yes? AUDIENCE: When x-- the commutator of x and p is itself an operator, right? PROFESSOR: RIght. AUDIENCE: So is that what we're saying? When we had operators before, we can't simply just cancel the f of x. I mean we're not really canceling it, but it just because I h bar is the only eigenvalue of the operator? PROFESSOR: Well, basically what we've shown by this calculation is that this operator, this combination is really the same as the identity operator. That's all we've shown, that some particular combination is the identity operator. Now this is very deep, this equation. In fact, that's the way Heisenberg invented quantum mechanics. He called it the matrix mechanics, because he knew that operators were related to matrices. It's a beautiful story how he came up with this idea. It's very different from what we're doing today that we're going to follow Schrodinger today. But basically his analysis led very quickly to this idea. And this is deep. Why is it deep? Depends who you ask. If you ask a mathematician, they would probably tell you this equation is not deep. This is scary equation. And why is it scary? Because whenever a mathematician see operators, they want to write matrices. So the mathematician, you show him this equation, will say OK, Let me try to figure out which matrices you're talking about. And this mathematician will start doing calculations with two by two matrices, and will say, no, I can't find two by two matrices that behave like these operators. I can't find three by three matrices either. And four by four. And five by five. And finds no matrix really can do that, except if the matrix is infinite dimensional. Infinite by infinite matrices. So that's why it's very hard for a mathematician. This is the beginning of quantum mechanics. This looks like a trivial equation, and mathematicians get scared by it. You show them for physicists there will be angular momentum. The operators are like this, and there's complicated into the [INAUDIBLE]. The three components of angular momentum have this commutation relation. And h bar here as well. Complicated. Three operators. They mix with each other. Show it to a mathematician, he starts laughing at you. He says that best, the simplest case, this is easy. This is complicated. It's very strange. But the reason this is easier, the mathematician goes and, after five minutes, comes to you with three by three matrices that satisfies this relation. And here there weren't. And four by four that satisfy, and five by five, and two by two, and all of them. We can calculate all of them for you. But this one it's infinite dimensional matrices, and it's very surprising, very interesting, and very deep. All right, so we move on a little bit more to the other observable. So after this, we have more general observable. So let's talk a little about them. That's another postulate of quantum mechanics that continues with this one, postulate number five. So once you measure, upon measuring an observable A associated with the operator A hat, two things happen. You measure this quantity that could be momentum, could be energy, could be position, you name it. The measured value must be a number. It's one of the eigenvalues of A hat. So actually those eigenvalues, remember the definition of the eigenvalues. It's there. I said many, but whenever you measure, the only possibilities that you get this number. So you measure the momentum, you must get this hk, for example. So observables, we have an associated operator, and the measured values are the eigenvalues. Now these eigenvalues, in order to be observable, they should be a real numbers. And we said oh, they can be complex. Well, we will limit the kind of observables to things that have real eigenvalues, and these are going to be called later on Hermitian operators. At this moment, the notes, they don't mention them. You're going to get them confused. So anyway, special operators that have real eigenvalues. So we mentioned here they will have to be a real. Have to be real. And then the second one, which is an even stranger thing that happens is something you've already seen in examples. After you measure, the whole wave function goes into the state which is the eigenfunction of the operator. So after measurement system collapses into psi a. The measure value is one over the eigenvalues a of A. And the system collapses into psi a. So psi a is such that A hat psi a is a psi a. So this is the eigenvector with eigenvalue a that you measured. So after you measure the momentum and you found that its h bar k, the wave function is the wave function of momentum h bar k. If at the beginning, it was a superposition of many, as Fourier told you, then after measuring, if you get one component of momentum, that's all that is left of the wave function. It collapses. This collapse is a very strange thing, and is something about quantum mechanics that people are a little uncomfortable with, and try to understand better, but surprisingly nobody has understood it better after 60 years of thinking about it. And it works very well. It's a very strange thing. Because for example, if you have a wave function that says your particle can be anywhere, after you measure it where it is, the whole wave function becomes a delta function at the position that you measure. So everything on the wave function, when you do a measurement, basically collapses as we'll see. Now for example, let's do an example. Position. So you have a wave function psi of x. You find measure and find the particle at x0. Measure and you find the particle at x0. So measure what? I should be clear. Measure position. So we said two things. The measured value is one of the eigenvalues of a, and after measurement, the system collapses to eigenfunctions. Now here we really need a little of your intuition. Our position eigenstate is a particle a localized at one place. What is the best function associated to a position eigenstate? It's a delta function. The function that says it's at some point and nowhere else. So eigenfunctions delta of x minus x0, it's a function as a function of x. It peaks at x0, and it's 0 everywhere else. And this is, when you find a particle at x0, this is the wave function. The wave function must be proportional to this quantity. Now you can't normalize this wave function. It's a small complication, but we shouldn't worry about it too much. Basically you really can't localize a particle perfectly, so that's the little problem with this wave function. You've studied how you can represent delta functions as limits and probably intuitively those limits are the best things. But this is the wave function, so after you measure the system, you go into an eigenstate of the operator. Is this an eigenstate of the x operator? What a strange question. But it is. Look, if you put the x operator on delta of x minus x zero, what is it supposed to do? It's supposed to multiply by x. So it's x times delta of x minus x zero. If you had a little experience with delta functions, you'd know that this function is 0 everywhere, except when x is equal to x0, so this x can be turned into x0. It just never at any other place it contributes. This x really can be turned into x0 times delta of x minus x0. Because delta functions are really used to do integrals. And if you do the integral of this function, you will see that it gives you the same value as the integral of this function. So there you have it. The operator acting on the eigenfunction is a number times this. So these are indeed eigenfunctions of the x operator. And what you measured was an eigenvalue of the x operator. Eigenvalue of x. And this is an eigenfunction of x. So we can do the same with the momentum. Eigenvalues and eigenfunctions, we've seen them more properly. Now we'll go to the sixth postulate, the last postulate that we'll want to talk about is the one of general operators, and general eigenfunctions, and what happens with them. So let's take now our operator A and its functions that can be found. So six, given an observable A hat and its eigenfunctions phi a of x. So an a runs over many values, many values. OK so let's consider this case. Now eigenfunctions of an operator are very interesting objects. You see the eigenfunctions of momentum were of this form. And they allow you to expand via the Fourier any wave function as super positions of these things. Fourier told you you can't expand any function in eigenfunctions of the momentum, or the result is more general. For observables in general you can expand functions, arbitrary functions, in terms of the eigenfunctions. Now for that, remember, an eigenfunction is not determined up to scale. You change multiplied by three, it's an eigenfunction still. So people like to normalize them nicely, the eigenfunctions. You construct them like this, and you normalize them nicely. So how do you normalize them? Normalize them by saying that the integral over x of psi a star of x psi b of x is going to be what? OK, basically what you want is that these eigenfunctions be basically orthogonal. Each one orthogonal to the next. So you want this to be 0 unless these two different eigenfunctions are different. And when they are the same, you want them to be just like wave functions, that their total integral of psi squared is equal to 1. So what you put here is delta ab. Now this is something that you can always do with eigenfunctions. It's proven in mathematics books. It's not all that simple to prove, but this can always be done. And when we need examples, we'll do it ourselves. So given an operator that you have its eigenfunctions like that, two things happen. One can expand psi as psi of x. Any arbitrary wave function as the sum. Or sometimes an integral. So some people like to write this and put them an integral on top of that. You can write it whichever way you want. It doesn't matter. Of coefficients times the eigenfunctions. So just like any wave could be a written, a Fourier coefficient [INAUDIBLE] Fourier function. Any state can be written a superposition of these things. So that's one. And two, the probability of measuring A hat and getting a. So a one of the particular values that you can get. That probability is given by the square of this coefficient. Ca squared. So this is P, the probability, to measure in psi and to get a. I think, actually, let's put an a0 there. So here we go. So it's a very interesting thing. Basically you expand the wave function in terms of these eigenfunctions, and these coefficients give you the probabilities of measuring these numbers. So we can illustrate that again with the delta functions, and we'll do it quick, because we get to the punchline of this lecture with the Schrodinger equation. So what do we have? Well, let me think of the operator X example. Operator X, the eigenfunctions are delta of x minus x0 for all x0. These are the eigenfunctions. And we'll write sort of a trivial equation, but it sort of illustrates what's going on. Psi of x as a superposition over an integral over x0 of delta of x minus x0. Psi of x0. Delta of x minus x zero. OK, first let's check that this make sense. Here we're integrating over x0. x0 is the variable. This thing shoots and fires whenever x is equal to x-- whenever x0 is equal to x. Therefore the whole result of that integral is psi of x is a little funny how it's written, because you have x minus 0, which is the same as delta of x0 minus x is just the same thing. And you integrate over x0, and you get just psi of x. But what have you achieved here? You've achieved the analogue of this equation in which these are the psi a. These are the coefficients Ca. And this is the sum, this integral. So there you go. Any wave function can be written as the sum of coefficients times the eigenfunctions of the operator. And what is the probability to find the particle at x0? Well, it's from here. The coefficients, the a squared. That's exactly what we had before. So this is getting basically what we want. So this brings us to the final stage of this lecture in which we have to get the time evolution finally. So how does it happen? Well it happens in a very interesting way. So maybe I'll call it seven, Schrodinger equation. So as with any fundamental equation in physics, there's experimental evidence and suddenly, however, you have to do a conceptual leap. Experimental evidence doesn't tell you the equation. It suggests the equation. And it tells you probably what you're doing is right. So what we're going to do now is collect some of the evidence we had and look at an equation, and then just have a flash of inspiration, change something very little, and suddenly that's the Schrodinger equation. Allan told me, in fact, still sometimes are disappointed that we don't derive the Schrodinger equation. Now let's derive it mathematically. But you also don't derive Newton's equations. F equal ma. You have an inspiration, you get it. Newton got it. And then you use it and you see it makes sense. It has to be a sensible equation, and you can test very quickly whether your equation is sensible. But you can't quite derive it. In 805 we come a little closer to deriving the Schrodinger equation, which we say unitary time evolution, something that I haven't explained what it is, implies the Schrodinger equation. And that's a mathematical fact. And you can begin unitary time evolution, define it, and you derive the Schrodinger equation. But that's just saying that you've substituted the Schrodinger equation by saying there is unitary time evolution. The Schrodinger question really comes from something a little deeper than that. Experimentally it comes from something else. So how does it come? Well, you've studied some of the history of this subject, and you've seen that Planck postulated quantized energies in multiples of h bar omega. And then came Einstein and said look, in fact, the energy of a photon is h bar omega. And the momentum of the photon was h bar k. So all these people, starting with Planck and then Einstein, understood what the photon is. The quantum of photons for photons, you have E is equal h bar omega, and the momentum is equal to h bar k. I write them as a vector, because the momentum is a vector, but we also write them in this because p equal h bar k, assuming you move just in one direction. And that's the way it's been written. So this is the result of much work beginning by Planck, Einstein, and Compton. So you may recall Einstein said in 1905 for that there seemed to be this quantum of light that carry energy h omega. Planck didn't quite like that. And people were not all that convinced. Experiments were done by Millikan in 1915, and people were still not quite convinced. And then came Compton and did Compton scattering. And then people said, yeah, they seem to be particles. No way out of that. And they satisfy such a relation. Now there was something about this that was quite nice, that these photons are associated with waves, and that was not too surprising, because people understood that electromagnetic waves are waves that correspond to photons. So you can also see that this says that E p is equal to h bar omega k as an equation between vectors. You see the E is the first, and the p is the second equation. And this is actually a relativistic equation. It's a wonderful relativistic equation, because energy and momentum form what is called a relativity of four vector. It's the four vector-- this is a little aside on relativity-- four vector. The index mew runs from 0, 1, 2, 3. Just like the x mews, which are t and x. Run from x0, which is t, x1, which is x, x2 which is y, three-- these are four vectors. And this is a four vector. This is a four vector. This all seemed quite pretty, and this was associated to photons. But then came De Broglie. And De Broglie had a very daring idea that even though this was written for photons, it was true for particles as well, for any particle. De Broglie says good for particles, all particles. And these particles are really waves. So what if he write-- he wrote psi of x and t is equal a wave associated to a matter particle. And it would be an e to the i kx minus omega t. That's a wave. And you know that this wave has momentum p equal h bar k. If k is positive, look at this sign. If this sign is like that, then k is positive. This is a wave that is moving to the right. So p being hk. If k is positive, p is positive, is moving to the right, this is a wave moving to the right, and has this momentum. So it should also have an energy. Compton said that this is relativistic because this all comes from photons. So if the momentum is given by that, and the energy must also be given by a similar relation. In fact, he mostly said, look, you must have the energy being equal to h bar omega. The momentum, therefore, would be equal to h bar k. And I will sometimes erase these things. So what happens with this thing? Well, momentum equal to hk. We've already understood this as momentum operator being h bar over i d dx. So this fact that these two must go together and be true for particles was De Broglie's insight, and the connection to relativity. Now here we have this. So now we just have to try to figure out what could we do for the energy. Could we have an energy operator? What would the energy operator have to do? Well, if the energy operator is supposed to give us h bar omega, the only thing it could be is that the energy is i h bar d dt. Why? Because you go again at the wave function. And you think i h bar d dt, and what do you get? i h bar d dt on the wave function is equal to i h bar. You take the d dt, you get minus i omega times the whole wave function. So this is equal h bar omega times the wave function, times the wave function like that. So here it is. This is the operator that realizes the energy, just like this is the operator that realizes the momentum. You could say these are the main relations that we have. So if you have this wave function, it corresponds to a particle with momentum hk and energy h omega. So now we write this. So for this psi that we have here, h bar over i d dx of psi of x and t is equal the value of the momentum times psi of x and t. That is something we've seen. But then there's a second one. For this psi, we also that i h bar d dt of psi of x and t is equal to the energy of that particle times x and t, because the energy of that particle is h bar omega. And look, this is familiar. And here the t plays no role, but here the t plays a role. And this is prescribing you how a wave function of energy E evolves in time. So you're almost there. You have something very deep in here. It's telling you if you know the wave function and it has energy E, this is how it looks later. You can take this derivative and solve this differential equation. Now this differential equation is kind of trivial because E is a number here. But if you know that you have a particle with energy E, that's how it evolves in time. So came Schrodinger and looked at this equation. Psi of x and t equal E psi of x and t. This is true for any particle that has energy E. How can I make out of this a full equation? Because maybe I don't know what is the energy E. The energy E might be anything in general. What can I do? Very simple. One single replacement in that equation. Done. It's over. That's the Schrodinger equation. It's the energy operator that we introduced before. Inspiration. Change E to E hat. This is the Schrodinger equation. Now what has really happened here, this equation that was trivial for a wave function that represented a particle with energy E, if this is the energy operator, this is not so easy anymore. Because remember, the energy operator, for example, was p squared over 2m plus v of x. And this was minus h squared over 2m d second dx squared plus v of x acting on wave functions. So now you've got a really interesting equation, because you don't assume that the energy is a number, because you don't know it. In general, if the particle is moving in a complicated potential, you don't know what are the possible energies. But this is symbolically what must be happening, because if this particle has a definite energy, then this energy operator gives you the energy acting on the function, and then you recover what you know is true for a particle of a given energy. So in general, the Schrodinger equation is a complicated equation. Let's write it now completely. So this is the Schrodinger equation. And if we write it completely, it will read i h bar d psi dt is equal to minus h bar squared over 2m d second dx squared of psi plus v of x times psi-- psi of x and t, psi of x and t. So it's an equation, a differential equation. It's first order in time, and second order in space. So let me say three things about this equation and finish. First, it requires complex numbers. If psi would be real, everything on the right hand side would be real. But with an i it would spoil it, so complex numbers have to be there. Second, it's a linear equation. It satisfies the proposition. So if one wave function satisfies the Schrodinger equation, the sum of wave functions, and another wave function does, the sum does. Third, it's deterministic. If you know psi at x and time equals 0, you can calculate psi at any later time, because this is a first order differential equation in time. This equation will be the subject of all what we'll do in this course. So that's it for today. Thank you. [APPLAUSE]
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_13_More_on_Scattering.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Any questions from last lecture from before spring break? No questions, nothing? Nothing at all? We you all still on vacation? Thank you. Yeah. AUDIENCE: Could you talk about coherence states? PROFESSOR: Ah. OK, well, that's a great question. So coherence states-- actually, how many people have looked at the optional problems? Nice, OK, good. Good, so coherence states were a topic that we touched on the problem sets and on the optional problems, and the optional problems are mostly on the harmonic oscillator and nice problems revealing some of the structure of the harmonic oscillator, and it generalizes quite boldly. But here's the basic idea of coherence states. Let me just talk you through the basic ideas rather than do any calculations. So what's the ground state of a harmonic oscillator? What does its wave function look like? AUDIENCE: Gaussian. PROFESSOR: It's a Gaussian, exactly. It's minimum uncertainty wave packet. How does it evolve in time? AUDIENCE: Phase. PROFESSOR: Yeah, phase. It's an energy eigenstate, it's the ground state, so it just evolves in time with the phase. So if we look at the wave function for the ground state, phi naught, it's something like e to the minus x squared over 2a squared with some normalization coefficient, which I'm not going to worry about. So this is a minimal uncertainty wave packet. Its position distribution is time independent, because it's a stationary statement. It's momentum distribution, which is also a Gaussian, the Fourier transform, is time independent. And so this thing up to its rotation by the overall phase just sits there and remains a Gaussian. Now, here's a question. Suppose I take my harmonic oscillator potential, and I take my Gaussian, but I displace it a little bit. It's the same ground state, it's the same state but I've just displaced it over a little. What do you expect to happen? How will this state evolve in time? So, we know how to solve that problem. We take this state, it's a known wave function at time 0. We expand it in the basis of energy eigenstates, each energy eigenstate evolves in time with a phase, so we put in that phase and redo the sum, and recover the time evolution of the full state. And we've done this a number of time on the problem sets. Yeah. AUDIENCE: Would the Gaussian, the displaced Gaussian evolve the same way, keep its width, if it had any other initial width other than the one of the ground state? PROFESSOR: Let me come back to that, because it's a little more of a precise question. So we know how to solve this problem practically, algorithmically. But here's a nice fact. I'm not going to derive any equations, that's part of the point of the optional problems, but here's a nice fact about this state. So it's clearly a minimum uncertainty wave packet because at time 0, because it's just the same Gaussian just translated over a little bit. So what we'd expect, naively, from solving is we expand this in Fourier modes and then we let this the system evolve in time. We let each individual Fourier mode-- or, sorry, Fourier mode-- we let each individual energy eigenstate evolve in time, pick up a phase, and now what we get is a superposition. Instead of sum over n at time 0, we'd have sum over n cn phi n of x. Now, as a function of time, we get e minus I omega nt, and these phases are going to change the interference from summing up all these energy eigenstates, and so the system will change in time. Because the way that the various terms in the superposition interfere will change in time. So very naively, what you might expect, if you just took a random function-- for example, if I took a harmonic oscillator potential and I took some stupid function that did something like this. What would you expect it to do over time? Well, due to all the complicated interference effects, you'd expect this to turn into, basically, some schmutz. Just some crazy interference pattern. The thing that's nice about a coherent state, and this is where it gets its name, is the way that all of these interference effects conspire together to evolve the state is to leave it a Gaussian that does nothing but translates in time. A coherent state is coherent because it remains coherently a Gaussian as it moves along. It oscillates back and forth. And in fact, the peak, the center of this Gaussian wave packet, oscillates with precisely the frequency of the trap. It behaves just as a classical particle would have had you displaced it from the center of the harmonic oscillator trap, it oscillates back and forth. Now, on the other hand, you know that its momentum is changing in time, right. And at any given point, as you've shown on the problem set that's due tomorrow-- or you will have shown-- at any given moment in time, the momentum can be understood as the overall phase. You could just change that momentum by the overall phase, so it's the spatial rate of change of the phase, e to the ikx. So you know that the way the phase depends on position is changing over time. So it can't be quite so simple that the wave function, rather than the probability distribution, is remaining a Gaussian over time. It's not, it's got all sorts of complicated phases. But the upshot is that the probability distribution oscillates back and forth perfectly coherently. So that seems a little magical, and there's a nice way to understand how that magic arises, and it's to understand the following. If I take a state, phi 0, but I translate it x minus x0. OK. So we know that this state, phi 0 of x, was the ground state, what does it mean to be the ground state of the harmonic oscillator? How do I check if I'm the ground state of the harmonic oscillator? Look, I walk up to you, I'm like, hey, I'm the ground state of the harmonic oscillator. You're suspicious. What do you do? AUDIENCE: Annihilate. PROFESSOR: Annihilate. Exactly. You act with the annihilation operator. Curse you. So you act with the annihilation operator and you get 0, right. What happens if I act with the annihilation operator on this? Is this the ground state? No, it's been displaced by x0. And meanwhile, I told you that it oscillates back and forth. So what happens if you act with the annihilation operator? Should you get 0? No, what are you going to get? AUDIENCE: Something weird. PROFESSOR: Some random schmutz, right? If you just take a and you act on some stupid state, you'll just get some other stupid state. Except for Gaussians that have been displaced, you get a constant times the same wave function, phi 0 of x minus x0. Aha. It turns out that these displaced Gaussians are eigenstates of the annihilation operator. What does that mean? Well, it means they're coherent states. And so the optional problems are a working through of the study of the eigenfunctions of the annihilation operator, the coherent states. Is that cool? It's a state, it's a superposition. You can think about it like this. It's a superposition of the energy eigenstate. Any state is a superposition of energy eigenstates, and a coherent state is just some particularly special superposition of energy eigenstates. So you can think about it literally as c0 phi 0 plus c1 phi 1 plus c2 phi 2, blah blah blah. What does the annihilation operator do, what does the lowering operator do? It takes any state and then lowers it with some coefficient. So a coherent state, an eigenstate of the annihilation operator, must be a state such that if you take c 0 phi 0 plus c1 phi 1 plus c2 phi 2 and you hit it with the annihilation operator, which will give you see c1 phi 0 plus c2 phi 1 divided by root 2 plus dot, dot, dot-- or times root 2 plus dot, dot, dot. It gives you the same state back. So that gave you a relation between the coefficients c0 and c1, c1 and c2, c2 and c3. They've all got to be suitable multiples of each other. And what you can show-- and this is something you work through, not quite in this order, on the optional problems-- what you can show is that doing so it gives you a translated Gaussian. OK. So physically, that's what a coherent state is. Another way to say what a coherent state is, it's as close to a classical object as you're going to get by building a quantum mechanical wave function. It's something that behaves just like a classical particle in that potential would have behaved, in the harmonic oscillator potential. Did that answer your question? OK. Anything else? So, Matt works with these for a living. Matt, do you want to add anything? All right. I like these. They show up all over the place. This is like the spherical cow of wave functions because it's awesome. It's an eigenfunction, it's an annihilation operator, it behaves like a classical particle. AUDIENCE: Is there an easy way to define coherent states in potentials where you don't have nice operators like a? PROFESSOR: No, I mean, that's usually what we mean by a coherent state. So the term coherent state is often used interchangeably to mean many different things, which in the case of the harmonic oscillator, are identical. One is a state which is a Gaussian. OK. So in the harmonic oscillator system, that's a particularly nice state because it oscillates particularly nicely and it maintains its probability distribution as a function of time roughly by translating, possibly, some phases. But people often use the phrase coherent state even when you're not harmonic. And it's useful to keep in mind that we're often not harmonic, nothing's truly harmonic. You had another question a moment ago though, and your question was about the width. Yeah. AUDIENCE: It just translates without changing its width, with or without changing its shape at all. If the Gaussian were of a different width, would that still happen? PROFESSOR: Yeah, it does, although the details of how it does so are slightly different. That's called a squeeze state. The basic idea of a squeeze state is this. Suppose I have a harmonic oscillator-- and this is actually one way people build squeeze states in labs-- so suppose I have a harmonic oscillator and I put the system in a ground state. So there's its ground state. And its width is correlated with the frequency of the potential. Now, suppose I take this potential at some moment in time and I control the potential, the potential is created by some laser field, for example. And some annoying grad student walks over to the control panel and doubles the power of the laser. All of a sudden, I've squeezed the potential. But my system is already in a state which is a Gaussian, it's just the wrong Gaussian. So what does this guy do? Well, this is funny. The true ground state, once we've squeezed, the true ground state would be something that's much narrower in position space. I didn't draw that very well, but it would be much narrower in position space, and thus its distribution would be much broader in momentum space. So the state that we put the system, or we've left the system in, has too much position uncertainty and too little momentum uncertainty. It's been squeezed compared to the normal state in the delta x delta p plane. It's uncertainty relation is still extreme, it still saturates the uncertainty bound because it's a Gaussian, but it doesn't have the specific delta x and delta p associated with the true ground state of the squeezed potential. OK, so now you can ask what does this guy do in time. And that's one of the optional problems, too, and it has many of the nice properties of coherent states, it's periodic, it evolves much like a classical particle, but it's uncertainties are different and they change shape. It's an interesting story, that's the squeezed state. Yeah. AUDIENCE: Do other potentials have coherent states that are Gaussian? PROFESSOR: It sort of depends. So other potentials have states that behave classically. Yeah. There are generally-- so systems that aren't the harmonic oscillator do have very special states that behave like a classical particle. But they're not as simple as annihilations by the annihilation operator. Come to my office and I'll tell you about analogous toys for something called supersymmetric quantum models, where there's a nice story there. OK. I'm going to cut off coherent states for the moment and move on to where we are now. OK, so last time we talked about scattering of a particle, a quantum particle of mass m against a barrier. And we made a classical prediction, which I didn't quite phrase this way. But we made a classical prediction that if you took this particle of mass m and threw it against a barrier of height v0, that the probability that it will transmit across this barrier to infinity is basically 0. So this is the classical prediction. It will not transmit to infinity until the energy goes above the potential. And when the energy is above the potential, it will slow down, but it will transmit 100% of the time. So this is our classical prediction. And so we sought to solve this problem, we did. It was pretty straightforward. And the energy eigenstate took the form-- well, we know how to solve it out here, we know how to solve it out here, because these are just constant potentials so it's just plain waves. Let's take the case of the energy is less than the potential. So this guy. Then out here, it's in a classically disallowed state, so it's got to be a exponential. If we want it to be normalized, well, we need it to be a decaying exponential. Out here it's oscillatory, because it's a classically allowed region. And so the general form of the wave function of the energy eigenstate is a superposition of a wave moving this way with positive momentum, a wave with negative momentum, a contribution with minus k, and out here it had to be the decaying exponential. And then by matching, requiring that the wave function was oscillatory and then exponentially decaying out here, and requiring that it was continuous and differentiably continuous-- that its derivative was continuous-- we found matching conditions between the various coefficients, and this was the solution in general. OK. Now, in particular, this term, which corresponds to the component of the wave function moving towards the barrier on the left hand side, has amplitude 1. This term, which corresponds to a wave on the left with negative momentum, so moving to the left, has amplitude k minus i alpha over k plus i alpha, where h bar squared k squared upon 2m is equal to-- that's just the energy, e-- and h bar squared alpha squared over 2m-- that's on this side-- is equal to v0 minus e. But this, notably, is a pure phase. And we understood that, so we'll call this parameter r, because this is the reflected wave and this is an amplitude rather than a probability, so we'll call it little r. And we notice that the probability that we get out-- good. So if we want to ask the question, what is the transmission probability, the probability that I get from far out here to far out here, what is that probability? So if I consider a state that starts out as a localized wave packet way out here and I send it in but with energy below the barrier, what's the probability that I'll get arbitrarily far out here, that I will subsequently find the particle very far out here? 0, right, exactly. And you can see that because here's the probability amplitude, the norm squared is the probability density as a function of the position. And it goes like a constant times e to the minus alpha x. For large x, this exponential kills us. And we made that more precise by talking about the current. We said look, the current that gets transmitted is equal to-- that's a funny way to write things. H bar-- sorry. The current that gets transmitted, which equal to h bar upon 2mi psi complex conjugate dx psi minus psi dx psi complex conjugate on the right. So we'll put right, right, right, right. This is just equal to 0, and the easiest way to see that is something you already showed on a problem set. The current vanishes when the wave function can be made real up to a phase. And since this is some number, but in particular it's an overall constant complex number, and this is a real wave function, we know that this is 0. So the transmitted current out here is 0. The flux of particles moving out to the right is 0. So nothing gets out to the right. So that was for the energy less than the potential, and we've now re-derived this result, the classical prediction. So the classical prediction worked pretty well. Now, importantly, in general, we also wanted to define the transmission probability a little more carefully, and I'm going to define the transmission probability as the current of transmitted particles on the right divided by the current of probability that was incident. And similarly, we can define a probability that the particle reflects, which is the ratio of the reflected current, the current of the reflected beam, to the incident current. So this is going to be the transmission probability and the reflection probability. Yeah. AUDIENCE: Is there a square on t? PROFESSOR: Thank you. OK. So let's do a slightly different example. In this example, I want to study the same system, the wall, but I want to consider a ball incident from the left-- a ball. An object, a quantum particle incident from the left, with energy greater than v. Greater than v0. So again, classically, what do you expect in this case. You send in the particle, it loses a little bit of energy going up the potential barrier, but it's still got a positive kinetic energy, and so it just keeps rolling. Just like my car making it up the driveway, just barely makes it. OK. So here again, what's the form of the wave function. Phi e can be put in the form on the left. It's going to be e to the ikx plus b to the minus ikx, and that's over here on the left. And on the right it's going in the form ce to the ik-- in fact, let me call this k1x and k2x-- or sorry, k1x, which is on the left. H bar squared k1 squared upon 2m is equal to e. And on the right, we're going to have k2x because it's, again, a classically allowed region, it's going to be oscillatory but with a different momentum. e to the ik2x plus de to the minus ik2x where h bar squared k2 squared upon 2m is equal to e minus v0, which is a positive number, so that's good. So here's our wave function. Now, before we do anything else, I want to just interpret this quickly. So again, this is like a wave that has positive momentum, and I'm going to say that it's like a contribution, a term in the superposition, that is moving to the right, and the way to think of it as moving to the right, well, first off, it's got positive momentum. But more to the point, this is an energy eigenstate. So how does this state evolve in time? It's an energy eigenstate with energy, how does it evolve in time. AUDIENCE: [INAUDIBLE] PROFESSOR: It rotates by an overall phase, exactly. So we just get this times an e to the minus i omega t, where h bar omega is equal to e, the energy. And that's true on the left or right, because the energy is just a constant. So this term goes as e to the ikx minus i omega t, or kx minus omega t. And this is a point of constant phase in this move to the right. As t increases-- in order for the phase to be constant or for the phase, for example, to be 0-- as t increases, x must increase. By assumption here, k is a positive number. On the other hand, this guy-- that was a k1-- the second term is of the form e to the minus ik1x, and when I add the omega t, minus i omega t, this becomes kx plus omega t. So point of constant phase, for example, phase equals 0, in order for this to stay 0 then, as t increases, x must become negative. x must decrease. So that's why we say this corresponds to a component of the wave function, a term in the superposition which is moving to the left. And this is one which is moving to the right. Sorry, my right, your left. Sorry, this is moving to your right and this is moving to your left. OK? It's that little z2, it's harder than it seems. OK. In this set up, we could imagine two different kinds of scattering experiments. We can imagine a scattering experiment where we send in a particle from the left and ask what happens. So if you send in a particle from the left, what are the logical possibilities? AUDIENCE: It can go through. PROFESSOR: It could go through, or? AUDIENCE: Reflect back. PROFESSOR: Reflect back. Are you ever going to get a particle spontaneously coming from infinity out here? Not so much. So if you're sending a particle in from the left, what can you say about these coefficients? d equals 0 and a is not 0, right. Because d corresponds to a particle on the right hand side coming in this way. So that's a different scattering experiment. So in particular, coming in from the left means d equals 0 and a not equal to 0. Coming in from the right means a equals 0 and d not equal to 0 by exactly the same token. Cool? So I want to emphasize this. a and d, when you think of this as a scattering process, a and d are in guys. In, in. And c and b are out. This term corresponds to moving away from the barrier, this term corresponds to moving away from the barrier, just on the left or on the right. OK? Out and out. Everyone happy with that? Yeah. AUDIENCE: According to that graph, if d is-- no, that one. Yeah. If d is bigger than [INAUDIBLE] then the transplants will be 1. Doesn't that mean that d is also 0? PROFESSOR: So this is the classical prediction. I've written it, classical prediction. So let me rephrase. The question was basically, look, that classical prediction implies that b must be 0 on the other scattering process, and that doesn't sound right. So is that true? No, it's not true, and we'll see it again in a second. So very good intuition. That was good. OK, good. So in general, if we have a general wave function, general superposition of the two states with energy e with a,b,c, and d all non-0, that's fine. That just corresponds to sending some stuff in from the right and some stuff in-- sorry, some stuff in from the left and some stuff in from the right. Yeah? But it's, of course, going to be easier if we can just do a simpler experiment. If we send in stuff from the left or send in stuff from the right. And if we solve those problems independently, we can then just superpose the results to get the general solution. Yeah. So it will suffice to always either set d to 0 or a to 0, corresponding to sending things in from the left or sending things in from the right. And then we can just take a general superposition to get the general answer. Everyone happy with that? So let's do that in this set up. So here's our wave function. And you guys are now adept at solving the energy eigenvalue equation, it's just the matching conditions for a,b,c, and d. I'm not going to solve it for you, I'm just going to write down the results. So the results-- in fact, I'll use a new border. The results for this guy, now for e greater than v, are that-- doo doo doo, what just happened. Right, OK, good. So let's look at the case d equal 0 corresponding to coming in from the left. So in the case of in from the left, c is equal to 2k1 over k1 plus k2, and b is equal to k1 minus k2 over k1 plus k2. And this tells us, running through the definition of j, the currents, and t and r, gives us that, at the end of the day, the reflection coefficient is equal to k1 minus k2 upon k1 plus k2 [INAUDIBLE] squared, whereas the transmission amplitude or probability is equal to 4k1 k2 over k1 plus k2 squared. And a good exercise for yourself is to re-derive this, you're going to have to do that on the problem set as a warm up for a problem. So at this point we've got an answer, but it's not terribly satisfying because k1 and k2, what's the-- so let's put this in terms of a more easily interpretable form. k1 and k2 are nothing other than code for the energy and the energy minus the potential. Right, so we must be able to rewrite this purely in terms of the energy in the potential e and v. In fact, if we go through and divide both top and bottom by k1 squared for both r and t, this has a nice expression in terms of the ratio of the potential to the energy. And the nice expression is 1 minus the square root of 1 minus v0 over e over 1 plus the square root of 1 minus v0 over e norm squared. And suddenly, the transmission amplitude has the form 4 root 1 minus v0 over e over 1 plus the square root of 1 minus v0 over e squared. I don't remember which version of the notes I posted for this year's course. In the version from 2011 there is a typo that had here e over v, and in the version from 2012, that was corrected in the notes to being v0 over e. So I'll check, but let me just warn you about that. On one or two pages of the notes, at some point, these guys were inverted with respect to each other. But the reason to write this out is now we can plot the following. We can now plot the quantum version of this plot. We can plot the actual quantum transmission as a function of energy, and the classical prediction was that at energy is equal to v0, we should have a step function. But now you can see that this is not a step function, and that's related to the fact that b was not actually 0, as you pointed out. Hold on one sec. Sorry, was it quick? AUDIENCE: The notes are wrong. PROFESSOR: OK, the notes are wrong, good, thank you. That's great. OK, so the notes are wrong. It should be v0 over e, not e over v0 on the notes that are posted. We're in the process of teching them up, so eventually a beautiful set of nice notes will be available. Extra elbow grease. OK. So what do we actually see? Here we got for e less than v0, we got in d that the transmission probability was exactly 0. So that's a result for the quantum result. But when e is equal to v0, what do we get? Well, when e is equal to v0, this is 1. And we get square root of 1 minus 1, which is square root of 0, over square root of 1 plus square root of 1 minus 1, which is 0. That's 0 over-- OK, good. So that's not so bad. Except for this 1 over this e, this is a little bit worrying. If you actually plot this guy out, it does this, where it asymptotes to 1. And to see that it asymptotes to 1, to see that it asymptotes to 1, just take e0 gigantic. If you know it's gigantic, this becomes v0 over e, which goes to 0, and if e is much larger than the potential. So this becomes square root of 1. And in the denominator, this becomes square root of 1, 1 plus 1, that's 2 squared, 4, 4 divided by 4, that goes to 1. So for large e much larger than v0, this goes to 1. So we do recover the classical prediction if we look at energy scales very large compared to the potential height. Yeah? OK. So that's nice. Another thing that's nice to note is that if you take this reflection probability and this transmission probability, then they sum up to 1. This turns out to be a general fact and it's a necessary condition in order for those to be interpretable as reflection in transition probabilities. The probability that it transmits and the probability that it reflects had better add to 1, or something is eating your particles, which is probably not what you're looking for. So this turns out to be true in this case, just explicitly. But you can also, as you will in your problem set, prove that from that definition r and t, it's always true. A check on the sensibility of our definition. If it weren't true, it wouldn't tell you that quantum mechanics is wrong, it would tell you that we chose a stupid definition of the transmission and reflection probabilities. So in this case, we actually chose quite an enlightened one. OK, questions at this point? Yeah. AUDIENCE: Does this analysis even hold for classical particles? If we're talking about the difference between-- like if I threw a baseball and I happened to throw it at a potential that had height one millionth of a joule less than the energy of the baseball, would we observe this also? PROFESSOR: No. AUDIENCE: No? PROFESSOR: No, for the following reason. So here's my classical system. Classical system is literally some hill. So where I was growing up as a kid, there was a hill not far from our house. I'm not even going to go into it, but it was horrible. If you have energy just ever so slightly greater than the hill-- he said from some experience-- if you have energy that's ever so slightly greater-- OK. I'm going resist the temptation. So one of these days. If you have ever so slightly greater than the hill, and you start up here, what does that tell you? What does it mean to say you have energy just ever so slightly greater than the potential energy at that point? AUDIENCE: Up more slowly. PROFESSOR: Little tiny velocity. OK, and I'm going to say I have a little tiny velocity this way. OK. So what happens? We follow Newton's equations. They are totally unambiguous, and what they tell you is, with 100% certainty, this thing will roll and roll and roll, and then all hell will break loose. And then if you're very lucky, your car gets caught in the trees on the side of the cliff, which is later referred to by the policeman who helps you tow it out as nature's guard rail. I was young, it won't happen again. And So with 100% certainty, Newton's equations send you right off the cliff. And for style points, backward. So now I'm going to do the second thing. Newton's laws satisfy time reversal invariance. So the time reversal of this is, this thing has this much energy, and it shoots up the cliff. And ever so slowly, it just eventually goes up to the top, where everything is fine. OK, but does it ever reflect back downhill? No. And does it ever-- classically, when you roll the thing-- does it ever actually not go off the cliff and hit the trees, but instead reflect backwards. I wish the answer were yes. But sadly, the answer is no. Now, if that car had been quantum mechanically small, my insurance would have been much more manageable. But sadly, it wasn't. OK, so this is a pretty stark and vivid-- this is a pretty stark difference between the quantum mechanical prediction and the classical prediction. Everyone cool with that? Other questions? Yeah. AUDIENCE: I understand why it works out that r plus t equals 1, but I'm not sure I understand the motivation to use the square with the ratios. PROFESSOR: Excellent, OK. So the reason to do it, so great, excellent. How to say. Here's one way to think about it. The first thing we want is we want a-- so this is a very good question, let me repeat the question. The question is, look, why is it squared? Why isn't it linear? So let's think about that. Ah, ah, ah. The reason it's squared is because of a typo. So let's think about-- [LAUGHTER] PROFESSOR: Let's step back for second and let's think, OK, it's a very good question. So let's think about what it should be. OK. For the moment, put an arbitrary power there. OK. And let's think about what it should be. Thank you so much for this question. I owe you, like, a plate of cheese or something. That's high praise, guys. France, cheese. So should it be linear or should it be quadratic? What is that supposed to represent? What is T, the capital T, supposed to represent? AUDIENCE: The probability of transmission. PROFESSOR: The probability of transmission, exactly. The wave function. Is the wave function, the value of the wave function at some point, is that a probability? It's a-- AUDIENCE: Probability of density. No, it's the square root. PROFESSOR: Is it a probability of density? It is a probability amplitude. It is a thing whose norm squared is a probability. So we want something that's quadratic in the wave function, right. Meanwhile, some time before, we wanted a definition of how much stuff is moving past a point in the given moment in time. What's the probability density moving past a point at a given moment in time. And that's where we got the current j from the first place. j was the probability density moving past a point at a given moment in time. So notice that j is quadratic in the wave function, so it's how much stuff is moving past any particular direction-- we chose it to be the incident to the reflected bit-- at a moment in time. So it's a probability density-- it's actually a current-- and it's quadratic in the wave function. So it has the right units and the right structure to be a probability. If we squared that, we would be in trouble, because it wouldn't be a probability, it would be a probability squared. And in particular, it wouldn't normalize correctly. So that typo was actually a bad typo. So what happened, now thinking back to-- you have a video someday, so you can scroll back. What happened was I didn't put the squared on the first, I put the squared on the second, and then someone said squared on the first? Like, yes. But the correct answer is no squared on the second. So there are no squareds. Thank you for the question. Very good question. In particular, the thing that was so awesome about that question was it was motivated by physics. It was like, look, why is this thing squared the right thing? That's two factors of the wave function, it's already quadratic. It should be like a probability. Why are you squaring it? Very good question. Yeah. AUDIENCE: So in the transmission probability, why do we know it's 0 if the energy is less than v0? I mean, is there any rule? PROFESSOR: Yeah, OK, good. We'll come back to that question in a little bit, but let me just quickly say that we calculated it, and we see explicitly that it's 0. Now you might say well, look, the true wave function has a little bit of a tail. So there's some [INAUDIBLE] probability you'll find it here if you do a measurement. But the question we want to ask in transmission is, if you send it in from far away over here, how likely are you to catch it far away over here? And the answer is, you are not. That make sense? OK, good. OK, so so much for that one. Now, here's a fun fact, I'm not going to go through this in any detail. We could have done exactly the same calculation-- OK, we could have done the same calculation sending something in from the right. And sending something in from the right, in on the right, corresponds to d not equal to 0 and nothing coming in from the left, that's a equals 0. So we could have done the case a equals 0, which is in from the right. And if you do that calculus, what you find is c is equal to k1 minus k2 upon k1 plus k2 d. And b is equal to 2k1 upon k1 plus k2 d. And this says that plugging these guys in, the reflection and the transmission are the same. In particular, physically, what does that mean? That means the reflection and transmission are the same uphill as downhill. Downhill is uphill, they're the same both ways. But that's truly weird, right? What's the probability to transmit quantum mechanically if you have just a little bit of energy and you send the particle in? How likely are you to go off the cliff. If we have energy ever so slightly greater the potential, and the transmission amplitude is the same downhill as it was uphill, how likely are you to fall off the potential? AUDIENCE: Always. PROFESSOR: Never. Because if energy is just slightly greater than v0, then the transmission probability is very, very low. The transmission to go from here to here is extremely low if the energy is close to the height of the barrier. So had I only been on a quantum mechanical hill, I would have been just fine. This is a really striking result, but this thing, the fact that you're unlikely to scatter uphill, that's maybe not so shocking. But you're really unlikely to scatter downhill, that is surprising. So I'll leave it to you to check that, in fact, the transmission, when we do it in from the right, the transmission and reflection probabilities are the same. Recalling that transmission means going this way, reflection means bouncing back to the right. Cool? Yeah. AUDIENCE: [INAUDIBLE] PROFESSOR: Uh, because, again. Jeez, today is just a disaster. Because there's a typo. This should be times a. God. And the reason you know it should be times a, first off, is that these should have the appropriate dimensions. So there should be an appropriate power of a. And the second thing is that if you double this stuff in, you'd better double this stuff out. So if you double a, you'd better double c. So you know there had better be a factor of a here. And that's just a typo again. I'm sorry, today is a bad day at the chalkboard. Thank you for that question. Yeah. AUDIENCE: Is there a physical reason for why r and t are the same and so different from what you would classically expect, or is that just the way the math works? PROFESSOR: There is. I want you to ponder that, and either at the end of today or at the beginning of tomorrow, we'll talk about that more in detail when we've done a little more technology. It's a good question and you should have that tingling sensation in your belly that something is confusing and surprising and requires more explanation, and we'll get there. But I want you to just think about it in the background first. Yeah. AUDIENCE: Is there any physical experiment where r and t don't sum up to 1? PROFESSOR: Mm mm. Nope. So the question is, is there any experiment in which r and t don't sum up to 1? And there are two ways to answer that. The first way to answer that is, it turns out that r and t adding up to 1 follows from the definitions. So one of the things you'll do on the problem set is you'll show that using these definitions of the transmission and reflection of probabilities, they necessarily add up to 1, strictly. They have to. It follows from the Schrodinger equation. The second answer is, let's think about what it would mean if r and t didn't add up to 1. If r and t didn't add up to 1, then that would say, look, if you throw a particle at a barrier, at some feature in a potential, some wall, then the probability that it goes this way at the end versus the probability that it goes that way at the end if you wait long enough is not 1. So what could have happened? What could have gotten stuck in there? But that doesn't even conserve momentum, not even approximately. And we know that the expectation of momentum is time independent for a free particle. In between while it's actually in the potential, it's not actually time independent, there's a potential, there's a force. AUDIENCE: I mean, if it's whole particles, can't some of them get annihilated or something? PROFESSOR: Ah, well OK. So if you're talking about multiple particles and interactions amongst multiple particles, then it's a slightly more complicated question. The answer there is still yes, it has to be that the total probability in is the total probability out. But we're only going to talk about single particles here. But it's always true that for-- we're going to take this as an assumption, that things don't just disappear, that the number of particles or the total probability is conserved. There's a third way to say this, this really isn't independent from the first. Remember that these were defined in terms of the current? The current satisfies-- the current concept of probability conservation equation dt of rho is equal to the gradient minus the gradient of the current. So the probability is always conserved, the integral of probability density is always conserved, it's time independent. AUDIENCE: So it's by [INAUDIBLE]. PROFESSOR: Yeah, it's by construction. Exactly. Yeah? AUDIENCE: Looking at your expression for the transition probability, I'm having trouble seeing how that works out to 0 when e is less than v0? PROFESSOR: Oh, this expression is only defined when e is equal to v0. Because we derived this-- excellent question-- we derived this assuming that we had an energy greater than v0 and then that the wave function had this form. You cannot use this form of the wave function if the energy is less than v0. If the energy is less than v0, you've got use that form of the wave function. And in this form of the wave function, we derived that the transition amplitude is 0, because the current on the right, the transmitted current, is 0. So this calculation is appropriate when e is less than v0 and this calculation is appropriate when e is greater than v0. So you're absolutely correct. You can't use this one e is less than v0, it gives you not the same answer. In fact, it gives you a complex-- it's kind of confusing. The factors cancel. So it's not really a probability at all, and indeed, this is just not the right quantity to use. AUDIENCE: All right, cool. PROFESSOR: Cool? All right. So there are a bunch of nice things I want to deduce from what we've done so far. So the first is, look, I pointed out that this can be derived just explicitly and it gives the same results as before. That's not an accident. If you take this system and you just reverse the roles of k1 and k2, what happens? Well that's just replaces e by e minus v0, we can do that by doing this, replacing e by e minus v0 and e minus v0 by e, it swaps the role of a and d, and it gives you exactly the same things back. So if you're careful about that, you never have to do this calculation. You can just do the appropriate transformation on that calculation and it gives you the exactly the same thing. It's the same algebraic steps. But the other thing that's nice is that you can actually do the same thing from here. So as long as you set the d equals 0-- so there's another term here that we neglected, the plus delta x, we got rid of it. But you can analytically continue this calculation by noting that look, if we just set alpha, we want minus alpha equals ik2, then the algebra is all going to be the same. We just have ik2 instead of minus alpha. So we can just replace alpha with minus ik2 everywhere in our expressions, being careful about exactly how we do so, being careful to take care of factors of i and such correctly. And you derive the same results for both cases, which is a nice check on the calculation. So often, when you get a little bit of experience with these, you don't actually have to do the calculation. Again, you can just take what you know from a previous calculation and write down the correct answer. So it's a fun thing to play with, exactly how do you do that. So I invite you to think through that process while you're doing your problem set. Another thing is the reflection downhill thing which is pretty surprising. But here's the thing that I really want to emphasize. What this calculation shows you is not so much that-- it's not just that transmission downhill is highly unlikely when the energy is very close to the height of the potential barrier. That's true, but it's not the most interesting thing about this calculation. The most interesting thing about this calculation, to my mind, is the fact that from the detailed shape of the transmission as a function of energy, we can deduce what the potential is. Think about what that tells you. If you do an experiment, you have a barrier, and you want to know the shape of the barrier. Is it straight, is it wiggly, does it have some complicated shape. How do you measure that? Well, you might measure it by just looking. But imagine you can't, for some reason. For whatever, it's in a box or you can't look at it. Maybe it's just preposterously small. How can you deduce what the shape of that hill is? Well, one way to do it is to send in particles as a function of energy, more and more energy, and measure the probability that they transmit. OK. Now if you do so and you get this graph as a function of e, what do you deduce? You deduce that the barrier that you're scattering off of is a square step with this height v0. Are we cool with that? So apparently, just look at the transmission amplitudes, the transition probabilities, you can deduce at least something of the form of the potential. Which is kind of cool. If you didn't know the potential, you could figure out what it was. And this turns out to be a very general statement that you can deduce an enormous amount, and as we'll see, you can, in fact, deduce basically everything you want of the potential from knowing about the transmission probabilities as well as the phase shift, the transmission amplitudes. So this is the basic goal of scattering. And so the way I want you to think about it is imagine, for example, that someone hands you an object. A box. And the box has an in port and it has an out port. And they allow you to send in particles as a function of energy and measure transmission and reflection, you can measure transmission and reflection. Just like I'm measuring transmission off of your faces right now, from the light from above. So suppose that you do so, put it on your test stand and you measure transmission. You measure transmission as a function of energy, and you observe the following. The transmission as a function of energy is small up until some point, and then at some point, which may be the minimum energy you can meaningfully probe, you get something like this. So here's the transmission as a function of energy. So what can you say? If this is all the information you have about what's going on inside the box, what can you deduce about the thing inside the box? One thing you can deduce is that it looks kind of like a potential with height around v0. It looks kind of like a potential step with some height v0. This is asymptoting to 1. However, it's not, because it has these oscillations in it. So there's more to the potential than just a barrier of height v0. What I want to show you is that you can deduce everything about that potential, and that's the point of scattering. So let's do it. So the goal here, again, to say it differently, is what's v0? What is v of x? Not just the height, but the total potential. So another way to say this, let me set up a precise version of this question. I want to be able to do the following. I want to take a system that has a potential which is constant up to some point which I'll call 0, and then again from some point, which I'll call L, is constant again. And inside, I don't know what the potential is. So in here, there's some unknown potential, v of x, which is some crazy thing. It could be doing anything. It could be some crazy-- it could have horns and whatever. It could be awful. But the potential is constant if you go far enough away, and the potential is constant if you go far enough away. A good example of this is a hydrogen atom. It's neutral but there's a clearly and complicated potential inside because the proton and the electron are moving around in there in some quantum state, anyway, and if you send something at it, far away, it's as if it's not there. But close by, you know there are strong electrostatic forces. And so the question is what you learn about those forces, what can you learn about the potential by throwing things in from far away, from either side. Now one thing we know already is that out here, the wave function always-- because it's a constant potential-- always takes the form e to the ikx plus b e to the minus ikx. And out here it takes the form c e to the ikx plus d e to the minus ikx. And again, this corresponds to moving in. d is in from the right. c is out to the right. b is out to the left. And a is in from the left. So again, there are basically four kind of scattering experiments we can do. We can send things in from the right, which corresponds to setting a to 0. We can send things in from the left, which corresponds to setting t equal to 0. And all the information about what happens in v is going to be encoded in what's coming out, the b and c coefficients. And the way to make that sharp is just to notice that the transmission probability, if we compute for this system, assuming it's forming the wave function asymptotically away from the potential, the transmission amplitude is just c over a squared when you're sending in from the left. And the reflection is equal to b over a norm squared. This squared is a squared. [INAUDIBLE] function over amplitude squared, good. To learn about the transmission and reflection coefficients, it's enough-- suffices-- to compute, to know b and c as a function of a and d. All of the scattering information is in those coefficients b and c for a and d. And here I'm assuming that I'm sending in a monochromatic wave with a single, well defined energy. I'm sending in a beam of particles with energy e. I don't know where they are, but I sure know what their momentum is. So some well defined beam of particles with energy e. And these probabilities are going to contain all the data I want. So this is the basic project of scattering. Questions? AUDIENCE: So basically, it only depends on the transmission-- it only depends on the edges? PROFESSOR: That's a good question. The question is, does the transmission depend only on what you do with the edges. And here's the important thing. The transmission depends crucially on what happens in here. For example, if this is an infinitely high barrier, nothing's going across. So this transmission depends on what's in here. But the point is we can deduce just by looking far away, we can deduce the transmission probability and amplitude just by measuring b and c far away, b and c far away. OK. So the transmission amplitude is something you measure when very far away. You measure-- if I throw something in from very far out here to the left, how likely is it to get out here very far to the right? And in order to answer that, if someone hands you the answer to that, they must have solved for what's inside. That the point. So knowing the answer to that question encodes information about what happened in between. AUDIENCE: So I guess initially, your potential's going to-- so say it's stuff you did earlier. The potential drops down to the [INAUDIBLE] in the box. Is that going to be problematic? PROFESSOR: Yeah, for simplicity, I'm going to assume that the potential always goes to 0 when we're far away, because that's going to be useful for modeling things like hydrogen and, exactly. Carbon, we're going to do diamond later in the semester, that'll be useful. But we could repeat this analysis by adding an extra change in the asymptotic potential, it doesn't really change anything important. Yeah. AUDIENCE: It looks like even for just the simple step up [INAUDIBLE] you can't tell from just the probabilities why the step is going up or going down. PROFESSOR: Ah, excellent. Excellent, excellent. So good, thank you for that question, that's really great. So already, it seems like we can't uniquely identify the potential from the transmission probability if the transmission probability is the same for step up or step down. So what's missing? AUDIENCE: Maybe the energy [INAUDIBLE] PROFESSOR: But we're working with an energy [INAUDIBLE]. So the energy is just a global constant. We'll see what's missing, and what's missing is something called the phase shift. So very good question, yes. AUDIENCE: It looks like when we did the example for the step, that t equals [INAUDIBLE] over a squared. PROFESSOR: Yeah, it is, because it's the ratio. That's why I wrote t is c over here. So t is the ratio of j over ji. And in fact, jt here is equal to c squared times-- so this is the probability density, so it's the probability density times the velocity, what's the effect of velocity here? H bar k 2 over m, whereas j incident is equal to a squared-- probability density times the momentum there, the velocity there-- which is h bar k1 upon m. So here the ratio of j transmitted to j incident is norm c squared k2 over norm a squared k1. AUDIENCE: So because it's not level-- PROFESSOR: Exactly, it's because it's not level. So here, they happen to be level, so they're only [INAUDIBLE] factor, cancel, and the only thing that survives is the amplitude. Good question. Yeah. AUDIENCE: For the lead box example, is it sufficient just to know what's reflected back to solve the situation? PROFESSOR: So for precisely for this reason, it's not sufficient to know t. It's not sufficient to know t and r. But, of course, once you know r, you know t, so you're exactly right. Once you know the reflection probability, you know the transmission probability, but there's one more bit of information which we're going to also need in order to specify the potential, which is going to be the phase shift. But you're right, you don't need to independently compute r and t, you can just compute one. AUDIENCE: You need sensors on both sides of the box, to answer my question. PROFESSOR: You don't need sensors on both sides of the box, but you need to do more than just do the counting problem. We'll see that. OK. So let's work out a simple example, the simplest example of a barrier of this kind. We want constant potential, and then ending at 0, and we want a constant again from L going off to infinity. So what's the easiest possible thing we could do? Step, step. We're just doing what we've done before twice. So this is an example of this kind of potential. It's sort of ridiculously simple, but let's work it out. So we want scattering, let's start out, we could do either scattering from the left or scattering from the right. Let's start out scattering from the left, so d equals 0, and let's study this problem. So what we know-- and let's also note that we have a choice to make. We could either study energy below the height of the potential or we can study energy above the height of the potential. And so for simplicity, I'm also going to start with energy greater than the height of the potential, v0, and then we'll do e less than v0 afterwards. It'll be an easy extension of what we've done. OK, so this will be our first case to study. So we know the form of the wave function out here, it's a e to the ikx, b to the minus ikx. We know that for the potential out here it's c e to the ikx and d e to the minus ikx. The only thing we don't know is the form of the potential in here. And in here it's actually very simple. It's got to be something of the form-- I think I called it f and g, I did. f e to the ik prime x-- I'm calling this k, so I'll just call this k prime x-- plus g e to the minus ik prime x. And the reason I chose k prime is because we're working with energy greater than the potential, so this is a classically allowed region. It's an oscillatory domain but with a different k prime. So here, k squared, h bar squared over 2m is equal to e, and h bar squared k prime squared over 2m is equal to e minus v0, which is positive when e is greater than v0. This analysis will not obtain when e is less than v0, we'll have to treat it separately. So now what do we do? We do the same thing we did before, we just do it twice. We'll do the matching conditions here, the matching conditions here. That's going to give us 1, 2, 3, 4 matching conditions. We have 1, 2, 3, 4, 5, 6 unknown coefficients, so we'll have two independent ones. That's great. We set d equal to 0 to specify that it's coming in from the left and not from the right, that's 5. And then we have normalization, which is 6, so this should uniquely specify our wave function. Yeah. Once we've fixed e, we have enough conditions. So I'm not going to go through the derivation because it's just an extension of what we did for the first. It's just a whole bunch of algebra. And let me just emphasize this. The algebra is not interesting. It's just algebra. You have to be able to do it, you have to develop some familiarity with it, and it's easy to get good at this. You just practice. It's just algebra. But once you get the idea, don't ever do it again. Once you get reasonably quick at it, learn to use Mathematica, Maple, whatever package you want, and use computer algebra to check your analysis. And use your physics to check the answer you get from Mathematica or Maple or whatever you use. Always check against physical reasonability, but use Mathematica. So posted on the website are Mathematica files that walk through the computation of the transmission and reflection amplitudes and probabilities for this potential, and I think maybe another one, I don't remember exactly. But I encourage you strongly to use computer algebra tools, because it's just a waste of time to spend three hours doing an algebra calculation. In particular, on your problem set this week, you will do a scattering problem similar to the bound state probably you did last week, the quantum glue problem-- which you may be doing tonight, the one due tomorrow. Which is two delta function wells and find the bound states, so that involves a fair amount of algebra. The scattering problem will involve a similar amount of algebra. Do not do it. Use Mathematica or computer algebra just to simplify your life. So if we go through and compute the-- so what are we going to do, we're going to use the matching conditions here to determine f and g in terms of a and b, then we'll use the matching conditions here to determine c-- d is 0-- c in terms of f and g. So that's going to give us an effective constraint relating a and b, leaving us with an overall unknown coefficient a, which we'll use for normalization. The upshot of all of which is the answers are that, I'm not even going to write down-- they're in the notes. Should I write this down? I will skip. So the upshot is that the transmission amplitude, as a function of k and k prime-- the transmission probability, I should say, is 1 over-- actually, I'm going to need the whole amplitude. Shoot. The transmission probability is equal to-- and this is a horribly long expression-- the transmission probability, which is c over a norm squared, is equal to 4 k squared k prime squared cosine squared of k prime l plus k squared plus k prime squared sine squared of k prime l under 4k squared k prime squared. Seriously. So we can simplify this out, so you can do some algebra. This is just what you get when you just naively do the algebra. I want to do two things. First off, this is horrible. There's a cosine squared, there's a sine squared, surely we can all be friends and put it together. So let's use some trig. But the second thing, and the more important thing, is I want to put this in dimensionless form. This is horrible. Here we have ks and we have ls, and these all have dimensions, and they're inside the sines and the cosines it's kl. That's good, because this has units of one over length, this has units of length, so that's dimensions. Let's put everything in dimensionless form. And in particular, what are the parameters of my system? The parameters of my system are, well, there's a mass, there's an h bar, there's a v0, and then there's an energy, and there's a length l. So it's easy to make a dimensionless parameter out of these guys, and a ratio of energies-- a dimensionless ratio of energies-- out of these guys. So I'm going to do that, and the parameters I'm going to use are coming from here. I'm going to define the parameters g0 squared, which is a dimensionless measure of the depth of the potential. We've actually run into this guy before. 2m l squared v0 over h bar squared. So this is h bar squared, 1 over l squared is k squared over 2m, so that's an energy. So this is a ratio of the height of the potential to the characteristic energy corresponding to length scale l. So the width, there's an energy corresponding to it because you take a momentum which has 1 over that width. You can build an energy out of that h bar k squared over 2m. And we have an energy which is the height of the barrier, we take the ratio of those. So that's a dimensionless quantity, g0. And the other dimensionless quantity I want to consider is a ratio of the energy, e, to v0, which is what showed up before in our energy plot. Or in our transmission plot over there, e over v0. So when we take this and we do a little bit of algebra to simplify our life, again, use Mathematica, it's your friend. The result is much more palatable. It's t-- again, for the energy greater than v0-- is equal to, still long. But 1 plus 1 over 4 epsilon, epsilon minus 1 sine squared of g0 square root of epsilon minus 1. And upstairs is a one. So remember this is only valid for e greater than v0, or equivalently, epsilon greater than 1. And I guess we can put an equal in. So when you get an expression like this, this is as easy as you're going to make this expression. It's not going to get any easier. It's 1 over a sine times a function plus 1. There's really no great way to simplify this. So what you need when you get an expression like this is try to figure out what it's telling you. The useful thing to do is to plot it. So let's just look at this function and see what it's telling us. Let's plot this t as a function of epsilon and for some fixed g0. Keep in mind that this only makes sense for e greater than v0 or for epsilon greater than 1, so here's 1. And we're going to remain agnostic as to what happens below 1. And just for normalization, we know that the transmission probability can never be greater than 1, so it's got to be between 1 and 0. So here's 0 and 0. We're remaining agnostic about this for the moment. So let's start thinking about what this plot looks like. First off, what does it look like at 1? So when epsilon goes to 1, this is going to 0, that's bad because it's in a denominator. But upstairs, this is going to 0, and sine squared of something when it's becoming small goes like-- well, sine goes like that thing. So sine squared goes like this quantity squared. So sine squared goes like g0 squared, this goes like 1 over-- this is going like g0 squared epsilon minus 1 square root quantity squared, which is epsilon minus 1. So this is going to 0 and this, the denominator, is going to 0 exactly in the same way. Epsilon minus 1 from here, epsilon minus 1 downstairs from here. So the epsilon minus ones precisely cancel. From the sine squared we get a g0 squared, and from here we get a 4 epsilon. So we get 1 plus g0 squared over 4 epsilon. But 4 epsilon, what was epsilon here? 1. We're looking at epsilon goes to 1, so this is just g0 squared over 4. So the height, the value of t-- so we're not looking in here. But at energy is equal to v0 or epsilon is equal to 1, we know that the transmission amplitude is not 0, but it's also strictly smaller than 1. Which is good, because if it were 6, you'd be really worried. So g0 squared, if g0 squared is 0, what do we get? 1. Fantastic, there's no barrier. We just keep right on going through, perfect transmission. If g0 is not zero, however, the transmission is suppressed. Like 1 over g0 squared. What does this actually look like with some value. And what is this value, it's just 1 over 1 plus 1/4 g0 squared. Cool? OK. And now what happens, for example, for a very large epsilon? Well, when epsilon is gigantic, sine squared of this-- well, sine squared is oscillating rapidly, so that's kind of worrying. If this is very, very large, this is a rapidly oscillating function. However, it's being divided by roughly epsilon squared. So something that goes between 0 and 1 divided by epsilon squared, as epsilon gets large, becomes 0. And so we get 1 over 1 plus 0, we get 1. So for very large values it's asymptoting to 1. So naively, it's going to do something like this. However, there's this sine squared. And in fact, this is exactly what it would do if we just had the 1 over epsilon times some constant. But in fact, we have the sine squared, and the sine squared is making it wiggle. And the frequency of the wiggle is g0, except that it's not linear in epsilon, it's linear in square root of epsilon. So as epsilon gets larger, the square root of epsilon is getting larger less than linearly. So what that's telling you is if you looked at root epsilon, you would see it with even period. But we don't have root epsilon, we're plugging this as a function of epsilon. So it's not even period, it's getting wider and wider. Meanwhile, there's a nice fact about this. For special values of epsilon, what happens to sine squared? It goes to zero. And what happens when this is zero? Yeah, t is 1. So every time sine is 0, i.e., for sufficient values when root epsilon is a multiple of pi determined by g0, transmission goes to 1. It becomes perfect. So in fact, instead of doing-- wow. Instead of doing this, what it does is it does this. So let's check that I'm not lying to you. So let me draw it slightly different. To check, so it's going to 0, and the period of the 0 is getting further and further along. That's because this is square root, not squared. So epsilon has to get much larger to hit the next period. That's why it's getting larger and larger spacing. However, the amplitude goes down from 1, that's when this gets largest. Well, at large values of epsilon, this is suppressed. As epsilon goes larger and larger, this deviation become smaller and smaller. So that's what this plot is telling us. This plot is telling us a bunch of things. First off, we see that at large energies, we transmit perfectly. That makes sense. This was a finitely high barrier. Large energies, we don't even notice it. It tells you at low energies-- well, we don't know yet what happens at very low energies. But at reasonably low energies, the transmission is suppressed. And if you sort of squint, this roughly does what we'd expect from the step barrier. However, something really special is happening at special values of epsilon. We're getting perfect transmission. We get perfect transmission of these points, t is equal to 1. Perfect. Star. Happy with a big nose. I don't know. Perfect transmission at all these points. So this leaves us with a question of why. Why the perfect transmission, what's the mechanism making transmission perfect. But it also does something really lovely for us. Suppose you see a spectrum, you see a transmission amplitude or transmission probability that looks like this. You may get crappy data. You may see that it's smudged out and you see all sorts of messy stuff. But if you know that there's perfect transmission at some particular epsilon 1, there's more perfect transmission at another epsilon 2, more perfect transmission at epsilon 3, and they scale, they fit to this prediction. What do you know? That you've got a finite high barrier, and it's probably pretty well approximated by a square step. So in your problem set, you're going to get experimental data, and you're going to have to match to the experimental data. You're going to have to predict something about the potential that created a particular transmission probability distribution. And knowing where these resonances are, where these points of perfect transmission, is going to be very useful for you. Yeah. AUDIENCE: Wasn't there another potential that will also create periodic 1s? PROFESSOR: A very good question. So I'm going to turn that around to you. Can you orchestrate a potential that gives you the same thing? That's an interesting empirical question. So, on the problem set, you'll study a double well potential. So the question was, how do I know isn't another potential that gives me the same answer? At this point, we don't know. Maybe there is another potential. In fact, you can do all sorts of things to make a potential that's very arbitrarily close that gives you an arbitrarily similar profile. So if they're significantly different, how different do they look? So then the students say, well, I don't know. What about a double well potential? So, in fact, we'll be doing a double well potential on the problem set. A good question. And we certainly haven't proven anything like this is the only potentially that gives you this transmission amplitude. What we can say is if you get transmission amplitude that looks like this, it's probably pretty reasonable to say it's probably well modeled. We seem to be reproducing the data reasonably well. So it's not a proof of anything, but it's a nice model. And we're physicists, we build models. We don't tell you what's true, we tell you what are good models. If a theorist ever walks up on you and he says, here's the truth, punch him in the gut. That's not how it works. Experimentalists, on the other hand. We'll just punch them too, I guess. [LAUGHTER] PROFESSOR: We build models. OK, so this leaves us with an obvious question, which we've answer in the case of energy grade in the potential. What about the case of the energy less than the potential. So we haven't filled out that part of the graph. Let's fill out that part of the graph. What happens if the energy is less than the potential? And for this, I'm going to use the trick that I mentioned earlier that, look, if the energy is less than the potential, that just means that in the intervening regime, in here, instead of being oscillatory, it's going to be exponentially growing [INAUDIBLE] because we're in a classically disallowed region. So if the energy is less than v0, here we have instead of ikx, we have minus alpha x. And instead of minus ikx we have plus alpha x. So if you go through that whole analysis and plug in those values for the k prime, the answer you get is really quite nice. We find that t for energy less than v0 is equal to-- and it's just a direct analytic continuation of what you get here-- 1 plus 1 over 4 epsilon 1 minus epsilon. So epsilon is less than 1 now. 1 sinh squared of g0 1 minus epsilon under 1. OK, so now this lets us complete the plot. And you can check, it's pretty straightforward, that because sinh, again, goes like its argument at very small values of its argument, we get g0 squared 1 minus epsilon. Sorry, this should be square root 1 minus epsilon. Really? Is that a typo? Oh no, it's in there. OK, good. So again, we get g0 squared 1 minus epsilon, the 1 minus epsilon cancels, and we get 1 over 4 g0 squared. Which is good, because if they disagreed, we'd be in real trouble. So they agree, but the sinh squared is just a strictly-- or that function is just a strictly decreasing function as we approach epsilon goes to 0. It goes mostly to 0. So this is what we see. We see an exponential region and then we see oscillations with residences where the period is getting wider and wider. But this should trouble you a little bit. Here, what are we saying? We're saying look, if we have a barrier, and we send in a particle with energy way below the barrier, that's kind of troubling. If we send in a particle with energy way below the barrier, there's some probability that it gets out. It goes across. So for e less than v0, classically no transmission, we get transmission. Now, do we get resonances when e is less than v0? No, that's an interesting thing. We'll talk about resonances and where they come from, and we'll talk about the more generalized notion of a nest matrix in the next lecture. But for now I just want to say a couple of things about it. So I want to ask-- so this is called tunneling, this transmission across a disallowed barrier, a classically disallowed barrier-- so this transmission across a disallowed barrier is called tunneling. And just a quick thing to notice is that if we hold e fixed and we vary l, we vary the width of the barrier, how does the tunneling amplitude depend on the width of the barrier? At fixed energy, if we vary the width of the barrier-- which is only contained in g0-- how does the transmission amplitude vary? And we find that for large l-- for l much greater than 1 for a typical scale in the problem-- the tunneling amplitude goes like e to the minus 2 alpha l where alpha just depends on the energy in the potential. So what we see is that the probability of transmitting, of tunneling through a wall, depends on the width of that wall. And it depends on it exponentially. The wider the wall, the exponentially less likely you are to tunnel across it. And this fits the simulation we ran where we saw that tunnelling through a very thin wall was actually quite efficient. And we also saw that tunnelling across a wall can have resonant peaks. At special values, it transmits perfectly. We saw that in the simulations. So at this point I can't resist telling you a very short story. I have a very good friend who is not a physicist, but who is a professor at MIT. So a smart person, and very smart. She's one of the smartest people I know. She broke her ankle. She's always losing stuff, she's constantly losing stuff. And she broke her ankle, and she was going to the rehab place, and she parked her car right in front of the rehab place. She parked her car right in front because she's got the broken ankle and she doesn't want to have to walk a long way to get the car, she parked right in front. She went in, she did her two hours of rehab, and she came outside, and her car was gone. And she's always misplacing her car, she's always misplacing her keys, she's always losing everything. But this time, this time she knew where it was. And it wasn't there. So first she thought, oh crap, my car's been stolen. How annoying, I've got a broken ankle. But she looks around, and there's her car on the other side of the street pointed the opposite direction. And she's like, finally. Finally. She was explaining this to me and a friend of mine who is also a physicist, she said finally, I knew I had incontrovertible proof that quantum effects happen to macroscopic objects. My car tunnelled across the street. [LAUGHTER] PROFESSOR: At this point, of course, my friend and I are just dying of laughter. But she said my car tunnelled across the street. But here's the problem, I couldn't tell anyone. Because if I told anyone, they would know that I was crazy. Clearly I'm crazy, because I lose my stuff all the time, but surely it's just gotten misplaced. So I went home, and I got home and I was sitting down to dinner with my partner and our daughter and I was burning up inside because I wanted to tell them this crazy thing happened. I know quantum mechanics, it happens. I couldn't tell them anything, so I was just fidgeting and dying. And finally my daughter said, mom, didn't you notice that we moved the car across the street? [LAUGHTER] PROFESSOR: See you next time.
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_17_More_on_Central_Potentials.txt
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The following content is provided under a Creative Commons License. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right, shall we get started? So, today-- well, before I get started-started-- so, let me open up to questions. Do y'all have questions from the last lecture, where we finished off angular momentum? Or really anything up to the last exam? Yeah? AUDIENCE: So, what exactly happens with the half l states? PROFESSOR: Ha, ha, ha! What happens with the half l states? OK, great question! So, we're gonna talk about that in some detail in a couple of weeks, but let me give you a quick preview. So, remember that when we studied the commutation relations, Lx, Ly is i h bar Lz . Without using the representation in terms of derivatives, with respect to a coordinate, without using the representations, in terms of translations and rotations along the sphere, right? When we just used the commutation relations, and nothing else, what we found was that the states corresponding to these guys, came in a tower, with either one state-- corresponding to little l equals 0-- or two states-- with l equals 1/2-- or three states-- with little l equals 1-- or four states-- with l equals 3/2-- and so on, and so forth. And we quickly deduced that it is impossible to represent the half integer states with a wave function which represents a probability distribution on a sphere. We observed that that was impossible. And the reason is, if you did so, then when you take that wave function, if you rotate by 2pi-- in any direction-- if you rotate by 2pi the wave function comes back to minus itself. But the wave function has to be equal to itself at that same point. The value of the wave function at some point, is equal to the wave function at some point. That means the value of the wave function must be equal to minus itself. That means it must be zero0. So, you can't write a wave function-- which is a probability distribution on a sphere-- if the wave function has to be equal to minus itself at any given point. So, this is a strange thing. And we sort of said, well, look, these are some other beasts. But the question is, look, these furnish perfectly reasonable towers of states respecting these commutation relations. So, are they just wrong? Are they just meaningless? And what we're going to discover is the following-- and this is really gonna go back to the very first lecture, and so, we'll do this in more detail, but I'm going to quickly tell you-- imagine take a magnet, a little, tiny bar magnet. In fact, well, imagine you take a little bar magnet with some little magnetization, and you send it through a region that has a gradient for magnetic field. If there's a gradient-- so you know that a magnet wants to anti-align with the nearby magnet, north-south wants to go to south-north. So, you can't put a force on the magnet, but if you have a gradient of a magnetic field, then one end a dipole-- one end of your magnet-- can feel a stronger effective torque then the other guy. And you can get a net force. So, you can get a net force. The important thing here, is that if you have a magnetic field which has a gradient, so that you've got some large B, here, and some smaller B, here, then you can get a force. And that force is going to be proportional to how big your magnet is. But it's also going to be proportional to the magnetic field. And if the force is proportional to the strength of your magnet, then how far-- if you send this magnet through a region, it'll get deflected in one direction or the other-- and how far it gets deflected is determined by how big of a magnet you sent through. You send in a bigger magnet, it deflects more. Everyone cool with that? OK, here's a funny thing. So, that's fact one. Fact two, suppose I have a system which is a charged particle moving in a circular orbit. OK? A charged particle moving in a circular orbit. Or better yet, well, better yet, imaging you have a sphere-- this is a better model-- imagine you have a sphere of uniform charge distribution. OK? A little gelatinous sphere of uniform charge distribution, and you make it rotate, OK? So, that's charged, that's moving, forming a current. And that current generates a magnetic field along the axis of rotation, right? Right hand rule. So, if you have a charged sphere, and it's rotating, you get a magnetic moment. And how big is the magnetic moment, it's proportional to the rotation, to the angular momentum, OK? So, you determine that, for a charged sphere here which is rotating with angular momentum, let's say, l, has a magnetic moment which is proportional to l. OK? So, let's put this together. Imagine we take a charged sphere, we send it rotating with same angular momentum, we send it through a field gradient, a gradient for magnetic field. What we'll see is we can measure that angular momentum by measuring the deflection. Because the bigger the angular momentum, the bigger the magnetic moment, but the bigger the magnetic moment, the bigger the deflection. Cool? So, now here's the cool experiment. Take an electron. And electron has some charge. Is it a little, point-like thing? Is it a little sphere? Is it, you know-- Let's not ask that question just yet. It's an electron. The thing you get by ripping a negative charge off a hydrogen atom. So, take your electron and send it through a magnetic field gradient. Why would you do this? Because you want to measure the angular momentum of this electron. You want to see whether the electron is a little rotating thing or not. So, you send it through this magnetic field gradient, and if it gets deflected, you will have measured the magnetic moment. And if you have measured the magnetic moment, you'll have measured the angular momentum. OK? Here's the funny thing, if the electron weren't rotating, it would just go straight through, right? It would have no angular momentum, and it would have no magnetic moment, and thus it would not reflect. Yeah? If it's rotating, it's gonna deflect. Here's the experiment we do. And here's the experimental results. The experimental results are every electron that gets sent through bends. And it either bends up a fixed amount, or it bends down a fixed amount. It never bends more, it never bends less, and it certainly never been zero. In fact, it always makes two spots on the screen. OK? Always makes two spots. It never hits the middle. No matter how you build this experiment, no matter how you rotate it, no matter what you do, it always hits one of two spots. What that tells you is, the angular momentum-- rather the magnetic moment-- can only take one of two values. But the angular momentum is just some geometric constant times the angular momentum. So, the angular momentum must take one of two possible values. Everyone cool with that? So, from this experiment-- glorified as the Stern Gerlach Experiment-- from this experiment, we discover that the angular momentum, Lz, takes one of two values. L, along whatever direction we're measuring-- but let's say in the z direction-- Lz takes one of two values, plus some constant and, you know, plus h bar upon 2, or minus h bar upon 2. And you just do this measurement. But what this tells us is, which state? Which tower? Which set of states describe an electron in this apparatus? L equals 1/2. But wait, we started off by talking about the rotation of a charged sphere, and deducing that the magnetic moment must be proportional to the angular momentum. And what we've just discovered is that this angular momentum-- the only sensible angular momentum, here-- is the two state tower, which can't be represented in terms of rotations on a sphere. Yeah? What we've learned from this experiment is that electrons carry a form of angular momentum, demonstrably. Which is one of these angular momentum 1/2 states, which never doesn't rotate, right? It always carries some angular momentum. However, it can't be expressed in terms of rotation of some spherical electron. It has nothing to do with rotations. If it did, we'd get this nonsensical thing of the wave function identically vanishes. So, there's some other form of angular momentum-- a totally different form of angular momentum-- at least for electrons. Which, again, has the magnetic moment proportional to this angular momentum with some coefficient, which I'll call mu0. But I don't want to call it L, because L we usually use for rotational angular momentum. This is a different form of angular momentum, which is purely half integer, and we call that spin. And the spin satisfies exactly the same commutation relations-- it's a vector-- Sx with Sy is equal to ih bar Sz. So, it's like an angular momentum in every possible way, except it cannot be represented. Sz does not have any representation, in terms of h bar upon i [INAUDIBLE]. It is not related to a rotation. It's an intrinsic form of angular momentum. An electron just has it. So, at this point, you ask me, look, what do you mean an electron just has it? And my answer to that question is, if you send an electron through a Stern Gerlach Apparatus, it always hits one of two spots. And that's it, right? It's an experimental fact. And this is how we describe that experimental fact. And the legacy of these little L equals 1/2 states, is that they represent an internal form of angular momentum that only exists quantum mechanically, that you would have never noticed classically. That was a very long answer to what was initially a simple question. But we'll come back and do this in more detail, this was just a quick intro. Yeah? AUDIENCE: So, for L equals 3/2, does that mean that there's 4 values of spins? PROFESSOR: Yeah, that means there's [? 4 ?] values of spins. And so there are plenty of particles in the real world that have L equals 3/2. They're not fundamental particles, as far as we know. There are particles a nuclear physics that carry spin 3/2. There are all sorts of nuclei that carry spin 3/2, but we don't know of a fundamental particle. If super symmetry is true, then there must be a particle called a gravitino, which would be fundamental, and would have spin 3/2, and four states, but that hasn't been observed, yet. Other questions? AUDIENCE: Was the [? latter of ?] seemingly nonsensical states discovered first, and then the experiment explain it, or was it the experiment-- PROFESSOR: Oh, no! Oh, that's a great question. We'll come back the that at end of today. So today, we're gonna do hydrogen, among other things. Although, I've taken so long talking about this, we might be a little slow. We'll talk about that a little more when we talk about hydrogen, but it was observed and deduced from experiment before it was understood that there was such a physical quantity. However, the observation that this commutation relation led to towers of states with this pre-existed as a mathematical statement. So, that was a mathematical observation from long previously, and it has a beautiful algebraic story, and all sort of nice things, but it hadn't been connected to the physics. And so, the observation that the electron must carry some intrinsic form of angular momentum with one of two values, neither of which is 0, was actually an experimental observation-- quasi-experimental observation-- long before it was understood exactly how to connect this stuff. AUDIENCE: So it wasn't--? PROFESSOR: I shouldn't say long, it was like within months, but whatever. Sorry. AUDIENCE: The intent of the experiment wasn't to solve-- AUDIENCE: No, no. The experiment was this-- there are the spectrum-- Well, I'll tell you what the experiment was in a minute. OK, yeah? AUDIENCE: [INAUDIBLE] Z has to be plus or minus 1/2. What fixes the direction in the Z direction? PROFESSOR: Excellent. In this experiment, the thing that fixed the fact that I was probing Lz is that I made the magnetic field have a gradient in the Z direction. So, what I was sensitive to, since the force is actually proportionally to mu dot B-- or, really, mu dot the gradient of B, so, we'll do this in more detail later-- the direction of the gradient selects out which component of the angular momentum we're looking at. So, in this experiment, I'm measuring the angular momentum along this axis-- which for fun, I'll call Z, I could've called it X-- what I discover is the angular momentum along this axis must take one of two values. But, the universe is rotationally invariant. So, it can't possibly matter whether I had done the experiment in this direction, or done the experiment in this direction, what that tells you is, in any direction if I measure the angular momentum of the electron along that direction, I will discover that it takes one of two values. This is also true of the L equals 1 states. Lz takes one of three values. What about Lx? Lx also takes one of three values, those three values. Is is every system in a state corresponding to one of those particular values? No, it could be in a superposition. But the eigenvalues, are these three eigenvalues, regardless of whether it's Lx, or Ly, or Lz. OK, it's a good thing to meditate upon. Anything else? One more. Yeah? AUDIENCE: [INAUDIBLE] the last problem [INAUDIBLE]. PROFESSOR: Indeed. Indeed. OK. Since some people haven't taken the-- there will be a conflict exam later today, so I'm not going to discuss the exam yet. But, very good observation, and not an accident. OK, so, today we launch into 3D. We ditch our tricked-out tricycle, and we're gonna talk about real, physical systems in three dimensions. And as we'll discover, it's basically the same as in one dimension, we just have to write down more symbols. But the content is all the same. So, this will make obvious the reason we worked with 1D up until now, which is that there's not a heck of a lot more to be gained for the basic principles, but it's a lot more knowing to write down the expressions. So, the first thing I wanted to do is write down the Laplacian in three dimensions in spherical coordinates-- And that is a beautiful abuse of notation-- in spherical coordinates. And I want to note a couple of things. So, first off, this Laplacian, this can be written in the following form, 1 over r dr r quantity squared. OK, that's going to be very useful for us-- trust me on this one-- this is also known as 1 over r dr squared r. And this, if you look back at your notes, this is nothing other than L squared-- except for the factor of h bar upon i-- but if it's squared, it's minus 1 upon h bar squared. OK, so this horrible angular derivative, is nothing but L squared. OK, and you should remember the [? dd ?] thetas, and there are these funny sines and cosines. But just go back and compare your notes. So, this is an observation that the Laplacian in three dimensions and spherical coordinates takes this simple form. A simple radial derivative, which is two terms if you write it out linearly in this fashion, and one term if you write it this way, which is going to turn out to be useful for us. And the angular part can be written as 1 over r squared, times the angular momentum squared with a minus 1 over h bar squared. OK? So, in just to check, remember that Lz is equal to h bar upon i d phi. So, Lz squared is going to be equal to minus h bar squared d phi squared. And you can see that that's one contribution to this beast. But, actually, let me-- I'm gonna commit a capital sin and erase what I just wrote, because I don't want it to distract you-- OK. So, with that useful observation, I want to think about central potentials. I want to think about systems in 3D, which are spherically symmetric, because this is going to be a particularly simple class of systems, and it's also particularly physical. Simple things like a harmonic oscillator In three dimensions, which we solved in Cartesian coordinates earlier, we're gonna solve later, in spherical coordinates. Things like the isotropic harmonic oscillator, things like hydrogen, where the system is rotationally independent, the force of the potential only depends on the radial distance, all share a bunch of common properties, and I want to explore those. And along the way, we'll solve a toy model for hydrogen. So, the energy for this is p squared upon 2m, plus a potential, which is a function only of the radial distance. But now, p squared is equal to minus h bar squared times the gradient squared. But this is gonna be equal to, from the first term, minus h bar squared-- let me just write this out-- times r dr squared r. And then from this term, plus minus h bar squared times minus 1 over h bar squared [? to L squared ?] [? over ?] r squared, plus L squared over r squared. So, the energy can be written in a nice form. This is minus h bar squared, 1 upon r dr squared r-- whoops, sorry-- upon 2m, because it's p squared upon 2m. And from the second term, L squared over r squared upon 2m plus 1 over 2mr squared L squared plus u of r. OK, and this is the energy operator when the system is rotational invariant in spherical coordinates. Questions? Yeah? AUDIENCE: [INAUDIBLE] is that an equals sign or minus? PROFESSOR: This? AUDIENCE: Yeah. PROFESSOR: Oh, that's an equals sign. So, sorry. This is just quick algebra. So, it's useful to know it. So, consider the following thing, 1 over r dr r. Why would you ever care about such a thing? Well, let's square it. OK, because I did there. So, what is this equal to? Well, this is 1 over r dr r. 1 over r dr r. These guys cancel, right? 1 over r times dr. So, this is equal to 1 over r dr squared r. But, why is this equal to dr squared plus 2 over r times dr? And the answer is, they're operators. And so, you should ask how they act on functions. So, let's ask how they act on function. So, dr squared plus 2 over r dr times a function-- acting as a function-- is equal to f prime prime-- if this is a function of r-- plus 2 over r f prime. On the other hand, 1 over r dr squared r, acting on f of r, well, these derivatives can hit either the r of the f. So, there's going to be a term where both derivatives hit f, in which case the rs cancel, and I get f prime prime. There's gonna be two terms where one of the d's hits this, one of the d's hits this, then there's the other term. So, there're two terms of that form. On d hits the r and gives me one, one d hits the f and gives me f prime. And then there's an overall 1 over r plus 2 over r f prime. And then there's a term were two d's hit the r, but if two d's hit the r, that's 0. So, that's it. So, these guys are equal to each other. So, why is this a particularly useful form? We'll see that in just a minute. So, I'm cheating a little bit by just writing this out and saying, this is going to be a useful form. But trust me, it's going to be a useful form. Yeah? AUDIENCE: Do we need to find d squared [INAUDIBLE] dr squared r. Isn't that supposed to be 1 over r? PROFESSOR: Oh shoot! Yes, that's supposed to be one of our-- Thank you. Thank you! Yes, over r. Thank you. Yes, thank you for that typo correction. Excellent. Thanks OK. So, anytime we have a system which is rotationally invariant-- whose potential is rotationally invariant-- we can write the energy operator in this fashion. And now, you see something really lovely, which is that this only depends on r, this only depends on r, this depends on the angular coordinates, but only insofar as it depends on L squared. So, if we want to find the eigenfunctions of E, our life is going to be a lot easier if we work in eigenfunctions of L. Because that's gonna make this one [? Ex ?] on an eigenfunction of L, this is just going to become a constant. So, now you have to answer the question, well, can we? Can we find functions which are eigenfunctions of E and of L, simultaneously? And so, the answer to that question is, well, compute the commutator. So, do these guys commute? In particular, of L squared. And, well, does L commute with the derivative with respect to r, L squared? Yeah, because L only depends on angular derivatives. It doesn't have any rs in it. And the rs don't care about the angular variables, so they commute. What about with this term? Well, L squared trivially commutes with itself and, again, r doesn't matter. And ditto, r and L squared commute. So, this is 0. These commute. So, we can find common eigenbasis. We can find a basis of functions which are eigenfunctions both of E and of L squared. So, now we use separation. In particular, if we want to find a function-- an eigenfunction-- of the energy operator, E phi E is equal to E phi E, it's going to simplify our lives if we also let phi be an eigenfunction of the L squared. But we know what the eigenfunctions of L squared are. E phi E is equal to-- let me write this-- of r will then be equal to little phi of r times yLm of theta and phi. Now, quickly, because these are the eigenfunctions of the L squared operator. Quick, is little l an integer or a half integer? AUDIENCE: [MURMURS] Integer. PROFESSOR: Why? AUDIENCE: [MURMURS] PROFESSOR: Yeah, because we're working with rotational angular momentum, right? And it only makes sense to talk about integer values of little l when we have gradients on a sphere-- when we're talking about rotations-- on a spherical coordinates, OK? So, little l has to be an integer. And from this point forward in the class, any time I write l, I'll be talking about the rotational angular momentum corresponding to integer values. And when I'm talking about the half integer values, I'll write down s, OK? So, let's use this separation of variables. And what does that give us? Well, l squared acting on yLm gives us h bar squared lL plus 1. So, this tells us that E, acting on phi E, takes a particularly simple form. If phi E is proportional to a spherical harmonic, then this is gonna take the form minus h bar squared upon 2m 1 over r dr squared r plus 1 over 2mr squared l squared-- but l squared acting on the yLm gives us-- h bar squared lL plus 1, which is just a constant over r squared plus u of r phi E. Question? AUDIENCE: Yeah. [INAUDIBLE] yLm1 and yLm2? PROFESSOR: Absolutely. So, can we consider superpositions of these guys? Absolutely, we can. However, we're using separation. So, we're gonna look at a single term, and then after constructing solutions with a single eigenfunction of L squared, we can then write down arbitrary superposition of them, and generate a complete basis of states. General statement about separation of variables. Other questions? OK. So, here's the resulting energy eigenvalue equation. But notice that it's now, really nice. This is purely a function of r. We've removed all of the angular dependence by making this proportional to yLm. So, this has a little phi yLm, and this has a little phi yLm, and nothing depends on the little phi. Nothing depends on the yLm-- on the angular variables-- I can make this phi of r. And if I want to make this the energy eigenvalue equation, instead of just the action of the energy operator, that is now my energy eigenvalue equation. This is the result of acting on phi with the energy operator, and this is the energy eigenvalue. Cool? So, the upside here is that when we have a central potential, when the system is rotationally invariant, the potential energy is invariant under rotations, then the energy commutes with the angular momentum squared. And so, we can find common eigenfunctions. When we use separation of variable, the resulting energy eigenvalue equation becomes nothing but a 1D energy eigenvalue equation, right? This is just a 1D equation. Now, you might look at this and say, well, it's not quite a 1D equation, because if this were a 1D equation, we wouldn't have this funny 1 over r, and this funny r, right? It's not exactly what we would have got. It's got the minus h bar squareds upon 2m-- whoops, and there's, yeah, OK-- it's got this funny h bar squareds upon 2m, and it's got these 1 over-- or sorry,-- it's got the correct h bar squareds upon 2m, but it's got this funny r and 1 over r. So, let's get rid of that. Let's just quickly dispense with that funny set of r. And this comes back to the sneaky trick I was referring to earlier, of writing this expression. So, rather than writing this out, it's convenient to write it in this form. Let's see why. So, if we have the E phi of r is equal to minus h bar squared upon 2m, 1 over r d squared r r, plus-- and now, what I'm gonna write is-- look, this is our potential, u of r. This is some silly, radial-dependent thing. I'm gonna write these two terms together, rather than writing them over, and over, and over again, I'm going to write them together, and call them V effective. Plus V effective of r, where V effective is just these guys, V effective. Which has a contribution from the original potential, and from the angular momentum, which, notice the sign is plus 1 over r squared. So, the potential gets really large as you get to the origin. Phi of r. So, this r is annoying, and this 1 over r is annoying, but there's a nice way to get rid of it. Let phi of r-- well, this r, we want to get rid of-- so, let phi of r equals 1 over r u of r. OK, then 1 over r squared-- or sorry, 1 over r-- dr squared r phi is equal to 1 over r dr squared r times 1 over r times u, which is just u. But meanwhile, V on phi is equal to-- well, V doesn't have any r derivatives, it's just a function-- so, V of phi is just 1 over r V on u. So, this equation becomes E on u, because this also picks up a 1 over r, is equal to minus h bar squared upon 2m dr squared plus V effective of r u of r. And this is exactly the energy eigenvalue equation for a 1D problem with the following potential. The potential, V effective of r, does the following two things-- whoops, don't want to draw it that way-- suppose we have a potential which is the Coulomb potential. So, let's say, u is equal to minus E squared upon r. Just as an example. So, here's r, here is V effective. So, u first, so there's u-- u of r, so let me draw this-- V has another term, which is h bar squared lL plus 1 over 2mr squared. This is for any given value of l. This is a constant over r squared, with a plus sign. So, that's something that looks like this. This is falling off like 1 over r, this is falling off like 1 over r squared. So, it falls off more rapidly. And finally, can r be negative? No. It's defined from 0 to infinity. So, that's like having an infinite potential for negative r. So, our effective potential is the sum of these contributions-- wish I had colored chalk-- the sum of these contributions is going to look like this. So, that's my V effective. This is my Ll plus 1 [INAUDIBLE] squared over 2mr squared. And this is my u of r. Question? AUDIENCE: [INAUDIBLE]. PROFESSOR: Good. OK, so this is u of r, the original potential. AUDIENCE: OK. PROFESSOR: OK? This is 1 over L squared-- or sorry-- lL 1 over 2mr squared. AUDIENCE: [INAUDIBLE]. PROFESSOR: Oh shoot! Oh, I'm sorry! I'm terribly sorry! I've abused the notation terribly. Let's-- Oh! This is-- Crap! Sorry. This is standard notation. And in text, when I write this by hand, the potential is a big U, and the wave function is a little u. So, let this be a little u. OK, this is my little u and so, now I'm gonna have to-- oh jeez, this is horrible, sorry-- this is the potential, capital U with a bar underneath it. OK, seriously, so there's capital U with a bar underneath it. And here's V, which is gonna make my life easier, and this is the capital U with the bar underneath it. Capital U with the bar underneath it. Oh, I'm really sorry, I did not realize how confusing that would be. OK, is everyone happy with that? Yeah? AUDIENCE: [INAUDIBLE]. PROFESSOR: Which one? AUDIENCE: Middle. Middle. PROFESSOR: Middle. AUDIENCE: Up, up. Right there! Up! There. PROFESSOR: Where? AUDIENCE: To the right. [CHATTER] Near the eraser mark. [LAUGHTER] PROFESSOR: So, these are the wave function. AUDIENCE: I know. PROFESSOR: That's the wave function. That is V. AUDIENCE: [CHATTER] PROFESSOR: Wait, if I erased, how can I correct it? AUDIENCE: [CHATTER] There! PROFESSOR: Excellent, so the thing that isn't here, would have a bar under it. Oh, oh, oh, oh, sorry! Ah! You wouldn't think it would be so hard. OK, good. And this is not [? related ?] to the wave function. OK, god, oh! That's horrible! Sorry guys, that notation is not obvious. My apologies. Oh, there's a better way to do this. OK, here's the better way to do this. Instead of calling the potential-- I'm sorry, your notes are getting destroyed now-- so instead of calling potential capital U, let's just call this V. Yeah. AUDIENCE: [LAUGHTER] No! PROFESSOR: And then we have V effective. No, no. This is good. This is good. We can be careful about this. So, this is V. This is V effective, which has V plus the angular momentum term. Oh, good Lord! This is V effective. This is V. V phi [INAUDIBLE] U. Good, this is V. AUDIENCE: [INAUDIBLE] There's no U-- PROFESSOR: There's no U underline, it's now just V, V effective. Oh! Good Lord! OK, wow! That was an unnecessary confusion. AUDIENCE: Top right. PROFESSOR: Top right. AUDIENCE: There is no bar. [? PROFESSOR: Mu. ?] AUDIENCE: Is that V or V effective? PROFESSOR: That's V. Although, it would've been just as true as V effective. So, we can write V effective. It's true for both. Because it's just a function of r. Oh, for the love of God! OK. Let's check our sanity, and walk through the logic. So, the logic here is, we have some potential, it's a function only of r, yeah? As a consequence, since it doesn't care about the angles, we can write things in terms of the spherical harmonics, we can do separation of variables. Here's the energy eigenvalue equation. We discover that because we're working in spherical harmonics, the angular momentum term becomes just a function of r, with no other coefficients. So, now we have a function of r plus the potential V, this looks like an effective potential, V effective, which is the sum of these two terms. So, there's that equation. On the other hand, this is tantalizingly close to but not quite the energy eigenvalue equation for a 1D problem with this potential, V effective. To make it obvious that it's, in fact, a 1D problem, we do a change of variables, phi goes to 1 over ru, and then 1 upon r d squared r phi becomes 1 over r d squared u, and V effective phi becomes 1 over r V effective u. Plugging that together, gives us this energy eigenvalue equation for u, the effective wave function, which is 1d problem. So, we can use all of our intuition and all of our machinery to solve this problem. And now we have to ask, what exactly is the effective potential? And the effective potential has three contributions. First, it has the original V, secondly, it has the angular momentum term, which is a constant over r squared-- and here is that, constant over r squared-- and the sum of these is the effective. And this guy dominates because it's 1 over r squared. This dominates at small r, and this dominates at large r if it's 1 over r. So, we get an effective potential-- that I'll check-- there's the effective potential. And finally, the third fact is that r must be strictly positive, so as a 1D problem, that means it can't be negative, it's gotta have an infinite potential on the left. So, as an example, let's go ahead and think more carefully about specifically this problem, about this Coulomb potential, and this 1D effective potential. AUDIENCE: Professor? PROFESSOR: Yeah? AUDIENCE: [INAUDIBLE]? PROFESSOR: Yes? AUDIENCE: Where does the 1 over r go? PROFESSOR: Good. So, remember the ddr squared term gave us a 1 over r out front. So, from this term, there should be 1 over r, here. From this term, there should also be a 1 over r. And from here, there should be a 1 over r. AUDIENCE: Ah! PROFESSOR: So, then I'm gonna cancel the 1 over r by multiplying the whole equation by r. Yeah? Sneaky, sneaky. So, any time you see-- any time, this is a general lesson-- anytime you see a differential equation that has this form-- two derivatives, plus 1 over r a derivative-- you know you can play some game like this. If you see this, declare in your mind a brief moment of triumph, because you know what technique to use. You can do this sort of rescaling by a power of r. And more generally, if you have a differential equation that looks like-- let me do this here-- if you have a differential equation that looks something like a derivative with respect to r plus a constant over r times phi, you know how to solve this. Let me say plus dot, dot, dot phi. You know how to solve this because ddr plus c over r means that phi, if there were nothing else, equals zero. If there were no other terms here, then this would say, ddr plus c over r is phi, that means when you take a derivative it's like dividing by r and multiplying by c. That means that phi goes like r to the minus c, right? But if phi goes like r to the minus c, that's not the exact solution to the equation, but I can write phi is equal to r to the minus c times u. And then this equation becomes ddr plus dot, dot, dot u equals zero. OK? Very useful little trick-- not really a trick, It's just observation-- and this is the second order version of the same thing. Very useful things to have in your back pocket for moments of need. OK? So, let's pick up with this guy. So, let me give you a little name for this. So, this term that comes from the angular momentum [? bit, ?] this originally came from the kinetic energy, right? It came from the L squared over r, which was from the gradient squared energy. This is a kinetic energy term. Why is there a kinetic energy term? Well, what this is telling you is that if you have some angular momentum-- if little l is not equal to 0--- then as you get closer and closer to the origin, the potential energy is getting very, very large. And this should make sense. If you're spinning, and you pull in your arms, you have to do work, right? You have to pull those guys in. You speed up. You're increasing your kinetic energy due to conservation of angular momentum, right? If you have rotationally invariance, as you bring in your hand you're increasing the kinetic energy. And so, this angular momentum barrier is just an expression of that. It's just saying that as you come to smaller and smaller radius, holding the angular momentum fixed, your velocity-- your angular velocity-- must increase-- your kinetic energy must increase-- and we're calling that a potential term just because we can. Because we've worked with definite angular momentum, OK? You should have done this in classical mechanics as well. Well, you should have done it in classical mechanics. So, this is called the angular momentum barrier. Quick question, classically, if you take a charged particle around in a Coulomb potential, classically that system decays, right? Irradiates away energy. Does the angular momentum barrier save us from decaying? Is that why hydrogen is stable? No one wants to stake a claim here? Is hydrogen stable because of conversation of angular momentum? AUDIENCE: No. PROFESSOR: No. Absolutely not, right? So, first off, in your first problems set, when you did that calculation, that particle had angular momentum. So, and if can radiate that away through electromagnetic interactions. So, that didn't save us. Angular momentum won't save us. Another way to say this is that we can construct-- and we just explicitly see-- we can construct a state with which has little l equals 0. In which case the angular momentum barrier is 0 over r squared, because there's nothing. Angular momentum barrier's not what keeps you from decaying. And the reason is that the electron can radiate away energy and angular momentum, and so l will decrease and decrease, and can still fall down. So, we still need a reason for why the hydrogen system, quantum mechanically, is stable. [? Why do ?] [? things exist? ?] So, let's answer that question. So, what I want to do now is, I want to solve-- do I really want to do it that way?-- well, actually, before we do, let's consider some last, general conditions. General facts for central potentials. So, let's look at some general facts for central potentials. So, the first is, regardless of what the [? bare ?] potential was, just due to the angular momentum barrier, we have this 1 over r squared behavior near the origin. So, we can look at this, we can ask, look, what are the boundary conditions at the origin? What must be true of u of r near the origin? Near u of r-- or sorry, near r goes to zero-- what must be true of u of r? So, the right way to ask this question is not to look at this u of r, which is not actually the wave function, but to look at the actual wave function, phi sub E, which goes near r equals 0, like u of r over r. So, what should be true of u? Can u diverge? Is that physical? Does u have to vanish? Can it take a constant value? So, I've given you a hint by telling you that I want to think about there being an infinite potential, but why? Why is that the right thing to do? Well, imagine u of r went to a constant value near the origin. If u of r goes to a constant value near the origin, then the wave function diverges near the origin. That's maybe not so bad, maybe it has a 1 over r singularity. It's not totally obvious that that's horrible. What's so bad about having a 1 over r behavior? So, suppose u goes to a constant. So, phi goes to constant over r. What's so bad about this? So, let's look back at the kinetic energy. P is equal t-- the kinetic energy is gonna be minus h bar squared p squared-- so the energy is going to go like, p squared over 2md squared. But here's an important fact, d squared-- the Laplacian-- of 1 over r, well, it's easy to see what this is at a general point. At a general point, d squared has a term that looks like 1 over rd squared r r. So, 1 over rd squared r on 1 over r. Well, r times 1 over r, that's just 1. And this is 0, right? So, the gradient squared of 1 over r, is 0. Except, can that possibly be true at r equals 0? No, because what's the second derivative at 0? As you approach the origin from any direction, the function is going like 1 over r, OK, so it's growing, but it's growing in every direction. So, what's its first derivative at the origin? It's actually ill-defined, because it depends on the direction you come in. The first direction coming in this way, the derivative looks like it's becoming this, from this direction it's becoming this, it's actually badly divergent. So, what's the second derivative? Well, the second derivative has to go as you go across this point, it's telling you how the first derivative changes. But it changes from plus infinity in this direction, to plus infinity in this direction. That's badly singular. So, this can't possibly be true, what I just wrote down here. And, in fact, d squared on 1 over r-- and this is a very good exercise for recitation-- is equal to delta of r. It's 0-- it's clearly 0 for r0 equals 0--- but at the origin, it's divergent. And it's divergent in exactly the way you need to get the delta function. OK, which is pretty awesome. So, what that tells us is that if we have a wave function that goes like 1 over r, then the energy contribution-- energy acting on this wave function-- gives us a delta function at the origin. So, unless you have the potential, which is a delta function at the origin, nothing will cancel this off. You can't possibly satisfy the energy eigenvalue equation. So, u of r must go to 0 at r goes to 0. Because if it goes to a constant-- any constant-- we've got a bad divergence in the energy, yeah? In particular, if we calculate the energy, we'll discover that the energy is badly divergent. It does become divergent if we don't have u going to 0. So, notice, by the way, as a side note, that since phi goes like, phi is equal to u over r, that means that phi goes to a constant. This is good, because what this is telling us is that the wave function-- So, truly, u is vanishing, but the probability density, which is the wave function squared, doesn't have to vanish. That's about the derivative of u, as you approach the origin from [? Lucatau's ?] Rule. So, this is the first general fact about central potential. So, the next one-- and this is really fun one-- good Lord! Is that, note--- sorry, two more-- the energy depends on l but not on m. Just explicitly, in the energy eigenvalue equation, we have the angular momentum showing up int the effective potential, little l. But little m appears absolutely nowhere except in our choice of spherical harmonic. For any different m-- and this was pointing out before-- for any different m, we would've got the same equation. And that means that the energy eigenvalue can depend on l, but it can't depend on m, right? So, that means for each m in the allowed possible values, l, l minus 1, [? i ?] minus l-- and this is 2l plus 1 possible values-- for each of these m's, the energy is the same. And I'll call this E sub l, because the energy can depend on l. Why? The degeneracy of E sub L is equal to 2l plus 1. Why? Why do we have this degeneracy? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah, exactly. We get the degeneracies when we have symmetries, right? When we have a symmetry, we get a degeneracy. And so, here we have a degeneracy. And this degeneracy isn't fixed by rotational invariance. And why is this the right thing? Rotational symmetry? So why, did this give us this degeneracy? But what the rotational degeneracy is saying is, look, if you've got some total angular momentum, the energy can't possibly depend on whether most of it's in Z, or most of it's in X, or most of it's and Y. It can't possibly depend on that, but that's what m is telling you. M is just telling you what fraction is contained in a particular direction. So, rotational symmetry immediately tells you this. But there's a nice way to phrase this, which of the following, look, what is rotational symmetry? Rotational symmetry is the statement that the energy doesn't care about rotations. And in particular, it must commute with Lx, and with Ly, and with Lz. So, this is rotationally symmetry. And I'm going to interpret these in a nice way. So, this guy tells me I can find common eigenfunctions. And, more to the point, a full common eigenbasis of E and Lz. Can I also find a common eigenbasis of ELz and Ly? Are there common eigenvectors of ELz and Ly? AUDIENCE: [CHATTER] PROFESSOR: No. Are there common eigenfunctions of Lz and Ly? AUDIENCE: No. PROFESSOR: No, because they don't commute, right? E commutes with each of these. OK, so, I'm just going to say, I'm gonna pick a common eigenbasis of E and Lz-- but I could've picked Lx, or I could've picked Ly, I'm just picking Lz because that's our convention-- but what do these two-- Once I've chosen this-- I'm gonna work with a common eigenbasis of E and Lz-- what do these two commutators tell me? These two commutators tell me that E commutes with Lx plus iLy and Lx minus iLy, L plus/minus. So, this tells you that if you have an eigenfunctions of E, and you act with a raising operator, you get another eigenfunction of E. And thus, we get our 2L plus 1 degeneracy, because we can walk up and down the tower using L plus and L minus. Cool? OK. So, this is a nice example that when you have a symmetry you get a degeneracy, and vice versa. OK. So, let's do some examples of using these central potentials. AUDIENCE: Professor? PROFESSOR: Yeah AUDIENCE: [INAUDIBLE]? PROFESSOR: It's 0. So, E with Lx is 0. E with Ly-- So, are you happy with that statement? That E with Lx is 0? AUDIENCE: Yeah. PROFESSOR: Yeah. Good. OK, and so this 0 because this is just Lx plus iOi. So, E with Lx is 0, and E with Ly is 0, so E commutes with these guys. And so, this is like the statement that L squared with L plus/minus equals 0. AUDIENCE: [INAUDIBLE]. PROFESSOR: Cool. OK, so, let's do some examples. So, the first example is gonna be-- actually, I'm going to skip this spherical well example-- because it's just not that interesting, but it's in the notes, and you really need to look at it. Oh hell, yes, I'm going to do it. OK, so, the spherical well. So, I'm going to do it in an abridged form, and maybe it's a good thing for recitation. AUDIENCE: Professor? PROFESSOR: Thank you recitation leader. So, in this spherical well, what's the potential? So, here's v of r. Not U bar, and not V effective, just v or r. And the potential is going to be this, so, here's r equals 0. And if it's a spherical infinite well, then I'm gonna say, the potential is infinite outside of some distance, l. OK? And it's 0 inside. So, what does this give us? Well, in order to solve the system, we know that the first thing we do is we separate out with yLms, and then we re-scale by 1 over r to get the function of u, and we get this equation, which is E on u is equal to minus h bar squared upon 2m dr squared and plus v effective-- well, plus [? lL ?] plus 1-- over r squared with a 2m and an h bar squared. And the potential is 0, inside. So we can just write this. So, if you just-- let me pull out the h bar squareds over 2m-- it becomes minus [INAUDIBLE] plus 1 over r squared. So, this is not a terrible differential equation. And one can do some good work to solve it, but it's a harder differential equation than I want to spend the time to study right now, so I'm just going to consider the case-- special case-- when there's zero angular momentum, little l equals 0. So, in the special case of a l equals 0, E-- and I should call this u sub l-- Eu sub 0 is equal to h bar squared upon 2m. And now, this term is gone-- the angular momentum barrier is gone-- because there's no angular momentum, dr squared ul. Which can be written succinctly as ul-- or sorry, u0-- prime prime, because this is only a function of r. So now, this is a ridiculously easy equation. We know how to solve this equation, right? This is saying that the energy, a constant, times u is two derivatives times this constant. So, u0 can be written as a cosine of kx-- or sorry-- kr plus b sine of kr, where h bar squared k squared upon 2m equals E. And I should really call this E sub 0, because it could depend on little l, here. So, there's our momentary solution, however, we have to satisfy our boundary conditions, which is that it's gotta vanish at the origin, but it's also gotta vanish at the wall. So, the boundary conditions, u of 0 equals 0 tells us that a must be equal to 0, and u of l equals 0 tells us that, well, if this is 0, we've just got B, but sine of kr evaluated at l, which is sine of kl, must be equal to 0. So, kl must be the 0 of sine, must be n pi over-- must be equal to n pi, a multiple of pi. And so, this tells you what the energy is. So, this is just like the 1D system. It's just exactly like when the 1D system. So now, to finally close this off. What does this tell you that the eigenfunctions are? And let me do that here. So, therefore, the wave function phi sub E0 of r theta and phi-- oh god, oh jesus, this is so much easier in [INAUDIBLE] so, phi [INAUDIBLE] 0 of r theta and phi is equal to y0m. But what must m be? 0, because m goes from plus L to minus L, 0. I'm just [INAUDIBLE] the argument. Y00 times, not u of r, times 1 over r times u. 1 over r times u of r. But u of r is a constant times sine of kr. Sine of kr, but k is equal to n pi over L. N pi over Lr. And what's Y00? It's a constant. And so, there's an overall normalization constant, that I'll call n. OK, so, we get that our wave function is 1 over r times sine of n pi over Lr. So, this looks bad. There's a 1 over r. Why is this not bad? At the origin, why is this not something I should worry about it? AUDIENCE: [MURMURS] PROFESSOR: Yeah, because sine is linear, first of all, [INAUDIBLE] argument. So, this goes like, n pi over L times r. That r cancels the 1 over r. So, near the origin, this goes like a constant. Yeah? So, u has to 0, but the wave function doesn't. Cool? OK. So, this is a very nice more general story for larger L, which I hope you see in the recitation. OK. Questions on the spherical well? The whole point here-- Oh, yeah, go. AUDIENCE: What do [INAUDIBLE] generally [INAUDIBLE]? PROFESSOR: That's true. So, good. So, let me rephrase the question, and tell me if this is the same question. So, this is strange. There's nothing special about the origin. So, why do I have a 0 at the origin? Is that the question? AUDIENCE: Yeah. PROFESSOR: OK. It's true. There's nothing special about the origin, except for two things. One thing that's special about the origin is we're working in a system which has a rotational symmetry. But rotational symmetry is rotational symmetry around some particular point. So, there's always a special central point anytime you have a rotational symmetry. It's the point fixed by the rotations. So, actually, the origin is a special point here. Second, saying that little u has a 0 is not the same as saying that the wave function has a 0. Little u has a 0, but it gets multiplied by 1 over r. So, the wave function, in fact, is non-zero, there. So, the physical thing is the probability distribution, which is the [? norm ?] squared of the wave function. And it doesn't have a 0 at the origin. Does that satisfy? AUDIENCE: Yes. PROFESSOR: OK. So, the origin is special when you have a central potential. That's where the proton is, right? Right, OK. So, there is something special about the origin. Wow, that was a really [? anti-caplarian ?] sort of argument. OK, so that's where the proton-- so, there is something special about the origin, and the wave function doesn't vanish there, even if u does. It may vanish there, but it doesn't necessarily have to. And we'll see that in a minute. Other questions? Yeah? AUDIENCE: So, what again, what's the reasoning for saying that the u of r has to vanish at [? 0 instead ?] [? of L? ?] PROFESSOR: Good. The reason that u of r had to vanish at the origin is that if it doesn't vanish at the origin, then the wave function diverges-- whoops, phi goes to constant-- if u doesn't go to 0, if it goes to any constant, non-zero, then the wave function diverges. And if we calculate the energy, we get a delta function at the origin. So, there's an infinite contribution of energy at the origin. That's not physical. So, in order to get a sensible wave function with finite energies, we need to have the u vanishes, because of the 1 over u. And the reason that we said it had to vanish at l, was because I was considering this spherical well-- spherical infinite well-- where a particle is stuck inside a region of radius, capital L, and that's just what I mean by saying I have an infinite potential. AUDIENCE: OK, thanks. PROFESSOR: Cool? Yeah. Others? OK. So, with all that done, we can now do the hydrogen-- or the Coulomb-- potential. And I want to emphasize that we often use the following words when-- people often use the following words when solving this problem-- we will now solve the problem of hydrogen. This is false. I am not about to solve for you the problem of hydrogen. I am going to construct for you a nice toy model, which turns out to be an excellent first pass at explaining the properties observed in hydrogen gases, their emission spectra, and their physics. This is a model. It is a bad model. It doesn't fit the data. But it's pretty good. And we'll be able to improve it later. OK? So, it is the solution of the Coulomb potential. And what I want to emphasize to you, I cannot say this strongly enough, physics is a process of building models that do a good job of predicting. And the better their predictions, the better the model. But they're all wrong. Every single model you ever get from physics is wrong. There are just some that are less stupidly wrong. Some are a better approximation to the data, OK? This is not hydrogen. This is going to be our first pass at hydrogen. It's the Coulomb potential. And the Coulomb potential, V of r, is equal to minus e squared over r. This is what you would get if you had a classical particle with infinite mass and charge plus b. And then another particle over here, with mass, little m, and charge, minus e. And you didn't pay too much attention to things like relativity, or spin, or, you know, lots of other things. And you have no background magnetic field, or electric field, and anything else. And if these are point particles, and-- All of those things are false that I just said. But if all those things were true, in that imaginary universe, this would be the salient problem to solve. So, let's solve it. Now, are all those things that I said that were false-- the proton's a point particle, the proton is infinitely massive, there's no spin-- are those preposterously stupid? AUDIENCE: No. PROFESSOR: No, they're excellent approximations in a lot of situations. So, they're not crazy wrong. They're just not exactly correct. I want to keep this in your mind. These are gonna be good models, but they're not exact. So, we're not solving hydrogen, we're gonna solve this idealized Coulomb potential problem. OK, so let's solve it. So, if V is minus e over r squared, then the equation for the rescaled wave function, u, becomes minus h bar squared upon 2mu prime prime of r plus the effective potential, which is h bar squared upon 2mll plus 1 over r squared minus e squared over r u is equal to e sub l u. So, there's the equation we want to solve. We've already used separation of variables, and we know that the wave function is this little u times 1 over r times yLm, for some l and some m. So, the first thing we should do any time you're solving an interesting problem, the first thing you should do is do dimensional analysis. And if you do dimensional analysis, the units of e squared-- well, this is easy-- e squared must be an energy times a length. So, this is an energy times a length. Also known as p squared l, momentum squared over 2m, 2 times the mass times the length. It's useful to put things in terms of mass, momentum, and lengths, because you can cancel them out. H bar has units of p times l. And what's the only other parameter we have? We have the mass, which has units of mass. OK. And so, from this, we can build two nice quantities. The first, is we can build r0. We can build something with units of a radius. And I'm going to choose the factors of 2 judiciously, h bar squared over 2me squared-- whoops, e squared-- so, let's just make sure this has the right units. E squared has units of energy times the length, but h bar squared over 2m has units of p squared l squared over 2m, so that has units of energy times the length squared. So, length squared over length, this has units of length, so this is good. So, there's a parameter that has units of length. And from this, it's easy to see that we can build a characteristic energy by taking e squared and dividing it by this length scale. And so then, the energy, which I'll call e0, which is equal to e squared over r0, is equal to 2me to the 4th over h bar squared. So, before we do anything else, without solving any problems, we immediately can do a couple of things. The first is, if you take the system and I ask you, look, what do you expect? If this is a quantum mechanical-- a 1d problem in quantum mechanics-- with a potential, and we know something about 1D quantum mechanical problems-- I guess, this guy-- we know something about 1D quantum mechanical problems. Which is that the ground state has what energy? Some finite energy. It doesn't have infinite negative energy. It's got some finite energy. What do you expect to be roughly the ground state energy of this system? AUDIENCE: [MURMURING] PROFESSOR: Yeah. Right. Roughly minus e0. That seems like a pretty good guess. It's the only dimensional sensible thing. Maybe we're off by factors of 2. But, maybe it's minus e0. So, that's a good guess, a first thing, before we do any calculation. And if you actually take mu e to the 4th over h bar squared, this is off by, unfortunately, a factor of 4. This is equal to 4 times the binding energy, which is also called the Rydberg constant. Wanna make sure I get my factors of two right. Yep, I'm off by a factor of 4. I'm off by a factor of 4 from what we'll call the Rydberg energy, which is 13.6 eV. And this is observed binding energy of hydrogen. So, before we do anything, before we solve any equation, we have a fabulous estimate of the binding energy of hydrogen, right? All the work we're about to do is gonna be to deal with this factor of 4, right? Which, I mean, is important, but I just want to emphasize how much you get just from doing dimensional analysis. Immediately upon knowing the rules of quantum mechanics, knowing that this is the equation you should solve, without ever touching that equation, just dimensional analysis gives you this answer. OK? Which is fabulous. So, with that motivation, let's solve this problem. Oh, by the way, what do you think r0 is a good approximation to? Well, it's a length scale. AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah! It's probably something like the expectation value of the radius-- or maybe of the radius squared-- because the expectation value of the radius is probably 0. OK, so, let's solve this system. And at this point, I'm not gonna actually solve out the differential equation in detail. I'm just gonna tell you how the solution goes, because solving it is a sort of involved undertaking. And so, here's the first thing, so we look at this equation. So, we had this differential equation-- this guy-- and we want to solve it. So, think back to the harmonic oscillator when we did the brute force method of solving the hydrogen system, OK? When we did the brute force method-- she sells seashells-- when the brute force method of solving, what did we do? We first did, we did asymptotic analysis. We extracted the overall asymptotic form, at infinity and at the origin, to get a nice regular differential equation that didn't have any funny singularities, and then we did a series approximation. OK? Now, do most differential equations have a simple closed form expression? A solution? No, most differential equations of some, maybe if you're lucky, it's a special function that people have studied in detail, but most don't have a simple solution like a Gaussian or a power large, or something. Most of them just have some complicated solution. This is one of those miraculous differential equations where we can actually exactly write down the solution by doing the series approximation, having done asymptotic analysis. So, the first thing when doing dimensional analysis too, let's make everything dimensionless. OK, and it's easy to see what the right thing to do is. Take r and make it dimensionless by pulling out a factor of rho, or of r0. So, I'll pick our new variable is gonna be called rho, this is dimensionless. And the second thing is I want to take the energy, and I will write it as minus e0, times some dimensionless energy, epsilon. So, these guys are my dimensionless variables. And when you go through and do that, the equation you get is minus d rho squared plus l l plus 1 over rho squared minus 1 over rho plus epsilon u is equal to 0. So, the form of this differential equation is, OK, it's not different in any deep way, but it's a little bit easier. This is gonna be the easier way to deal with this, because I don't have to deal with any stupid constant. And so now, let's do the brute force thing. Three, asymptotic analysis. And here, I'm just going to write down the answers. And the reason is, first off, this is something you should either do in recitation, or see-- go through-- on your own, but this is just the mathematics of solving a differential equation. This is not the important part. So, when rho goes to infinity, which terms dominate? Well, this is not terribly important. This is not terribly important. That term is gonna dominate. And if we get that d rho squared plus u, rho goes to infinity, these two terms dominate. Well, two derivatives is a constant. You know what those solutions look like, they look like exponentials, with the exponential being brute-- with the power-- the exponent, sorry, being root epsilon. So, u is going to go like e to the minus square root epsilon rho. For normalize-ability, I picked the minus, I could've picked the plus, that would've been divergent. So, as rho goes to 0, what happens? Well, as rho goes to 0, this is insignificant. And this totally dominates over this guy. On the other hand, if l is equal to 0, then this is the only term that survives, so we'd better make sure that that behaves gracefully. As rho goes to 0, asymptotic analysis is gonna tell us that u goes like rho. Well, two derivatives, we pulled down a rho squared, and so two derivatives in this guy, we pulled down an l, then an l plus 1. So, this should go like rho to the l plus 1. There's also another term. So, in the same way that there were two solutions to this guy asymptotically-- one growing, one decreasing-- here, there's another solution, which is rho to the minus l. That also does it, because we get minus l, then minus minus l minus 1, which gives us the plus l l plus 1. But that is also badly diversion at the origin, it goes like 1 over 0 to the l. That's bad. So, these are my solutions. So, this tells us, having done this in analysis, we should write that u is equal to rho to the l plus 1 times e to the minus root epsilon rho times some remaining function, which I'll call v, little v. Little v of rho, and this, asymptotically, should go to a constant near the origin and something that vanishes slower than an exponential at infinity. So then, we take this and we do our series expansion. So, we take that expression, we plug it in. At that point, all we're doing is a change of variables. We plug it in, and we get a resulting differential equation. Rho v prime prime plus 2 1 plus l minus root epsilon rho v prime plus 1 minus 2 root epsilon l plus 1 v equals 0. So, this is the resulting differential equation for the little v guy. And we do a series expansion. V is equal to sum over, sum from j equals 0 to infinity, of a sub j rho to the j. Plug this guy in here, just like in the case of the harmonic oscillator equation, and get a series expansion. Now, OK, let me write it out this way. And the series expansion has a solution, which is a sub j plus 1. And this is, actually, kind of a fun process. So, if you, you know, like quick little calculations, this is a sweet little calculation to take this expression. Plug it in and derive this recursion relation, which is root-- or 2 root-- epsilon times j plus l plus 1 minus 1 over j plus 1 j plus 2 l plus 2 aj. So, here's our series expansion. And in order for this terminate, we must have that some aj max plus 1 is equal to 0. So, one of these guys must eventually vanish. And the only thing's that's changing is little j. So, what that tells us is that for some maximum value of little j, root 2 epsilon times j maximum plus l plus 1 is equal to minus 1. But that gives us a relationship between overall j max, little l, and the energy. And if you go through, what you discover is that the energy is equal to 1 over 4 n squared, where n is equal to j max plus l plus 1. And what this tells is that the energy is labeled by an integer, n, and an integer, l, and an integer, m-- these are from the spherical harmonics, and n came from the series expansion-- and it's equal to minus e0 over 4 n squared, independent of l and m. And so, by solving the differential equation exactly, which in this case we kind of amazingly can, what we discover is that the energy eigenvalues are, indeed, exactly 1/4 of e0. And they're spaced with a 1 over n squared, which does two things. Not only does that explain-- so, let's think about the consequence of this very briefly-- not only does that explain the minus 13.6 eV, not only does that explain the binding energy of hydrogen as is observed, that it does more. Remember in the very beginning one of the experimental facts we wanted to explain about the universe was that the spectrum of light of hydrogen went like 30 over 4 n squared. This was the Rydberg relation. And now we see explicitly. So, we've solved for that expansion. But there's a real puzzle here. Purely on very general grounds, we derived earlier that when you have a rotationally invariant potential-- a central potential-- every energy should be degenerate, with degeneracy 2l plus 1. It can depend on l, but it must be independent of m. But here, we've discovered-- first off, we've fit a nice bit of experimental data, but we've discovered the energy is, in fact, not just independent of m, but it's independent of l, too. Why? What symmetry is explaining this extra degeneracy? We'll pick that up next time.
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_11_Dispersion_of_the_Gaussian_and_the_Finite_Well.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So today is going to be our last pass at bound states. So starting next week or actually starting next lecture, we're going to look at scattering. Scattering's going to be great. But we need to close out bound states. So today's topic is the finite well, finite the potential well. We've sort of sketched this when we looked at qualitative structure of wave functions of energy eigenstates. But we're going to solve the system today. So good. So the system we're interested in is going to be-- The system we're interested in is going to be a system with a finite depth and a finite width. And I'll go into detail and give you parameters in a bit. But first I want to just think about how do we find energy eigenfunctions of a potential of this form, v of x, which is piecewise constant. So first off, is this is a terribly realistic potential? Will you ever in the real world find a system that has a potential which is piecewise constant? Probably not. It's discontinuous. Right? So it's rather unphysical. But it's a very useful toy model. So for example, if you take a couple of capacitor plates, then you can induce a situation where the electric field is nonzero in between the capacitor plates and zero outside of the capacitor plates. Right. So at a superficial level, this looks discontinuous. It looks like the electric field is-- But actually, you know that microscopically there are a bunch of charges, and everything is nice and continuous except for the behavior right at the charges. So but it's reasonable to model this as a step function for an electric field. So this is going to be an idealization, but it's going to be a very useful idealization, the constant potential. OK. So what's the equation? What are we trying to do? We want to find the energy eigenstates for this because we want to study the time evolution. And the easiest way to solve the Schrodinger equation, the time evolution equation, is to expand an energy eigenstates. So the equation we want to solve is energy eigenvalue times phi e of x is equal to minus h bar squared on 2m phi e phi prime plus v of x. And I'm going to put it in the form we've been using. Phi prime prime e of x is equal to 2m upon h bar squared v of x minus e. OK. So this is the form that I'm going to use today to solve for the energy eigenvalue equations. e is some constant. Do you expect the allowed energies to be arbitrary? No. They should be discrete. Yeah, exactly. So we expect that there should be discrete lowest energy state, some number of bound states. And then, eventually, if the energy is greater than the potential everywhere, the energies will be continuous. Any energy will be allowed above the potential. So we'll have a continuum of states above the potential. And we'll have a discrete set of bound states-- Probably, it's reasonable to expect some finite number of bound states just by intuition. --from the infinite well. So we expect to have a finite number of discrete energies and then a continuous set of energies above zero. So if this is the asymptotic value potential of zero. OK. And this is intuition gained from our study of qualitative structure of energy eigenfunctions. So we are going to talk today about the bound states. And in recitation, leaders should discuss the continuum above zero energy. OK. So to solve for the actual energy eigenfunctions and the energy eigenvalues, what we need to do is we need to solve this equation subject to some boundary conditions. And the boundary conditions we're going to want to solve are going to be finite. So it's normalizable infinity. The solution should be vanishing far away. And the wave function should be everywhere smooth. Well, at least it should be continuous. So let's talk about what exactly boundary conditions we want to impose. And so in particular, we're going to want to solve for the energy eigenfunctions in the regions where the potential is constant and then patch together solutions at these boundaries. We know how to solve for the energy eigenfunctions when the potential is constant. What are the energy eigenfunctions? Yeah. Suppose I have a potential, which is constant. v is equal to 0. What are the energy eigenfunctions of this potential? AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah, e to the ikx. Yeah. Now, what if I happen to tell you-- So we're at h bar squared k squared upon 2m is equal to the energy. Suppose I happen to tell you that here's the potential. And I want to find the solution in this region where the energy is here less than the potential. What are the solutions? That's a second order differential equation. There should be two solutions. What are the solutions to that differential equation when the energy is less than the potential? AUDIENCE: Decaying and growing. PROFESSOR: Decaying and growing exponentials. Exactly. e to the plus alpha x and e to the minus alpha x. And the reason is these are sinusoidal, and these have the opposite concavity. They are growing and dying exponentials. Cool? OK. So we've studied that. Yeah? AUDIENCE: Shouldn't that be a phi e of x? PROFESSOR: Sorry. Oh, oh, yes, indeed. Sorry. Thank you. Phi e of x. It's early, and I'm still working on the coffee. It won't take long. Good. So we know how to solve the energy eigenvalue equation in all these regions where the potential is constant. So our job is going to be to find a solution where we patch them together at these interfaces. We patch them together. And what condition should we impose? So the basic condition is going to be continuity of the wave functions phi e of x. And so what are the conditions that we need? Well, if v of x-- Here's the way I like to think about this. Suppose v of x is continuous. So if the potential is continuous, then what can you say? You can say that phi prime prime is a continuous function times phi of x. So very roughly, if we look at a region where phi isn't varying very much, so if we have a potential that's varying in some way, then phi prime prime, in a region where it's not varying much compared to its value as a function of x, does something smooth because it's varying with the potential. And so phi prime, which is just going to be the integral of this-- The integral of a smooth function is again a smooth function. --and phi, the integral of that function is also going to be a nice, smooth function. OK? I've drawn it badly, but-- So the key thing here is that if the potential is continuous, then the energy eigenfunction has a second derivative, which is also continuous. That means its first derivative is continuous. So that means the function itself is continuous. Everyone agree with this? Questions? So in regions where the potential's continuous, the wave function and its first two derivatives all have to be continuous. On the other hand, suppose the potential has a step. v of x has a step discontinuity. OK. So the potential does one of those. So what does that tell you about phi prime prime? It's a function of x. So for example, let's look at that first step. Suppose the potential is that first step down by some amount. Then phi prime prime is going to decrease precipitously at some point. And the actual amount that it decreases depends on the value of phi because the change in the potential time is the actual value of phi. And if that's true of phi prime prime what can you say of phi prime? So this is discontinuous. Phi prime of x, however, is the integral of this discontinuous function. And what does it do? Well, it's linearly increasing in this region because its derivative is constant. And here, it's linearly increasing less. So it's not differentiably smooth, but it's continuous. And then let's look at the actual function phi of x. OK. What is it doing? Well, it's quadratic. And then here it's quadratic a little less afterwards. But that still continuous because it's the integral of a continuous function. Everyone we cool with that? So even when we have a step discontinuity in our potential, we still have that our derivative and the value of the function are continuous. Yeah? However, imagine the potential has this delta function. Let's just really push it. What happens if our potential has a delta function singularity? Really badly discontinuous. Then what can you say about phi prime prime as a function of x? So it has a delta function, right? So the phi prime prime has to look like something relatively slowly varying, and then a step delta function. So what does that tell you about the derivative of the wave function? It's got a step function. Exactly. Because it's the integral of this, and the integral is 0, 1. Whoops. I missed. So this is a delta. This is a step. And then the wave function itself is, well, it's the integral of a step, so it's continuous. Sorry. It's certainly not differentiable. Its derivative is discontinuous. Well, it's differentiable but its derivative's not supposed to be. It is not continuous, indeed. So this is continuous. So we've learned something nice, that unless our potential is so stupid as to have delta functions, which sounds fairly unphysical-- We'll come back to that later in today's lecture. --unless our potential has delta functions, the wave function and its first derivative must be smooth. Yeah. This is just from the energy eigenvalue equation. Now, we actually argued this from the well definedness and the finiteness of the expectation value of the momentum earlier in the semester. But I wanted to give you this argument for it because it's going to play a useful role. And it also tells us that if we do have a delta function singularity in the potential, then the upshot is that the wave function is going to be continuous, but its first derivative will not. Its first derivative will jump at the wave function. And that means that the wave function-- Let me draw this slightly differently. --the wave function will have a kink. Its derivative will not be continuous. OK. So anywhere where we have a delta function in the potential, we will have a kink in the wave function where the first derivative is discontinuous. Cool? Yeah. AUDIENCE: What does that mean as far as the expectation value of the momentum? PROFESSOR: Ah, that's an excellent question. So what do you expect to happen at such a point? AUDIENCE: Well, your momentum blows up. PROFESSOR: Yeah, exactly. So we're going to have some pathologies with expectation values in the momentum. Let's come back to that when we talk about the delta function potential, which should be at the end of today. Hold that question in the back of your head. It's a good question. Other questions? OK. So what we're going to do now is we're going to use, you know, math to find the energy eigenfunctions and eigenvalues for the finite well potential, i.e. we're going to solve this equation contingent on the boundary conditions that the wave function and its derivative are smooth everywhere. And in particular, they must be smooth here and here. I mean they've got to be smooth everywhere. We know what the solution is inside here. We know what the solution is in here. All we have to worry about is what happens at the interface. And we're going to use smoothness of the wave function and its derivative to impose conditions that allow us to match across that step. Everyone cool with that? OK. So let's do it. By the way, just a quick side note. Let me give you a definition. I've used this phrase many times, but I haven't given you a definition of it. So I've used the phrase, bound state. And its opposite is called a scattering state. So here's what I mean by a bound state. Intuitively, a bound state, if you think about this classically, imagine I have a potential well. And I have a marble, and I let go from here. This marble is bound. Right? It's never getting out of the well. It's stuck. Yeah? And so I would call that a bound marble. On the other hand, a marble that I give a huge kick to, big velocity so that it can get out, well then it's not bound to this potential well. So I'll call that a scattering state just to give it a name. And next lecture we'll see why we call it a scattering state. The important thing is that bound configurations of a classical potential are things basically in a well that are stuck. OK. So the quantum version of that statement is the following. Suppose I take this potential, and I treat it quantum mechanically, and I consider a state with total energy like this. Well, among other things, the total energy is less than the value of the potential asymptotically far away. So what is the form of a wave function in this region with this energy? Exponential, right? It's e to the minus some alpha x, where alpha squared is roughly the difference, alpha x. And out here it's going to be e to the plus alpha x. And that's for normalizability. We want to have a single particle in this state. So what that tells us is that the wave function falls off in these classically disallowed regions exponentially. And so the probability of measuring the particle at an arbitrarily far position goes to zero. And it goes to zero exponentially rapidly. Cool? So I will call a bound state, a quantum bound state, an energy eigenstate, such that the probability falls off exponentially as we go far away from wherever we think is an interesting point, like the bottom of the well. Cool? A bound state is just a state which is exponentially localized. If you put it there, it will stay there. Yeah. And it's important that when I say a bound state I'm talking about energy eigenstates. And the reason is this. Bound state equals energy eigenstate. The reason is that, consider by contrast a free particle, so a free particle with constant potential. What are the wave functions? What are the energy eigenfunctions? Well, they're plane waves. Right. So are these bound states? No. Good. OK. On the other hand, I claimed that I can build a wave packet, a perfectly reasonable wave packet, which is a Gaussian. OK. This is some wave function, psi of x times 0. It's a Gaussian. It's nice and narrow. Is that a bound state? Well, it's localized at this moment in time. But will it remain, and in particular its probability distribution, which is this thing norm squared is localized in space-- Sorry. This was zero. --it's localized in space. The probability of it being out here is not just exponentially small, it's Gaussian so it's e to the minus x squared. It's really not out here. But what happens if I let go? What happens if I look at the system at time t? It's going to spread out. It's going to disperse. We're going to talk about that in more detail later. So it's going to spread out. And eventually, it will get out arbitrarily far away with whatever probably you like. So the probability distribution is not time invariant. That is to say it's not a stationary state. It's not an energy eigenstate. Saying something is bound means that it never gets away. So bound states are specifically energy eigenstates that are strictly localized, that fall off at least exponentially as we go away from the origin. Cool? It's just terminology, what I mean by a bound state. Questions? OK. So let's talk about the finite well. So I need to give you definitions of the parameters. Let's draw this more precisely. Here's my well. Asymptotically, the potential is zero. The potential depth, I'm going to call minus v naught. OK. And I'm going to center the well around zero. And I'll call the sides minus l and l. And I want to find bound states of this potential, just like we found bound states of the harmonic oscillator, i.e. states with energy e, which is less than zero. So these are going to give us bound states because we're going to have exponential fall offs far away. So a couple of things to note. The first is on, I think, problem set three or four you showed that if you have a potential, which is symmetric, which is even, under x goes to minus x, then every energy eigenfunction, or at least every bound state energy eigenfunction, every energy eigenfunction can be written as phi e symmetric or phi e anti-symmetric. So it's either even or odd. It's either even or odd under the exchange of x to minus x. So when our potential is symmetric, the wave function or the energy eigenfunctions are either symmetric or anti-symmetric. OK. So we want to solve for the actual eigenfunctions. So we want to solve that equation. And we have this nice simple fact that we know the solutions in this region. We know the general solution in this region. We know the general solution in this region. So I'm going to call these regions one, two, and three, just to give them a name. So in region one-- That's actually sort of stupid. Let's call this inside, left, and right. Good. OK. So let's look at this equation. We have two cases. If the energy is greater than the potential in some region, then this is of the form phi prime prime is equal to energy greater than potential. This is a negative number. And so this is a minus k squared phi. And we get exponentials. And if, on the other hand, e is less than v of x, then phi prime prime is equal to plus alpha squared phi. I should say oscillator. And in particular here, I want the k squared is equal to 2m upon h bar squared. It's just the coefficient v minus e. And alpha squared is equal to 2m over h bar squared e minus v. OK. So let's apply that here. So in this region, we're going to get oscillations because we're in a classically allowed region, where the energy is greater than the potential. So we'll get oscillatory solutions. And the salient value of k inside, is k is equal to the square root of 2m over h bar squared times v minus e. So that's minus v naught. What did I do? I did. It's e minus v. I thank you. Yes. And I want the other one to be l so it'd be minus v. Good. Good. Excellent. So root 2 over h bar squared. And now we have e minus v naught, which is the actual value of e, which is negative. Right? But minus v naught or plus v naught is positive and greater in magnitude. So this is a nice positive number, and k is the square root of it. This is controlling how rapidly the wave function oscillates in this region. Similarly, out here we have alpha is equal to-- Well, here the potential is zero. So it's particularly easy. Alpha is equal to the square root of 2m upon h bar squared of-- Now z minus e is zero minus e, which is a negative number. So we can just write e absolute value. So we can write the general solution of this eigenfunction, of this eigenvalue equation, as phi e of x is equal to-- Let's break it up into inside and outside. Well, inside we know, since it's constant with this value of k, we get superpositions of oscillatory solutions. It' a second-order difference equation. There are two solutions and two integration constants. So first we have a cosine of kx plus b sine kx. This is inside. And then on the left we have a combination of exponentially growing and exponentially decreasing. So the exponentially growing is e to the alpha x plus de the minus alpha x. And on the right, similarly by symmetry, we have some combination of e-- But I don't want to call it the energy, so I'll call it the curly e. --to the alpha x plus fe to the minus alpha x. OK. So that's the general solution. We solve the problem as a superposition of the two oscillatory solutions or a superposition of the two exponentially growing and damped solutions or exponentially growing and collapsing functions on the left and right. Questions? So a couple of things to note at this point. So the first is we have boundary conditions to impose. We have boundary conditions at these two interfaces. But we also have boundary conditions off in infinity. What are the boundary conditions at infinity? Yeah, exactly. It should vanish. So we want the system to be normalizable. So normalizable is going to say that phi goes to zero at minus infinity. Phi of x goes to minus infinity goes to zero, which it equals. And phi of x goes to plus infinity should also be zero. OK. And then we're also going to have the conditions at the left boundary, and we're going to have conditions at the right boundary. [LAUGHTER] All right. So what are the boundary condition at the left boundary condition? So first off, what are the boundary conditions we want to impose at the left and right boundaries? AUDIENCE: Continuous. PROFESSOR: Continuous, and the derivative should be continuous. Exactly. So we have that phi is continuous, and phi prime-- Good god. --phi prime is continuous. Similarly, phi continuous, phi prime continuous. OK. Do we have enough boundary conditions to specify our function? So we have now for our solution, we have six undetermined coefficients. And we have six boundary conditions. So that looks good. Are they all independent? Ponder that one. So in particular, let's start with the normalizable. So in order for phi to go to zero at minus infinity deep out on the left, what should be true? Yeah, d goes to zero. Oops, equals zero. And on the right? Yeah, that curly e equals 0, which is nice so I don't ever have to write it again. So that's zero. And that's zero. OK. That's good. We can take advantage though of something nice. We know that the wave function has to be either symmetric or anti-symmetric. Right? So we can exploit that and say, look, the wave function is going to be different from our boundary conditions, but it's a true fact, and we can take advantage of it. We can use the parody of the well. I can never-- So we can use the parody of the potential to say that the system is either symmetric or anti-symmetric. And these are often said as even or odd because the function will be, as a function of x, either even or odd. You either pick up a minus sign or a plus sign under taking x to minus x. So if the system is even, what can we say about these coefficients? What must be true of b, for example? AUDIENCE: It's zero. PROFESSOR: Yeah. So b equals 0. And what else? AUDIENCE: It equals f. PROFESSOR: This equals f. Yeah, good. OK. This equals f. And if the system is anti-symmetric, then a equal to 0. And we see that c is equal to minus f. Yeah? So that's a useful simplification. So it's easy to see that we could do this either way. We could do either symmetric or anti-symmetric. I'm going to, for simplicity in lecture, focus on the even case. b is equal to 0, and c is equal to minus f. Sorry, c is equal to f. So plus c. So now we're specifically working with the even solutions. And on your problem set, you'll repeat this calculation for the odd functions. So we're going to focus on the even solutions. And now what we have to do is we have to impose the boundary conditions for phi and phi prime. So that's easy enough. We have the function. All we have to do is impose that the values are the same. So for example, let's focus on the left boundary conditions. Sorry. Let's focus on the right because I don't want to deal with that minus sign. So let's focus on the right boundary conditions. So this is x is equal to plus l. So when x is equal to plus l, what must be true? Phi and phi prime must be continuous. So what's phi? So phi is equal to-- Well, from inside, it's a cosine of kl. Yeah. And b is gone because we're only looking at the even functions. On the right, however, it's equal to c e to the minus alpha l because we're evaluating at the right boundary. Yeah? OK. So this is cool. It allows us to determine c in terms of a. And if we solve that equation for c in terms of a, we'll get an eigenfunction, phi even, with one overall normalization coefficient, a. And then we can fix that to whatever it has to be so that everything integrates to one. Yeah? So that seems fine. It seems like we can solve for c in terms of a. c is equal to-- This is weight. Well, OK. So c is equal to a cosine e to the plus alpha l. On the other hand, we also have a condition on the derivative. And the condition on the derivative is that phi prime is continuous. And the derivative of this, well that's easy. It's the derivative of cosine. So this is going to be minus sine. But we pull out a factor of k, because we're taking derivative with respect to x. Minus k is sine of kx. Evaluate it out, sine of kl, l. And this is going to be equal to minus alpha c e to the minus alpha l. AUDIENCE: You forgot an a. PROFESSOR: Oh, yes. There should be an a. Thank you. OK. So but now we've got a problem because this says that c is equal to minus a times k over alpha sine of kl times e to the alpha l. And that's bad because c can't be equal to two different numbers at the same time. There's a certain monogamy of mathematical equations. It just doesn't work. So how do we deal with this? Well, let's think about what these equations would have meant. Forget this one for the moment, and just focus on that first expression. I'm going to rewrite this slightly. a cosine of kl. OK. So what does this expression say? Well, it seems like it's just saying if we fix c to be equal to this value, for fixed value of kappa l and alpha, then there's a solution. However, what is k? What are k and alpha. k and alpha are functions of the energy. So it would seem from this point of view, like for any value of the energy, we get a solution to this equation. Everyone see that? But we know that can't possibly be right because we expect the solutions to be discrete. We don't expect any value of energy to lead to a solution of the energy eigenvalue equation. There should be only discrete set of energies. Yeah? AUDIENCE: Did you pick up an extra minus sign in the expression for c? PROFESSOR: You do. Thank you. The sign's cancel. Yes, excellent. I've never written this equation in my life. So thank you. Yes, extra minus sign. So what's going on here? Well, what we see is that we've written down the general form of the solution. Here were imposing that we've already imposed the condition that we're normalizable at infinity. Here, we're imposing the continuity condition on the right. And if we impose just the continuity condition for the wave function, we can find a solution. Similarly, if we impose only the continuity condition for the derivative, we can find a solution for arbitrary values of the energy. But in order to find a solution where the wave function and its derivative are both continuous, it can't be true that the energy takes just any value because it would tell you that c takes two different values. Right? So there's a consistency condition. For what values of energy or equivalently, for what values of k and alpha are these two expressions equal to the same thing? Cool? So we can get that by saying, look we want both of these equations to be true. And this is easy. I can take this equation and divide it by this equation. And I will lose my coefficients c. I will lose my coefficients a. What do we get? On the right hand side, we get-- And I'm going to put a minus sign on everything. So minus, minus, minus. So if we take this equation and we divide it by this equation, on the right hand side, we get alpha, because the c exponential drops off. And on this side, we lose the a. We get a k. And then we get sine over cosine of kl, also known as tangent of kl. Here we have a kl. Here we have a k. These are all dimensionful things. Let's multiply everything by an l. And this is nice and dimensionless. Both sides are dimensionless. So we get this condition. This is the consistency condition, such that both the wave function and its derivative can be continuous at the right boundary. OK? And this is a pretty nontrivial condition. It says, given a value of k, you can always determine the value of alpha, such as this equation as true. But remember that k and alpha are both known functions of the energy. So this is really an equation, a complicated, nonlinear equation for the energy. So this is equal to a horrible expression, a condition, badly nonlinear, in fact, transcendental condition on the energy. And where's it coming from? It's coming from normalizability and continuity everywhere. And a useful thing to check, and I invite you to do this on your own, is to check that the boundary conditions at the left wall give the same expression. Yeah. AUDIENCE: For our final form of that equation, is there a reason that we prefer to multiply both sides by l than divide both sides by k? PROFESSOR: Yeah. And it'll be little more obvious in a second. But here's the reason. So let's divide through by l. This is the form that we got. What are the units? What are the dimensions of this expression? k is a wave number, so it has units of 1 upon length. Right? And that's good because that's 1 upon length times the length, and you'd better have something dimensionless inside a tangent. But it seems there are two things to say about this. The first is it seems like l is playing an independent role from k in this equation. But this is dimensionless. These are both dimensionful units of 1 over length. So we can make the entire expression dimensionless and make it clear that k and l don't have an independent life. The dimensionless quantity, kl, times the tangent of that dimensionless quantity is equal to this dimensionless quantity. So the reason that this is preferable is twofold. First off, it makes it sort of obvious that k and l, you can't vary them independently in this sense. But the second is that it makes it nice and dimensionless. And you'll always, whenever possible, want to put things in dimensionless form. I mean it's just multiplying by l. So it's obviously not all that deep. But it's a convenient bit of multiplication by l. Other questions? OK. So where are we? So I'd like to find the solutions of this equation. So again, just to-- Let me write this slightly differently where k squared is equal to 2m upon h bar squared v0 plus e. And alpha squared is equal to 2m upon h bar squared e, the positive value of e. So this is a really complicated expression as a function of e. So I'd like to solve for the actual energy eigenvalues. I want to know what are the energy eigenvalues of the bound states of the finite potential well, as a function of l, for example. Sadly, I can't solve this equation. It's a transcendental equation. It's a sort of canonically hard problem to solve. You can't write down a closed from expression for it. However, there are a bunch of ways to easily solve it. One is take your convenient nearby laptop. Open up Mathematica, and ask it to numerically find solutions to this. And you can do this. It's a good exercise. I will encourage you to do so on your problem set. And in fact, on the problem set, it asks you to do a calculation. And it encourages you do it using Mathematica. Let me rephrase the statement in the problem set. It would be crazy for you to try to do it only by hand. You should do it by hand and on computer because they're both easy. And you can check against each other. They make different things obvious. This should be your default is to also check on Mathematica. The second thing we can do is we can get a qualitative solution of this equation just graphically. And since this is such a useful technique, not just here, but throughout physics to graphically solve transcendental equations, I'm going to walk through it a little bit. So this is going to be the graphical solution. And we can extract, it turns out, an awful lot of the physics of these energy eigenstates and their energies through this graphical technique. So the first thing is I write this in nice, dimensionless form. And let me give those dimensionless variables a name. Let me call kl is equal to z, just define a parameter z. And alpha l is a parameter y. And I want to note that z squared plus y squared is equal to a constant, which is, I will call if you just plug these guys out, kl squared plus al squared. That's easy. Kl squared is this guy times l squared. Al squared is this guy times l squared. And so the e and the minus e cancel when we add them together. So we just get 2mv0 over h bar squared times l squared. So 2m upon h bar squared l squared v0. And I'm going to call this r naught in something of a pathological abusive notation. OK. So this is our expression. And I actually want to call this r0 squared. I know. I know. It's awful. But the reason I want to do this is that this is the equation for a circle. Yeah? And a circle has a radius. The thing that goes over here is r squared. OK. So at this point, you're thinking like, come on, circle. So let's plot it. So how are we going to solve this equation? Here's what I want to solve. I have now two equations relating z and y. We have that from this equation z tangent z is equal to y. And from this equation we have that z squared plus y squared is a constant r0 squared. Where r0 squared depends on the potential and the width in a very specific way, on the depth of the potential and the width in a very specific way. So we want to find-- Bless you. --simultaneously, we want to find simultaneous solutions of these two equations. Yeah? So that's relatively easy. So here's y, and here's z. So this equation has solutions. Any time that y plus z squared is equal to r0 squared, that means any time we have a circle. So solutions for fixed values of r0 lie on circles. Oh, I really should have drawn this under here. Sorry. y and z. So those are the circles. Notice that I'm only focusing on y and z, both positive. Why? Not yz, but W-H-Y . Why am I focusing on the variables y and z being positive? Because we started out defining them in terms of k and l, which were both positive, and alpha and l, which were both positive. Can we find solutions to this equation that have x and y negative? Sure. But they don't mean anything in terms of our original problem. So to map onto solutions of our original problem, we want to focus on the positive values of y and z. Cool? OK. So that's this one. The solutions lie on circles. So given a value of y, you can find a solution of z. But we want to also find a solution of this equation. And this equation is a little more entertaining to plot. Here's y. Here's z. So what does z tangent z do? Oh, shoot. I want to plot y vertical. Otherwise, it's going to a giant pain. Happily, this plot can be left identical. Let's plot y vertically. So the reason I want to plot y vertically is that this is z tangent z. So first off, what does tangent z look like? Yeah. This is awesome. Yeah, it looks like this. Yes, exactly. So tangent is sine over cosine. Sine is zero, and cosine is one. So it does this, as you go to a value where the argument, let's call the argument z. So if we just plot tangent-- OK. So when z is equal to pi over 2, then the denominator cosine vanishes, and that diverges. Oops. OK. So here's pi over 2. Here's pi. Whoops. Pi, pi over 2, and here's 3pi over 2, and so on. Now, we're only interested in the first quadrant. So I'm just ignore down here. OK. So this is pi over 2. This is pi, 3pi over 2. OK. But this is not what we're interested in. We're not interested in tangent of z. We're interested in z tangent z. And what does z tangent z look like? Well, it's basically the same. Right? z tangent z, it has an extra factor of zero here and remains extra small at the beginning. But it still curves off roughly like this. And z is just nice and linear, nice and regular throughout this. So it doesn't change the fact that we have a divergence at pi over 2. And it doesn't change the fact that it vanishes again at pi and becomes positive again. It just changes the shape of the curve. And in fact, the way it changes the shape of the curve is this becomes a little fatter around the bottom. It's just a little more round. And when we get out to large values of z, it's going to have a more pronounced effect because that slope is, at every example where it crosses z, that slope is getting larger because the coefficient of z is getting larger. OK. So it's just going to get more and more sharp. But anyway, with all that said, here's 0. Here's pi. Here's pi over 2. Here's pi. Here's 3pi over 2. The second plot we want to plot, y is z tangent z. We know how to plot this. Cool? And what we want to find are simultaneous solutions of this, values of y and z, for which this equation is solved and this equation is solved for the same value of y and z. This is a graphical solution. So let's combine them together. And the combined plots look like this. First we have pi over 2. So let's plot the tangents. And then we have these circles for various values of r. So for a particular value of r, for example, suppose this is the value of r. This is r0. So how many solutions do we have? One. One set of common points where at y and z solve both equations. So we immediately learn something really lovely. What happens to the radius of that circle as I make the well deeper? Yeah, as I make the well deeper, that means v0 gets larger and larger magnitude, the radius gets larger. So does the circle. So if I make the well deeper, I make this the circle larger. Will I still have a solution? Yeah, I'll still have a solution. But check this out. Now, I'll have a new solution. And you can even see the critical value of the depth and the width of the well. In order to have exactly a new bound state appearing, what must the value of r0 be? Well, it's got to be that value, such that r0 squared is pi. Yeah? And similarly, let me ask you the following question. As I make the well deeper and deeper and deeper, holding the width, and make it deeper and deeper and deeper, does the number of states increase or decrease? It increases. If you make it deeper and deeper, the radius of that circle is getting bigger and bigger. There are more points where this circle intersects this point. So here's another one. We've got one here, one here, one here, three solutions. And the number of solutions just goes. Every time we click over a new point by increasing the radius of the circle, we get a new solution. We get another bound state. But here's the thing that I really want to focus on. Let's make the well less and less deep. Let's make it shallower and shallower. At what depth do we lose that first bound state? We never do. Right? There is no circle so small that it doesn't intersect this curve. In a 1D, finite well potential, there is always at least one bound state. There are never zero bound states. This will turn out not to be true in three dimensions, which is kind of interesting. But it's true in one dimension that we always have at least one bound state. And in fact, you can decorate this. You can use this and fancy it up a bit to argue that in any potential in 1D, there's always at least one bound state unless the potential is constant, I mean any potential that varies and goes to zero infinity. Yeah? And so we still don't have any numbers. But we know an awful lot about the qualitative structure of the set of energy eigenvalues of the spectrum of the energy. Questions? Yeah? AUDIENCE: So what happens if r is bigger than pi or y is bigger than pi and you get two solutions? PROFESSOR: Great. So when we have two solutions, what does that mean? Well, you've got to bound states, two different energies. Right? It's a good question. Every solution here corresponds to some particular value of y and some particular value of z. But those values of y and z are just telling you what k and alpha are. And so that's determining the energy. So a different value of k is going to give you a different value of the energy. So we can just eyeball this in particular. Let's look at alpha. Alpha is just e. Alpha squared is just e. Yes? So here's a quick question. If alpha is just e, and alpha l is y-- So this is our y value. y is roughly alpha, the width, which means it's roughly the energy square root. So this value, the vertical value of each of these intersection points on a given circle corresponds to the square root of the energy times some coefficients. So which state has the largest value of the energy? Absolute value, which state is most deeply bound on this circle? Yeah, the first one. Right? Because it's got the largest value of alpha. So this is nice. We see that the first state always has a higher value of alpha than the second state, which always has a higher value of alpha than the third state. And every time we add a new state, we make the depth of these guys the binding energy of the already existing states. We make it just a little bit deeper. We make them a little more tightly bound. And only eventually then do we get a new bound state appearing. And what's the energy of that new bound state when it appears? Zero energy. It's appearing just at threshold. OK. So we'll explore that in more detail in the problem set. But for now, let me leave it at that. Questions? Other questions? Yeah. AUDIENCE: You said that this can be generalized to any nonconstant function that you'd like. So there's always going to be at least one bound state. What about, like with delta function? PROFESSOR: Excellent question. What about the delta function? We're going to come back to that in just a few minutes. It's a very good question. So the question is, look, if any potential that goes to zero infinity and wiggles inside, if any potential like that in 1D has a bound state, what about the delta function? We briefly talked about that. So we're going to come back to that in just a few minutes. But it's a pressing question. OK. Other questions? Yes. AUDIENCE: So the energy is zero, but that's not possible. PROFESSOR: Thank you. OK. Good. So let me talk about that in a little more detail. So I wasn't going to go into this, but-- So when new bound states appear, so let's consider a point where our r0 is, let's say, it's just the right value so that r0 is equal to pi. OK. And we see that we're just about to develop a new bound state. So let's think about what that bound state looks like. So this is the new bound state. And I'm going to put this in parentheses because it's got bound state. And we say at threshold. OK. At threshold, i.e. at the energy is roughly zero, and r0 is equal to pi. So this is really what we mean. This new state, when r0 is pi and we have a solution on that second branch. Cool? So what does this wave function look like? What does it look like when you have a wave function that just appeared? It's just barely bound. Well, first off, what does it mean to be bound? Let's just step back and remember for now. What does it mean to be a bound state? It means you're an energy eigenfunction and you're localized. Your wave function falls off at infinity. Now, if it falls off at infinity, do these guys fall off at infinity? These wave functions, sure, they fall of with an exponential damping. And in particular, let's look at the right hand side of the well. This new bound state is appearing just at zero energy. So out here, what is the wave function? It's e to the minus alpha x. But what's alpha? Zero, right? There it is, zero. So this is e to the minus alpha x where alpha is equal to 0. This is constant. So what does the wave function look like? Well, the wave function, again over the same domain-- Here's 0, l. And here's the value zero. We know that in this domain it's oscillatory, and in this domain, it's constant. And actually, since we know that it's the first excited state, we know that it does this. So if we make the well ever so slightly deeper, ever so slightly deeper, which means making the radius of the circle ever so slightly larger, we will get a nonzero value for the alpha of this solution. Right? It'll be just tiny. But it'll be nonzero. So we make the well just a little tiny bit deeper. We get something. OK, good. So what's going to happen to the wave function? Well, instead of going flat. This is going to curve. It's got a little more energy. There's a little more curvature. So it curves just a little tiny bit more. And then it matches onto a very gradually decaying exponential. OK. So what's happening as we take this bound state, the second bound state, and we make the well a little more shallow? We make the well a little more shallow. It's a little less curved inside. And the evanescent tails, the exponential tails become longer and longer and longer and broader until they go off to infinity, until they're infinitely wide. And is that normalizable anymore? No, that's not normalizable. So the state really isn't strictly localized at that point. It's not really a normalizable state. And just when the state ceases to be normalizable, it disappears. We make the well just a little deeper, and there's no state there at all. OK. So this tells you a very nice thing. It's a good bit of intuition that when states are appearing or disappearing, when states are at threshold as you vary the depth, those threshold bound states have exceedingly long evanescent tails, and they're just barely bound. OK? This turns out to have all sorts of useful consequences, but let me move on. Did that answer your question? Good. Yeah? AUDIENCE: So in this case, the radius, as you say, is proportional to the length. Right? But also we have intuition that as we increase the length of the well, the energy's going to keep increasing-- PROFESSOR: Fantastic. OK. So what's up with that? Right. So the question is this. We already have intuition that is if we take a finite well and we make it a little bit wider, the ground state energy should decrease. The energy of the ground state should get deeper and deeper, or the magnitude should increase, another way to say it. Right? That was our intuition. So let's check if that's true. What happens if I take the ground state some value of the radius, and then I make the well a little bit wider. Well, if I make the well a little bit wider, what happens to r0, the radius of the circle? Well, if I double the length of the well, the width of the well, then it will double r0, and it will double the radius of the circle. So if I make it wider, r0 gets bigger, and we go to a bigger circle. And what happened to the energy of this state? Yeah, it got deeper and deeper and deeper. And meanwhile, as we make it wider, as we make the well wider, the circle is getting bigger again. And we're going to get more and more states. So as we make the well wider, holding the depth fixed, we get more and more states. As we make the well deeper, holding the width fixed, we get more and more states. And so how do you trade off? If I make it twice as wide, how much-- So here's a good question. Suppose I take a well, and it has n states. Suppose I then make it twice as wide. What must I do to the energy so that it still has the same states with the same energies? AUDIENCE: Divide it by 4. PROFESSOR: Yep, exactly. I've got to divide it by 4. Because I've doubled the length, that means the radius has gone up by four. But I get exactly the same solutions if I just bring this out. Well, that's almost true. So if I put a factor of 1/4 here, that's almost true, except for the value of y is unchanged, but the alpha hasn't changed. Sorry. The alpha has changed because there's an l. So y is fixed. But alpha's changed because of the l. So the reason that it's useful to write things in these dimensionless forms is that you can see the play off of the various different dimensionful parameters in changing the answer. Other questions. OK. So a couple of comments. So the first is let's just check to make sure that this makes sense. We have already solved this problem. We solved this problem a while ago, but we solved it in a particular limit. We solved it in the limit that the potential one's arbitrarily deep. Right? So when the potential one's are arbitrarily deep, holding the width fixed, that was the infinite well. That was the first problem we solved. So let's make sure that we recover this in that limit. So what happens as we make the well arbitrarily deep, holding the length fixed or the width fixed. So if we make this arbitrarily deep, v0 is getting arbitrarily large. That means r0 is getting-- We've got a huge circle. So what do these solutions look like when we have a huge circle. Let me not do that here. Let me do that here. So if we make the potential nice and deep, Let's think about what that plot looks like. So again, that first plot looks identical with the tangents. So on and so forth. And what I want to do is I want to dot, dot, dot. OK. So this is way up there. So these guys are basically vertical lines at this point. So for very, very large values of y, and in particular, for very large values that are of order the gigantically deep radius r0, what does the circle look like? So what does the second equation look like, the second curve? Well, again it's circles. But now it's a gigantic circle. Yeah, exactly. If it's a gigantic circle, it's basically flat. It's not exactly flat, but it's almost flat. It's a circle. So what are the values? And here's the key question. What are the values of-- Did do that right? Yeah, OK. Good. So what are the values of the curve, where we get a solution? The values now of z, are just exactly on these vertical lines, on the separatrices. The value of z is at pi over 2. and 3pi over 2. And then another one at 5pi over 2, and so on and so forth. Right? So what we find is that the allowed values of z-- Sorry. kl, yes, good. --allowed values of z are equal to 2n plus 1 over pi. Whoops 2n plus 1 over 2 times pi. So let's just check that. So n is 0. That's 0 1/2 pi. n is 1. That's 3/2 pi. Good. So these are the values of z, which says that kl is equal to 2n plus 1 upon 2 pi or k is equal to 2n plus 1 over 2l, which is the width of this well because it's from minus l to l pi. So is this the correct answer for the infinite square well? Are these the allowed values of k inside the well for the infinite square well? Almost. Instead of 2n plus 1, it should just be n plus 1. We seem to be missing about half of the energy eigenvalues. AUDIENCE: That's only the even ones. PROFESSOR: Yeah, thank you. This is only the even ones. We started out saying, oh, look. Let's look only at the even ones. Where do you think the odd ones are going to be? Ah, well the odd ones, so this should be k even. So what about k odd? Well, we know the answer already-- Whoops. Odd, that's an odd spelling. --should be equal to 2n over 2l plus 2 over 2l. Whoops. 2 capital L, pi. OK. This is our guess just from matching on to the infinite square well. So what does that mean? Well, that means it should be this one and this one. So when you go through this exercise on your problem set and you find the solutions for the odd, and you repeat this analysis for the odd ground states. What should you expect? Well, you should expect to find this. And what do you think the curves are that you're going to use int he graphical solution to do your transcendental equation? Yeah, it's really tempting to say, look. It's just going to be something shifted, like this. OK. So these are going to be the odd question mark, question mark. OK. So you'll check whether that's correct intuition or not on your problem set. OK. Questions? Yeah? AUDIENCE: What about the lower end of these states, where it won't have gone off high enough? Is that [INAUDIBLE]? PROFESSOR: Excellent. So that's a really good question. How to say it? What we've done here to make this an infinite well-- So the question, let me just repeat the question. The question is, look, what about all the other states? OK, it's true that r0 as gigantic. But eventually, we'll go to a large enough z, where it's the circle coming down over here too. So what's up with that? What are those states? Where are they? What do they mean in terms if the infinite square well? Well, first off, what are the energies of those states? AUDIENCE: Very low. PROFESSOR: They're very low in magnitude, which means they're close to what in absolute value? AUDIENCE: Zero. PROFESSOR: Zero. They're close to zero. So they're at the top of the finite well. These are the states bound at the top of the finite well. These are the states bound at the bottom the finite well. But how many states are bound at the top of the finite well when we take the limit that the well goes infinitely deep? Yeah, none of them. Right? So when we make the well infinitely deep, what we're saying is, pay no attention to the top of the well. Look only at the bottom of the well. And if it's really deep, it's a pretty good approximation. So that's what we're doing here. We're saying, look. Pay no attention. There is no top of the well. There's just the bottom. And look at the energy eigenvalues. Does that make sense? So what we're saying is if you have a preposterously deep well, the energies of a preposterously deep well should be a good approximation to the low-lying energies of an infinitely deep well. Because it's way up there, what difference can it make? And that's what we're seeing work out. Did that answer your question? AUDIENCE: Wait. I might have this backward. But when it says the high energy instead of looking right at the top of the well-- PROFESSOR: Yeah. OK, good. So this is an important bit of intuition. So when we say this is the energy zero, and the potential has a minimum at minus v0, and we're measuring the energies relative to zero, then the states at the top of the potential well are the states with energy close to zero. And the states at the bottom of the potential well are those states with the energy of order v0. cool? And what are the energies of all these states? They're order of v0. Right? Because this is-- Are they exactly v0? No, because this isn't linear. It's actually a circle. So there's going to be a correction, and the correction is going to be quadratic. If you work out that correction, it's correct. The depth above the bottom of the potential is correct for the energy of the corresponding infinite well problem. I'll leave that to you as an exercise. Other questions? OK. So there's another limit of this system that's fun to think about. So this was the infinite well limit. What I want to do is I want to take advantage of the observation we made a second ago that as we make the well deeper, we get more states. As we make the well more narrow, we get fewer states. To trade that off, consider the following limit. I want to take a potential well, which has a ground state. What does the ground state look like? So the ground state wave function is going to be-- So here's zero. And here's exponentially growing. Here's exponentially decreasing. And if it's a ground state, how many nodes will it have inside? How many nodes will the ground state have? AUDIENCE: Zero. PROFESSOR: Zero. Good. OK. So the ground state will look something like this. We more conventionally draw it like this. But just for fun, I'm going to draw it in this fashion. In particular, it has some slope here. And it has some slope here. Oh, shoot. Did I? Yes, I did. Dammit. I just erased the one thing that I wanted to hold onto. OK. So there's my wave function. It has some particular slope here. It has some particular slope here. And this is the ground state wave function, with some energy. I don't know. I'll call it this. Now, what I want to do is we've already shown that as we make the well more and more shallow and more and more narrow, the energy of the ground state gets closer and closer to zero. But there remains always a bound state. There is always at least one bound state. We proved that. Proved, as a physicist would. So I want to do that. I want to take this seriously. But here's the limit I want to consider. Consider the limit that we make the potential v goes to infinity, v0, while making l go to zero. So what I want to do is I want to take this thing, and I want to make it deeper and deeper, but more and more narrow. If I do this repeatedly, eventually I will get a delta function. And I will get a delta function if I hold the area of this guy fixed. Yeah? So if I do so, holding the area under this plot fixed, I will get a delta function. Everyone cool with that? So let's think, though, quickly about what's going to happen to the ground state wave function. So as I make the potential, let's take this wave function, and let's look at this version of the potential. So as I make the potential deeper, what happens to the rate of the oscillation inside or to the curvature inside? It increases. Right? So the system is oscillating more, it changes more rapidly, because phi double dot or phi double prime is equal to v minus e phi. It oscillates more rapidly. So to make it deeper, the system tends to oscillate more rapidly. However, as we make it more narrow, the system doesn't have as far to oscillate. So it oscillates more rapidly, but it doesn't oscillate as far. So what's going to happen? Well, as we make it more and more narrow and deeper and deeper, we again have the wave function coming in. And now it oscillates very rapidly. Let's do it again. The wave function comes in, and it oscillates very rapidly. And the it evanescent tail out. And now as we have a delta function, exponential damping, it oscillates extremely rapidly over an arbitrarily short distance and gives us the kink that we knew at the very beginning we should expect when the potential is a delta function. Right? From our qualitative structure of the wave function at the very beginning we saw that when we have a delta function potential, we should see a kink in the wave function. Because again, if we have phi prime prime is delta function discontinuous, phi prime is the integral of this. This is a step, and phi is continuous. And so here we have a step function. We get a discontinuity in the second derivative. Here we have a delta function in the potential, and we get a discontinuity in the first derivative if we get a kink in the wave function. Yeah? AUDIENCE: Would we get a jump there? PROFESSOR: Sorry. Say again. For e1? AUDIENCE: Yeah. Would we get a jump? PROFESSOR: Very good question. So let me do this more seriously. Let's do this more carefully. So the question is, for the first excited state, do we get a jump? Do we get a discontinuity? What do we get for the first excited state? Right? So let's talk about that in detail. It's a very good question. Example v is equal to minus v0 delta of x. Now, here I want to just warn you of something. This is totally standard notation for these problems, but you should be careful about dimensions. What are the dimensions of v0, the parameter of v0? It's tempting to say energy. That's an energy. That's an energy. But wait. What are the dimensions of the delta function? AUDIENCE: Length. PROFESSOR: Whatever length right? Because we know that delta of alpha x is equal to 1 over norm alpha delta of x. So if I write delta of x, which is always a slightly ballsy thing to do because this should really be dimensionless, but if I write delta of x, then this has units of 1 over length, which means this must have units of length times energy. OK. Just a little warning, when you check your answers on a problem, you always want to make sure that they're dimensionally consistent. And so it will be important to make sure that you use the energy times the length for the dimensions of that beast. So my question here is, is there a bound state? So for this example, for this potential, the delta function potential bound state, which again is this guy, is there a bound state? So again, we just ran through the intuition where we made the potential deep and deeper and deeper, v0 divided by epsilon over width epsilon. So v0, in order for this to be an energy has to be an energy times a length because we're going to divide it by length. So this is going to give us a delta function potential in the limit. We have an intuition that we should get a bound state with a kink. But let's check that intuition. We want to actually solve this problem. So we'll do the same thing we did before. We now write the general solution in the places where the potential is constant, which is on the left and on the right. And then we want to impose appropriate boundary conditions at the interface and at infinity, where these are going to be normalizable, and this is whatever the right boundary conditions are. So we are going to have to derive the appropriate boundary conditions. So let's just do that quickly. So phi with definite e is equal to a-- So in this region in the left, it's either growing or decreasing exponential. So ae to the alpha x plus be to the minus alpha x. And this is x less than 0. And ce to the alpha x plus de to the minus alpha x for x greater than 0. So first off, let's hit normalizability. What must be true for normalizability? Yeah, they'd better be converging to zero here and converging to zero here, which means that c had better vanish and b had better vanish. OK. So those guys are gone from normalizability. And meanwhile, if this is symmetric, what is going to be true of the ground state? It's going to be symmetric. So a must be equal to d. Great. So a is now just some overall normalization constant, which we can fix from normalization. So it looks like this should be the solution. We have an exponential. We have an exponential. But there's one more boundary condition to fix. We have to satisfy some matching. We have to satisfy the boundary conditions at the delta function. So what are those? What are those matching conditions? So we can get that from the energy eigenvalue equation, which says that phi prime prime is equal to h bar squared upon 2m. Sorry. Get your dimensions right. So it's 2m over h bar squared v minus e. In this case, v is equal to minus v0 delta function. That's very strange. So minus 2m over h bar squared v0 delta of x minus e-- I pulled out the minus. --so plus e phi. So this must be true at every point. This of course, is zero everywhere, except for at the origin. So what we want to do is we want to turn this into a boundary condition. And we know what the boundary condition is. If v is a delta function, that means that phi prime prime is also a delta function or proportional to a delta function. That means that phi prime is a step function. And how did I get that? I got that by integrating phi prime prime. You integrate a delta function, you get a step function. Well, that's cool. How do we figure out what step function discontinuity gives us? Let's integrate. Let's integrate right across the delta function and figure out what the discontinuity is. So let's take this equation, integrate it from minus epsilon to epsilon, where epsilon is a very small number. And that's epsilon to epsilon. So what is this going to give us on the left? Well, integral of a total derivative is just the value of the thing at a value to the point. So this is going to be phi prime at epsilon minus phi prime and minus epsilon. What does that mean? The difference between the derivative just after the origin and just before the origin. This is the discontinuity for very small epsilon. This is the discontinuity of the derivative at the origin at the delta function. And we already expected it to have a step discontinuity. And there it is. And how big is it? Well on the right hand side we have 2m minus 2m upon h bar squared. And we're going to get two terms. We get a term from integrating the first term. But over this narrow window, around, let's say epsilon was here, over this narrow window we can treat the wave function as being more or less constant. But in any case, it's continuous. And this is a delta function. So we know what we get from the integral of the delta function. We just get the value v0 phi at the delta function. Phi of the zero at the delta functions, so phi 0. We get a second term, which is plus the energy integrated against phi. The energy's a constant. And phi is continuous. So this, whatever else you can say about it, is roughly the constant value of phi at the origin times the energy times the width, which is epsilon. So plus-order epsilon terms. Everyone cool with that? So now what I'm going to do is I'm going to take the limit as epsilon goes to zero. So I'm just going to take that the discontinuity just across zero. So this is going to give me, of this gives me the change in the slope at the origin. OK. The derivative just after the origin minus the derivative just before the origin is equal to-- These order epsilon terms go away. --minus 2m upon h bar squared v0 phi. So that's my continuity. That's the condition for continuity of the derivative and appropriate discontinuity of the first derivative at the origin. And so this, when we plug-in these values of this form for the wave function, when we take a derivative, all we're going to do is we're going to pick up an alpha. And so when we work all of this out-- I'm not going to go through the algebra. You're going to go through it on the problem set. --when we take this condition, when we impose this condition with this wave function, it gives us a very specific value for alpha. This is only solvable if alpha is equal to mv0 upon h bar squared. Good. So let's just check the units. So this is momentum times length, momentum times length. This is mass. This is an energy times a length. So this has overall units of, p squared over m, overall units of 1 upon length, which is what we wan. So that's good. So we get alpha is equal to mv0 upon h bar. And that gives us the form of the potential. And it also tells us that the energy, plugging this back in, is equal to minus h bar squared alpha squared upon 2m, which we could then plug-in the value of alpha and solve for v0. So what we found is that there is a single bound state of the delta function potential, which we could have gotten by just taking this limit. It's a fun way to rederive the same result. It's a nice check on your understanding. So we find that there's a single bound state of the delta function potential. Now, what about an odd bound state? We assumed at an important point that this was even. What if I assume that it was odd? One node, what if we had assumed that it was odd? What would be true of the wave function? Well for odd, this would be a, and this would be minus a. So the value of the wave function at the origin is what? Zero. So that tells us the value of the wave function is zero. What's the discontinuity? Zero. So it's as if there's no potential because it has a zero right at the delta function. Yeah? But that means that this wave function, an odd wave function, doesn't notice the delta function potential. So is there a bound state? No. So how many bound states are there? Always exactly one for the single isolated delta function. On your problem set, you're going to use the result of the single isolated delta function, and more broadly you're going to derive the results for two delta functions. So you might say, why two delta functions? And the answer is, the two delta function problem, which involves no math-- Right? It's a totally straightforward, simple problem. You can all do it right now on a piece of paper. The two delta function problem is going to turn out to be an awesome model for the binding of atoms. And we're going to use it as intuition on your problem set to explain how quantum mechanical effects can lead to an attractive force between two atoms. See you next time. [APPLAUSE]
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_18_Hydrogen_and_its_Discontents.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: OK, so let me just quickly remind you of what we've done what we did last time. So unfortunately, today is going to be a little bit of what I wanted to do this time and a little bit of what I would have wanted to have been done in recitation yesterday, because all recitations were cancelled. So today is going to include a few steps along the way that I skipped over last time. OK, so to review from last time, we talked about central potentials. The energy operator had a radial derivative term, an angular momentum term, and a potential term. So this would be true regardless of whether we were central or not. But if we have a central potential, where this only depends on the magnitude of the radius vector, then we can use separation of variables, and write an energy eigenstate as a radial part-- which we will sneakily pull out a factor of 1 over the radius-- as we saw last time, that was particularly convenient. It took care of this r and made everything nice and simple. So a radial part times a spherical harmonic. The reason we use a spherical harmonic is that neither of these two terms depend on the angular coordinates. The angular momentum, we know what its eigenfunctions are, so if we use a spherical harmonic, this angular momentum becomes just a constant. And having separated in this form, and pulling out that sneaky factor of 1 over r, the energy eigenvalue equation reduces to a simple 1-D effective energy eigenvalue equation with a simple dr squared, and an effective potential, where the effective potential is the original central potential plus an angular momentum term, often referred to the angular momentum barrier-- which is proportional to-- as Matt says, as Professor Evans says, in 804, one is forced to say h bar squared upon 2m repeatedly. And he pointed out that it would be much more efficient if I would just come up with a simple variable for saying it or sound. So I will attempt to henceforth call it pfft. [LAUGHTER] So the effective potential is pfft, l, l plus 1 upon r squared plus the central part of the potential-- props to Professor Evans. And this is true in general for central potential. The last thing we notice of this is that since the energy eigenvalue equation has been reduced to this form, which depends on l, but is independent of m-- m appears nowhere here-- the energy can't possibly depend on m, although it can depend on l. It's just e of l. And thus, for every different value of m, I will get the same energy-- fixed l, different m. And there are how many values of m for a fixed l? 2l plus 1. So the degeneracy of El is 2l plus 1. Right? That's general for any central potential. It only requires spherical symmetry. So then we say the example of the cooling potential, where the potential is minus E squared upon r. And we solved-- or I quickly reviewed the solution of the energy eigenvalue equation. I'll write it that way-- where r of what I will call-- in place of E I, will simply write n. Rnl is equal to e to the minus r over 2 r naught n times r over r naught to the l plus 1 times some simple function v sub n of l of r over r naught, with the energy nlm being equal to E naught-- or I should say e Rydberg in our notation of last time-- upon n squared minus bound state. So let's talk about this quickly. This term-- the fact that it goes exponentially-- falls exponentially at large distances-- came from the asymptotic analysis at r goes to infinity. And this is just saying it's a bound state. The probability density falls off exponentially as we get to large radius. This term came from the asymptotic analysis near r goes to zero. And near r goes to zero, the only term that matters is in the effective-- this is one upon r. This is 1 over r squared, so this gets large more rapidly. This term is important, and that l, l plus 1 gave us r to the l plus 1. We work in dimensionless variables, rho. And here, I replace rho with r over r naught, where r naught was the characteristic scale we computed last time. And finally, this function V sub nl satisfied-- it again satisfied the energy eigenvalue equation, but having pulled out these factors such that asymptotically, its regular. So this function must be regular, smooth-- no divergences, no poles-- as r goes to infinity and to 0. OK. And then we solved this by series expansion. And from termination of the series expansion, we got that the energy-- we got a relationship between the energy and l and n, and in particular the relationship-- this came from termination of the series, just like it did for the harmonic oscillator. And the termination series gave us that the energy was equal to minus a constant over n squared. With one additional condition, which is that n had to be greater than l. n had to be an integer greater than l. And l, of course, has to be greater than or equal to 0. So let me just quickly tell you how this arises, because this is an important fact about hydrogen which I want you to understand. So the way that arises is we got the energy from a series expansion for this remaining function of rv and l of r. And that series expansion started out by saying let v be of the form sum over j of a sub j, r over r naught, the dimensionless variable rho, to the j, to the j-th power, so it's a power series. And when we plug this whole form, together with this series expansion, into that differential equation, we get-- just like we did before for a harmonic oscillator-- we get a set of relations between the various coefficients in that series. And we solve that by saying a sub j plus 1-- by deriving a recursion relation-- is equal to a sub j times 2, square root of epsilon times j plus l plus 1 minus 1 upon j plus 1, j plus 2l plus 2. OK, fine. We get this recursion relation. It looks absolutely awful, but it tells you that if you know a0, say, is a constant which is not 0-- if it's zero, then they're all 0-- then this function is identically 0, because a j plus 1 is a j times some constant. So this had better not vanish-- a sub 0. So the first term, the constant term doesn't vanish, which is good. It tells us V doesn't vanish, so we've already learned something. If V doesn't vanish identically at the origin, then we can determine a1 from a0 by multiplying by this. j is 0 in this, and l is whatever number we fix from ylm. And so on. We can deduce all the higher a sub l's. It's a series expansion. This is the recursion relation for it. However, note the following thing-- as j gets large, so we go to higher and higher orders in the series, then j is much larger than whatever fixed number l is, and much larger than whatever fixed number 2 is. So this asymptotes to aj times roughly 2 epsilon j upon j plus 1 j. At very large j, this just goes to 2 epsilon over j. Now, let's think about what that means. If I have that aj plus 1 is roughly equal to aj times root epsilon times 2 over j plus 1, I guess, then this tells me that every time I increase j, I get a factor of j plus 1 and another factor of 2 root epsilon. This tells me that a sub j is equal to a sub 0 times 2 root epsilon to the j over j factorial. But those are the expansion coefficients of an exponential-- quantity to the n over n factorial. But that's bad, because if this series is the exponential series with argument 2 root epsilon, then this series becomes E to the 2 root epsilon r over r0. But that's bad, because that would exactly swamp this term. That would defeat the asymptotic analysis we did at the beginning. If this term, V, is growing so large twice as rapidly as this is going to 0, then the net is diverging at infinity. That's bad. So in order for this-- for the series expansion-- to describe a good function that really does vanish at infinity, we need that this doesn't happen. So it must be true that j doesn't get large enough that the exponential in V overwhelms the decaying exponential from the asymptotic analysis. Everyone cool with that? So it has to terminate. But if it terminates-- let's look at the coefficient here. If it terminates, this says that 2 root epsilon times j plus-- and if it terminates, that says that there's some j maximum. Well, let me write it over here. If it terminates, that means there's some value j such that aj plus 1 is 0. But if aj plus 1 is zero, how can that possibly be? It must be that for that maximum value of j, the numerator vanishes-- specifically for that maximum value of j. Yeah? Well, what's the condition for that numerator to vanish for a specific value of j? I'll j sub m, or j max, the maximum value, such that a sub j max plus 1 vanishes. So j plus l plus 1 must be equal to 1. Or said differently, epsilon-- which is the energy in units of e0, must be equal to 1 over 4 g max plus l plus 1, quantity squared. Cool? So that means that our solutions, or states psi, are now labeled by three integers. they're labeled by l And by m from spherical harmonics, but they're also labeled by the value j max. And j max can be any number from 0 to infinity, right? Strictly, it shouldn't be infinite, but any finite number, any countable, any integer-- because eventually the series terminates at j max. So it's labeled by these three numbers, and the energy of lmj max is equal to e0 minus over 4. And this e0 over 4 we already saw was e Rydberg, 13.6 dv, times what value? Well, here's the energy in units of e0-- jm plus l plus 1, quantity squared. Where L is greater than or equal to 0, j sub m is greater than or equal to 0, and 1 is 1. But this is sort of cumbersome. We can just as well call this quantity n, where n is greater than l as an integer. It can't be equal to l, but it could be 1 greater than l, or any greater integer. Is that cool? So instead of labeling it by jm, the point at which the series terminates, we'll label it by n, and you can deduce what jm is by subtracting I plus 1 from n. That tells you where the series terminates. Notice that this tells you something nice. This power series for V, if n is equal to l plus 1, has how many terms in it? So say it out loud, just think about it a second. If n is equal to l plus 1, how many terms are there, that are non-vanishing in the power series? I won't call on you, but raise your hand when you have an answer. What's the value of j max if n is equal to l plus 1? So what term is the first one to vanish in the series? OK, so how many terms are non-vanishing? Yeah, just the one-- just the first term. So when l is as large as it can possibly be-- when l is equal to n minus 1, V is a constant function. It's just constant. When l is one less, when l is n minus 1, or n is l plus 2, then V has two terms in it-- a constant, and a linear. So the greater the difference between l and n, the more terms there are in the series. OK? So anyway, this leads to this form of the energy. But as is usually the case, this sort of a brute force analysis doesn't give us a whole lot of insight into why we get the qualitative features we do. In particular, the qualitative feature that stands out most obviously to me-- I don't know about you guys-- is the fact that the degeneracy is now much, much larger. In particular, we know from basic principles of a central potential-- as we just reviewed-- that anytime a potential is symmetrically invariant, is spherically symmetric-- any time the potential is spherically symmetric, rotationally invariant-- then there must be degeneracy of 2l plus 1 for every energy eigenstate-- there must be. The l could be zero, but if there's any angular momentum, then that degeneracy must be 2l plus 1. But this is much more degenerate. And in fact, how degenerate is it? Well, for any given En, the degeneracy is equal to-- well, what are the possible values of l? So we have some state for every different value of l. l could be equal to 0, and it can go up to n minus 1, right? Because, again, n is some integer which could vanish, l plus 1. So as small as this can be is 0. So n can be l plus 1, or anything greater. I.e.: l can go up to n minus 1, but it can't be any greater. Then for every value of l, there is a state n-- there's a value of n which could be minus l all the way up to l in integer steps. So how many states are there? Well there's one state for every value of n, l, and m. So it's this sum. The sum on m for minus l to l is 2l plus 1. So this is the sum n minus 1, l equals 0, of 2l plus 1. And this is equal to, kind of beautifully, n squared-- just arithmetic series. So we have this huge degeneracy, which is much larger than 2l plus 1. It's the sum over 2l plus 1 from l to 0 to n minus 1. Where is this degeneracy coming from? Why? So quantum mechanically, we know the following-- we've learned the following-- and I hope it's under your fingernails at this point-- that when you see a degeneracy, you should expect that degeneracy to follow from some symmetry. There's some symmetry of the system that is leading to a degeneracy by virtue of the fact that the generators of that symmetry-- for example, for rotation it's angular momentum-- commute with the energy. And when you have an operator that commutes with the energy operator, as we've seen, you can generally construct new states given a single energy eigenstate by acting suitably with that operator. So we should expect there to be some new symmetry. But it's very hard to see what symmetry. I mean, this is hydrogen. This is central potential. It's just some stupid central potential. What's so special about the hydrogen system as opposed to, for example, the harmonic oscillator? Which hopefully you will see in recitation at some point. The harmonic oscillator in three dimensions has degeneracy 2l plus 1. It's a central potential. It has to have degeneracy 2l plus 1-- but not n squared. Where did that come from? What's so special about the Kepler problem? So there must be some symmetry, but it's not obvious what it is. But now let's use a note from classical mechanics. Noether's Theorem tells us when we have a symmetry, we have a conserved quantity. We also saw this in the quantum mechanical version for expectation values. We have a conserved quantity. And in quantum mechanics, having a conserved quantity means that the energy commutes with some quantity. Because that controls the time evolution of the expectation value. So there must be some quantity, which I've written by a question mark, that commutes with the energy operator, specifically in the case of the harmonic oscillator. So what is that quantity? So actually, this quantity in classical mechanics was studied, because the same thing is true in the Kepler case. In the Kepler case, the orbits close, and they have simple ellipses, and they have all sorts of nice properties that beg for an explanation in terms of symmetry. And it was pointed out by a number of people-- Laplace, Runge, Lenz, various people-- and this is often called the Runge-Lenz vector, but I think you can blame it on any number of people-- that there's a vector that in classical mechanics, for the Kepler problem-- or for the Coulomb problem, is conserved. And that quantity is p cross l-- momentum cross angular momentum-- minus m e squared-- and let me write m sub electrons-- it's the mass rather than the quantum number little m-- e squared, r vector. So it turns out this quantity is classically conserved, and if you make a quantum operator out of it, then e with A is equal to 0. Add hat. This was enormously non-obvious. If it's obvious to you, you're a freak. It's really, really not obvious. You can go through and do the calculation. And when I say really not obvious, I mean-- well, I'll show you what I mean in just a second. So you can go through and do the calculation classically. Is this quantity conserved in the Kepler problem using Newton's law? And the answer is yes. It's amazing. Quantum mechanically, you can do the same thing. But I'm going to warn you for anyone who has the chutzpah to try-- because I encourage you to, but it takes a little bit of brawn-- think about this operator for second. Does this classical quantity have an obvious interpretation as a quantum operator? No, because l is r cross p. But now we have prp. There's an ordering ambiguity. So you have to decide which order do you put that p and that r, and the other p-- ppr, rpr, prr-- right? You can write out which components you mean here. For example, you mean for the z component of a, you're going to get px ly, and ly px. But within that, how do you order the p's and the r's, because ly contains px, and in particular, it contains x. So there's an ordering ambiguity. So just be a little bit careful about this as a quantum mechanical operator if you do play with this. But if you do it, and you're thoughtful about it, it's pretty easy to see that it can be computed and checked that, in fact, the commutator vanishes. So at this point, it's tempting to say, aha! There's an extra conserved quantity. We declare victory. The problem is, this didn't give us the symmetry. This just told us there's a conserved quantity. What's the symmetry behind this? What's the symmetry that insures that this is a conserved quantity? It's obviously not rotations. It's something else. And so there are many answers to this question. One answer involves an explicit expression for in phase space, the change of variables dx-- and this is a variation that depends on f of x and p. So there's a little vector parameter, and the change of the physical coordinates depends on both the coordinates and the momenta. This is a slightly strange thing, because it's like a change of variables where the position gets mapped into some non-linear function of the positions and momenta. That's a little bit weird, but you can do this. But I don't find it a terribly satisfying answer. And here's the answer I find most satisfying-- it's not terribly useful for our present purposes, but you're going to run into this again in 805, when you do what's called the operator method for the Coulomb potential, and it turns out to be an enormously useful machine. We're not going to use it in 804, but I want to advertise it for you. I want to give you a description, though, that isn't the description you're going to get in 805. The description is the following-- 1935, guy named Fock, crazy guy, amazing, amazing physicist-- played a huge role in the development of quantum field theory. I'm not exactly sure why he was thinking about this problem, but he was, and this was a very sort of Soviet thing to do at the time, I guess, to be hard core and mathematical. So off he was going, computing things, and he observed the following fact-- how he observed this, again, crazy long story. He observed the following fact-- if you take a Kepler problem and you work in momentum space-- so you write everything in terms of the momentum-- it turns out that the Kepler problem, or he referred to as the Kepler problem, but I'll refer to it as the Coulomb problem in 3D, three spatial dimensions-- this is really crazy-- is exactly equivalent. There's a change of variables-- it's kind of complicated, but it's just a projection, it turns out. It's exactly equivalent to the problem of a free particle in four dimensions, constrained to the surface of a sphere-- to a three sphere in four dimensions. So you take a three sphere in four dimensions. You take a marble. You kick it, but make sure that it's stuck to the sphere. It will move in perfect circles, right? Obviously-- it's moving on a sphere, it's just going to move in great circles. OK, that's nice and simple. What's the symmetry group of that system? Well, the sphere is invariant under any rotation in four dimensions. STUDENT: [INAUDIBLE]. PROFESSOR: Ah, SO4-- there's a symmetry SO4-- special orthogonal transformation. Special means it doesn't change length. Orthogonal means that it's a rotation. So it's orthogonal rotations in four dimensions, and so now, this should sound familiar, because the angular momentum group in 3D is SO3. It's rotations that don't change the length in three dimensions. OK? How many elements does it have? How many conserved quantities go along with this? Well, how many conserved quantities go along with rotations in three dimensions? What's conserved by virtue of rotational invariance in three dimensions? Angular momentum-- how many components of angular momentum are there? Three, right, there are three. So it turns out that if you calculate in SO4 how many conserved quantities are there, there are six. And when you do this mapping back down from the free particle in 4D to the Coulomb problem in 3D, three of them are the l's, and the other three of them are this A vector-- comes out spot on. So what it's telling you is that the spectrum of hydrogen-- or at least the I shouldn't say of hydrogen, the spectrum of the Coulomb system-- has an enhanced symmetry. It's the symmetry group of rotations in four dimensions. Why? I don't know, it's just kind awesome. But I think that's about the best that can be said. It's just true. So for certain very special potentials, we find that this sort of thing happens. It happens in other systems, too. You discover that there are accidental symmetries in your system. They shouldn't be referred to as accidental symmetries. They are enormously nontrivial functions of the rest of the system, and we're going to break the symmetry in a minute, and you'll see why we shouldn't refer to it as accidental. But in any case, the answer to the question, why do we have this giant degeneracy in the Coulomb problem-- why is it n squared rather than 2l plus 1-- is that there is an enhanced symmetry group. There's more symmetry then you would have thought. And there's a conserved quantity associated with it. And that conserved quantity corresponds quantum mechanically to an operator that commutes with the energy operator. OK? Yeah. AUDIENCE: [INAUDIBLE]. PROFESSOR: Indeed, indeed. AUDIENCE: So how do you get three conserved components [INAUDIBLE]. PROFESSOR: Excellent. What do I mean in classical mechanics for a quantity to be conserved? I mean that the time rate of change of that quantity, under the classical dynamics, under classical evolution, is preserved. So what's the quantum mechanical version of that? So the mechanical version of that is, well, can we say that d dt of some operator-- or of some observable a is 0. Does this makes sense? No, because it turns out, in the language we've been using-- in the Schrodinger evolution description that we've been using-- this doesn't really make sense, because this thing is an operator. That's not a quantity. Quantity is something you measure. So what do you measure? Do you measure the operator? In a state, you measure its expectation values. You might measure its eigenvalues, but the more general thing is expectation values. So this is something that depends on time, because the wave function depends on time-- psi of t. OK, so the best to say something is conserved, is to say d dt of A is 0. But we've seen what the condition is for the expectation value to not change in time. In general, for any operator A, the time rate of change of its expectation value is given by the commutator of E with A. I can do the symbols-- good. OK, so in particular, if this commutator vanishes, if the commutator vanishes-- equals 0--, this is equal to-- AUDIENCE: [INAUDIBLE]. PROFESSOR: Sorry? Well, the wave function can be time variant, and I'm going to assume for the moment that the energy operator and the A operator are not, themselves, explicitly time dependent. Yes, that's an important assumption. So let me just assume for simplicity that neither the energy nor the operator A that I'm looking at are explicitly time dependent. Like, it shouldn't be, like, x plus tp. That would be annoying. So let me assume that that's not the case. You do have to deal with that sometimes. For example, if you are dealing with a system that changes in time, where there's a background magnetic field, or something like that, that changes in time. But module that sort of detail. And also, there's an i h bar, so where do the i h bars go? Do the units make sense here? A, A, good. E and d dt-- whoops. So where do the i h bars go? Well, it can go on either side. You have to tell me if they go up or down. What is the Schrodinger equation? Yeah, exactly, i h bar, OK so we need an i h bar. [LAUGHTER] OK, this spring has got to us all. So now, in the case that e with A as 0, imagine we have an operator a that commutes equal to 0. Then d dt of the expectation value of A in any state is equal to 0. And that's what I'll call conserved. So classically, to be a conserved quantity means that it's time independent. Quantum mechanically, to be a conserved quantity means that it commutes with the energy. Now, going back to the question-- the question was, look for angular momentum. There are only two numbers you can know at any moment in time. You can't know three. You can know the total angular momentum, little l. And you can know the angular momentum in a particular direction, which we conventionally call z-- little n, right? But you can't know some other angular momentum, like angular momentum in x direction simultaneously. So what does it mean to say that there are three conserved quantities? I can't know three conserved quantities at the same time. They're not even well defined at some moment in time. The answer is there are three operators-- lx, and ly, and lz-- which I will simply write as l vector-- all of which commute with E. So while it's true that systems don't have definite values of lx, ly and lz simultaneously, nonetheless lx, ly and lz all commute with the energy operator. And as a consequence, their time expectation values are all 0 in a central potential. So the time derivatives of their expectation are all 0s in a central potential. They're all constant in a central potential. Yeah. AUDIENCE: [INAUDIBLE]. PROFESSOR: You mean 1 over r? STUDENT: Yeah, 1 over r. PROFESSOR: So the question is, if you don't have 1 over r, do you get extra-special things that happen for other potentials that aren't 1 over r, but something else? AUDIENCE: [INAUDIBLE]. PROFESSOR: It's not quite a unique case. So the question is-- this is a very good question. So the question is this-- this is a great question-- look, there's something peculiar about 1 over r classically. The orbit's closed. You get ellipses. You don't get progressing ellipses. That's what you get if you have an n harmonic. If you have an extra 1 over r to the sixth term, or something, a small correction to 1 over r, which we do in the real solar system. It's a many body system instead of a two-body system, so r orbit doesn't close. It's close, but-- so that's special. And here we see that there's an enhanced symmetry. Are these things related to each other? So the answer is almost, in the following sense-- for the harmonic oscillator, we also have it close, but there isn't a Runge-Lenz vector. However, the orbits close, so there's something special about that one, too. Is there a conserved quantity there? And it turns out the answer to that is yes. There's an enhanced symmetry there as well. But it's not like this. It doesn't lead to this gigantic degeneracy. There is just extra structure that's nice about the harmonic oscillator. But for the harmonic oscillator, of course, it also closes. It's not this orbits this. So there's a nice structure there, but it's not quite so simple. OK, yeah, one last question. AUDIENCE: [INAUDIBLE] Kepler's problem. I was wondering if it's at all correct thinking [INAUDIBLE] symmetry or degeneracy [INAUDIBLE] equal time instead of equal area? PROFESSOR: Yeah, it's not unrelated. I don't know how to make that connection sharp, but the thing I can say sharp-- so the question is, look, is that conserved quantity somehow related to the fact that there's equal time in equal-- equal areas swept out in equal time at all points in the orbit? Which is a true thing about a Kepler problem. That's a good observation. I don't know how to make that connection, but it is true in the following sense-- that both of those facts are guaranteed by the fact that you have closed orbits with a particular ratio. And that is, indeed, can be blamed on the conservation of the Runge-Lenz vector. I don't know how to make that connection direct and explicit, but indeed it is. So that's fun question. I'll have to think about that. That's a good one. One of the totally awesome things about teaching 804 is that every single time, someone asks a question I've never even thought of before-- forget never answered. Never even thought of. I never thought of that. It's a great question. OK, so everyone cool with the degeneracy? We have this gigantic degeneracy. So knowing what the degeneracy is, or where it's coming from, at least more or less-- is progress. But if you really understand why something is conserved, then you can break that conservation. You can break that degeneracy. So here's my question, is it possible-- can we, or indeed must we, can we break or lift this degeneracy? So first, let me tell you what those words mean. So we have this gigantic degeneracy of the n squared of the Coulomb potential. Can we break that degeneracy or lift it? What I mean by lift the degeneracy, what I mean by lift the degeneracy is the following-- suppose I have a system that's degenerate, so it's got two energy eigenstates with the same energy eigenvalue. So this is energy. Then if I can kick the system in some way, or modify it, or bend it such that the energy levels are no longer to degenerate, one state has been lifted above the other. And so we use the word to lift a degeneracy. Now you can see that they're two different states by measuring the energies. It's also said to split a degeneracy, or to break a degeneracy. So those are just words. OK, so can we break or lift this degeneracy? So sure. How do we break a degeneracy? How do we lift a degeneracy? We need to break the symmetry. So if we don't have the symmetry, we won't get the degeneracy. We've seen this before. So first off, let's think about what are the two symmetries that we want? Well, we got this degeneracy of 2l plus 1 from spherical symmetry. And we got this extra degeneracy of n squared from conservation of the Runge-Lenz vector. And conservation of the Runge-Lenz vector came from the fact that we had exactly the Coulomb system. So anything that isn't exactly the Coulomb system, but is still rotationally invariant, should still have the 2l plus 1 degeneracy, but not the n squared degeneracy-- like the harmonic oscillator, for example. So let's change the system. Let's find a small correction to the system that leaves it mostly the Coulomb problem, but with some small correction. So what comes to mind? So one good way to think about a physical, natural way to break this degeneracy, to break the symmetry, is to think a real system that's described by the Coulomb potential. So the system that we usually talk about is hydrogen. Hydrogen is the proton. It's got a plus charge, and it's got electron. And it's attractive with a minus E squared upon r. So in hydrogen, is this the energy operator for hydrogen? Is this a good description of the energy operator for hydrogen? It's a good start, but it's not exactly right, for a bunch of reasons. And in fact, in your first problem set, when you computed the time it takes for hydrogen, classically, to decay from the classical Bohr radius, one of the things you discovered is that the electron starts out being non-relativistic, but very rapidly becomes relativistic. And rapidly is 10 to the minus 15 seconds. This is a non-relativistic, p squared upon 2m kinetic energy. That's probably not a perfect approximation. So the first example is going to be relativistic corrections. Now, will relativistic corrections break rotational symmetry? They can't, because relativity doesn't say, this is the z direction. In fact, it does quite the opposite. Relativity, among other things, says, don't get too hung up on what's the z direction. But relativistic corrections will not break the symmetry. So if we include relativistic corrections to the kinetic energy, this will change E, change the energy operator. But it will preserve the rotational symmetry. E with l is equal to 0. So we won't break this rotational symmetry, so we'll preserve the 2l plus 1. So let's check to see that this is true. So to check to see that this is true, let's actually estimate out the correction to the energy due to the first nontrivial relativistic effect in the atom. The first nontrivial, relativistic effect can be calculated [INAUDIBLE] kinetic energy in special relativity, of a particle of mass m. So that's easy. It's the total energy is the square root of m squared c to the 4th plus p squared c squared minus mc squared. That's the rest energy. That's the total energy, so this is the kinetic energy. But this is equal to-- well, if we pull out a factor of m squared c squared, this is m c squared times the square root of 1 plus p squared over n squared c squared minus 1. And now if we Taylor expand this-- because if the momentum is small compared to mc squared-- or mc, I should say-- the momentum is small compared to mc which is, morally speaking, at low velocity is, the velocity is small compared to c, then this is 1 plus a small number, and we can Taylor expand this to get 1 plus 1/2 a small number. And the 1s will cancel, giving us only 1/2. So this is going to be equal to 1/2 of that small number, p squared over m squared c squared, times mc squared upstairs, giving us the first term, which is p squared upon 2m. That's the familiar classical kinetic energy. Rock on. What's next correction? The next correction is the next term for this Taylor series, which is going to involve this quantity squared, so it's going to be p to the 4th over m to the 4th, c the 4th times mc to the 4th. The coefficient is a minus 1/8, so we get minus p to the 4th over 8m cubed c squared. And that's down by a factor of m squared c squared from this guy, n of p squared upstairs-- which is what you'd get-- p squared m squared, over c squared, OK? So the correction to the kinetic energy for a given momentum p is to correct it down by p to the 4th term. Everyone cool with that? Yeah. AUDIENCE: Why do you make the assumption [INAUDIBLE]. PROFESSOR: Because you, like Schrodinger, did problem Set One. And problem Set One said, if you have an electron at the Bohr radius classically, its velocity is about 1/400 of the speed of light. 200? It's small. It's a couple of orders of magnitude smaller than the speed of light. So that number, that ratio there, is 1 upon 200, or 400-- I don't remember-- squared. That's a very small number, so it's a reasonable approximation. AUDIENCE: [INAUDIBLE]. PROFESSOR: Because what is the typical-- that's an excellent question. So why am I talking about it as if the electron is sitting at that radius? And the answer is, well, look-- what is the average momentum? What is the average value of p squared in, let's say, the ground state of the hydrogen atom? What's the average value of p? What must be the average value of p in the ground state of the hydrogen atom? Zero on two grounds-- first off, p is a vector, and the ground state of hydrogen is rotationally invariant. It doesn't break anything. Secondly, it's not going anywhere. It's a bound state, so it's just sitting there. It's got an average momentum of 0. If it carried momentum, it would be cruising. So it's got average momentum 0. OK, fine, but what's the average momentum squared? What's the average of p squared? We can actually do out that calculation. You're going to have to do that calculation on the problem set. And it's a small number. And it's, in fact, almost exactly what you'd guess classically from the Bohr radius. So you have to know the h bar to compute the Bohr radius in the first place. So the answer is, to the best of anyone's ability to define what's the momentum, or the typical scale of momentum, that typical scale of momentum for the electron in the ground state of hydrogen is extraordinarily small compared to the speed of light by a factor of 100, 200, or 400. Does that answer your question? AUDIENCE: Yes. PROFESSOR: It's a very good question. OK, so here's the kinetic energy. And by the way, another way to say that this approximation must be good is that we've already computed-- ignoring the relativistic correction-- we computed the ground state of energy, and it was 13.6 Ev, which is pretty good, since the binding energy of hydrogen is about 13.6 Ev. So apparently, that was a good approximation. And you might be disgusted by answer analysis like that, but that's what physics is. You write down some stupid, cockamamie model that you know is wrong, but if it does a good job of fitting the data, you declare triumph, and you call Stockholm. [LAUGHTER] So it depends on the problem you've just solved approximately. So here's the first correction. And now, what this tells us is therefore, the energy of hydrogen-- or at least a better model of hydrogen-- I'll call it hydrogen tilde-- is equal to E Coulomb minus p to the 4th over 8m cubed c squared. OK so this, however, we know is small compared to the kinetic energy, because 13.6 is pretty close to 13.6. Notice I didn't include the rest of the significant digits. So this must be in the insignificant digits. So this must be small. So here's what I want to ask-- we know that the energy of the Coulomb problem, nlm, is equal to e0 squared minus e0 squared over n squared, independent of l and m. What's the energy in this toy model of hydrogen that includes the first relativistic correction-- nlm. Well, to answer that question we have to resolve the energy eigenvalue equation, right? We have to refined the energy eigenstates, and we have to solve that. And this is going to be a much worse problem, because it's going to involve a p to the 4th term, which is going to involve four derivatives. That sounds horrible. Can anyone think of a better way to just approximate, estimate the magnitude and the value of the leading correction from this quantity? Yeah. AUDIENCE: Some variation on take an eigenvalue, plug it in, see what the error is, correct for that error-- something like that. PROFESSOR: There's an iterative method for doing that. It's a very good guess, but there's an even easier way to estimate it. That's something we will call perturbation theory when we do this in 805. And doing that systematically is exactly the right answer. However I want you to do just the leading part of that. Yeah. AUDIENCE: Dimensional analysis. PROFESSOR: Oh, dimensional analysis would be awesome. That's fantastic, brilliant. So how would you compute p? OK, so we didn't calculate p. You can do a dimensional analysis and get a p, because you have an e, you have an l, you can probably do it. But here's the problem with that, though-- this is the correct first answer to any question of this kind-- correct first answer is plug in dimensional analysis, and get an estimate for the order of magnitude. But we want more than the order of magnitude here. We're interested in splitting the degeneracy due to this interaction. So what that means is we want to see that different values use of l lead to different energy. What we care about is the l dependence that comes out of this correction. And we're not going to be able to get the l dependence from dimensional analysis, because l is an integer. It is dimensionless. Yeah, so that will give you the correct magnitude, and that is the first thing you do. What's the second thing you do? AUDIENCE: Stick in the expectation value of p to the 4th. PROFESSOR: Excellent. Let's just stick in the expectation value p to the 4th. That's not exactly the right thing to do, but if this is small, that's probably pretty reasonable. And what you'll find when you do it systematically using perturbation theory, as was pointed out earlier, we can take the entire system, the entire energy eigenvalue equation-- we can think of this 1 over m cubed c squared as a small number, and we can do perturbation theory on that small number. We can Taylor expand everything in that small number. And when you Taylor expand the exact equation, what you discover is the leading term in that Taylor expansion is the expectation value p to the 4th over 8m cubed c squared. OK, so this is a fantastic guess. OK, so let's compute it. So E hydrogen is going to be equal to E nlm, which I'm just to write as minus E0 over m squared. OK, so what is the correct answer here? So it's this quantity minus-- so this is making it more tightly bound. Notice the sign, because p to the 4th is strictly positive if p is real. So minus E0 over m squared, and if you go out and you do this expectation value, which I really should have put on the problem set-- oh, I still can. It hasn't been posted yet, so it'll be posted this afternoon. But I won't. It's not that bad, actually. [LAUGHTER] Matt, what do you think? AUDIENCE: No! [LAUGHS] PROFESSOR: We'll discuss this afterwards, OK? Which may or may not appear on your problem set. So if you go ahead and you do this calculation, what you find is the correction is En squared, so E0 squared over n to the 4th-- this time it really is n to the 4th-- and now you have to worry about dimensional analysis. This is two energies. We want one energy, divided by-- wait for it-- mc squared, because it's a relativistic correction. That shouldn't be too surprising. Times the quantity 4n over-- and here where life gets awesome-- l plus 1/2 minus 3. [LAUGHTER] PROFESSOR: You laugh, but I feel delighted by this for so many reasons. One is, that's beautiful. What a crazy combination of symbols. But the second and more important one is, look it has an l in it, and the l is downstairs, right? That's good. It has more leverage that way. But so we've got this l appearing, so the energy depends on l. Good, we've broken the degeneracy. Different values of l give different energies. So let's do that graphically. So in particular, what this tells us is that now while the Coulomb energy commuted with the Runge-Lenz vector, it must be true that the energy of this model of hydrogen, which includes the first relativistic correction, does not commute with the Runge-Lenz vector. And indeed, if you check this-- which is effortful, but not intellectually difficult-- indeed, it doesn't vanish. And the failure of this conservation equation, the failure this commutational relation to vanish, tell us that it's possible for the system to not be degenerate. And lo and behold, it is not degenerate. OK? Yeah. AUDIENCE: So I just wanted to clarify something that-- did you say before that the actual first term when you encounter relativity just includes substituting the non-relativistic p to the 4th? PROFESSOR: Yes. Sorry, I should have said that. There are other things are important for relativity that we'll come to in a little bit. But if you just look at the kinetic energy, this is the first non-trivial correction. Yeah. AUDIENCE: So we still have the 2l plus 1 degeneracy. PROFESSOR: We still have the 2l plus 1 degeneracy, exactly. Because does this correction violate rotational invariance? No, that's p squared. That's a scalar under rotation. It's just the number, not a vector. But E still commutes with l. And you see that because this is independent of little m. This is the mass of the electron. So this is independent of the quantum number little m. It only depends on the quantum number little l. So indeed, it is still degenerate. So let's go through this and see the degeneracy explicitly. And let me do that by drawing you the following diagram. So what happens if we have E1, which is the ground state? It has some energy which is negative, which is minus E0. And when we include this first correction, what does it do? Well, it decreases it by some amount. Everyone cool with that? It doesn't change the number. It doesn't split any degeneracy, because there was only one ground state in the first place, because l had to be 0 when n is 1. l has to be one less than n or smaller. And if n is 1, l has to be 0. It can't be any smaller than 0, so it's 0. There's just a one state, with degeneracy 2, 0 plus 1, which is 1. OK so that's fine. It just changes the ground state by some amount. Let's consider the second excited level. So in the second excited level, we have the states-- and let me name this slightly differently, so this has energy minus E0. The ground state had l equals 0, and m equals 0. It had to. For the second excited state, n equals 2, how many states are there? 2 squared is 4, so there are four states, two, three, four. And what are their values of l and m? Well, if n is 2, l can be either 0 or 1. If it's 0, then m has to be 0. If it's 1, m could be 1, or it could be 0, or it could be minus 1. So this, again, is lm, lm, lm. So those are the four states with energy E2 in the hydrogen atom. And now when we ask, what are the energies of these three states, when we include this so-called fine structure, what is this going to do? Well, the ground state is going to get pushed down by some amount. I should draw this differently to make it more vivid. This gets pushed down by some amount, with some value of l. l here is 0. But for these three states, which all have the same value of little l, 1, the denominator is larger. So the amount by which it gets pushed down is less. So these three states get pushed down less, but the same amount as each other. That cool? So these three states-- degeneracy is 3, and this state degeneracy is 1. Now, notice that 1 plus 3 is equal to 4 is equal to n squared. So these states split into multiplets-- groups of states-- which have to have the same energy by rotational invariance. If they have l equas 1, there must be three states with the same energy-- there must be. And states with l equals 0, there's just the one state. And if we did the same thing for E3, let's just think about what would happen. So I'm just going to write the l. I'm not going to include the m. We have the l equals 0 times 1. We have the l equals 1 state, times 3. And we have l equals 2 state. And how many states are there in the l equals 2 state? Times 5, good. And again, this gets pushed down to some value. This gets pushed down to some value. This gets pushed down to some value, where this is one, three, and five. And again, these add up to 9, which is 3 squared-- the degeneracy. So by turning on the relativistic corrections, we break this degeneracy. We lift this degeneracy. You break the symmetry, you lift the degeneracy. Now, unfortunately, this isn't one we can control by turning a dial in the lab. No experiment that I can do in the lab will change the value of the speed of light. Well, actually there are experiments that will change the value of the speed of light. If you work in medium, the effective speed of light is different. So that's going to change relativistic effects. So that's interesting. That tells you that you can actually vary this continuously if you can vary continuously the index of refraction of the ambient system. I invite you to think about how you might do that experimentally, because it's a fun question. OK, however, what I'd really like is something where Professor Evan in his lab can actually turn a dial, and change the spectrum in a way that explicitly breaks the symmetry. Meanwhile, I also want an example that breaks the rotational invariance. I want to break the 2l plus 1 states up into 2l plus 1 states with different energies. I want to make them depend on the angular momentum in some direction. What must I do in order to lift that degeneracy? AUDIENCE: Add a magnetic field. PROFESSOR: Fantastic. If you add a magnetic field, why is that going to break the degeneracy? AUDIENCE: [INAUDIBLE]. PROFESSOR: Give me an answer that depends, not on your knowledge of chemistry, but give me an answer that depends on symmetries. Why doe turning on a magnetic field have the potential to split the degeneracy of the 2l plus 1 states? AUDIENCE: You don't have rotational symmetry. PROFESSOR: Thank you, exactly. You have no rotational symmetry, exactly. So that's pretty good. So let's break the 2l plus 1 by breaking rotational symmetry. There are many ways we could break rotational symmetry. We could turn on an electric field. We could turn on a magnetic field. We could put a shovel in the room. But anything that breaks rotational symmetry will do it, but a very convenient way is to do it with a magnetic field. Magnetic field is particularly beautiful. So this is effect is now call the Zeeman effect. I believe it was Faraday-- one of the classic electromagnetists of the 19th century attempted to look at the spectral lines coming off glowing gas, and see if they changed by turning on an electric or a magnetic field. It turns out to be a tricky experiment to do, because you need to have a very well-controlled magnetic field, otherwise what you get is just a bunch of schmutz. So to get a clean spectrum, you have to have a very uniform magnetic field, which is not trivial. It has to be time independent. So Zeeman, at the very end of the 19th century-- in fact, this experiment was done, I believe, in '96. So I think the paper came out in '97 but the experiment was done in '96. Zeeman did this experiment. He said look, if i turn on a magnetic field next to a hot, glowing gas and look the spectrum that comes off, anything happen to it? Why would you think that anything might happen? And the answer to this one is really quite nice. Look, if we compute the energy, the energy is going to be equal to the Coulomb energy plus a correction, which is the following-- if you think of the atom as an electron that carries some angular momentum in the presence of a proton, if you think of the electron as a thing carrying some angular momentum, and a thing with a charge-- it has a charge and angular momentum. It's going to have magnetic moment, mu. And the magnetic moment is going to be equal to the current of the charge times the area swept out divided by c, because we work with sensible units. And this is equal to minus-- for the case of the atom-- minus mu, the magnetic moment of the system dotted into the magnetic field that you turn on in the background. So this is a background magnetic field. I am turning on an electromagnet, which induces a magnetic field in some known direction, with some known magnitude. So my dial is the magnitude of that B field. I could make it large, or I could make it small. And it's in a known direction. And the point is, our electron because it's orbiting, because it carries some angular momentum and it carries some in charge, has some magnetic moment. So it behaves like a little bar magnet. And that bar magnet wants to anti-align with the background magnetic field that I've turned on. So there's an energy penalty to not being aligned. This is the usual dipole-dipole potential from electrostatics. OK and mu here is equal to minus E over 2mc-- angular momentum. So this can be computed from a simple model of the atom. Whatever, it's easy to compute. So this is slightly annoying, because if E is in some arbitrary direction, let's just simplify our life and say that's equal to E Coulomb minus mu B. And I want to turn on a magnetic field of known amplitude in a known direction. Let's call it the z direction. So this is mu z. But this mu z, that's just mu z is lz times minus E over 2mc. So this is going to be plus E over 2mc, Bz-- I'm going to put this all together, Bz Lz. Everyone cool with that? So where this is coming from, again, is just the angular momentum is telling you the velocity. The velocity is telling you the current. And then the area is the radius. So that's what's giving you the radius and the angular momentum. So this is our energy. What does this tell you about the energy eigenvalues? They depend on m, right? And do we have to solve the energy eigenvalue equation again? No, because our energy eigenfunctions of Ec are proportional to ylm, so they're already eigenfunctions of lz. So when I take this beast, and I act on the wave functions I erased, but on the 1 upon little r, capital R, sub nl, y sub lm-- when I act on the ylm's, this is just going to give me h bar m, right? So this is just a constant when acting on the Coulomb energy eigenfunction. So this is a much easier problem. That way, we don't even have to make any approximations. We can just say exactly energy Zeeman is equal to minus E0 upon n squared, nlm plus E h bar upon 2mc, Bz-- the background magnetic field, and this is the m of electron mass-- times little m. And I'm going to write little m sub l, just to distinguish little m sub l from m sub E. So when we turn on the prediction here, when we turn on the magnetic field a la Zeeman, our energy levels will get split according to the different values of m. Now if instead of working with the pure Coulomb system, if we were in fact a little clever, and we'd already noticed that there's a fine structure-- because they already knew that there was a fine structure at this point-- they could see the different spectra in lines coming off of hot gas. So it's just an experimental fact you can't turn off the fine structure. This is going to be equal to e nlm of our correction to the hydrogen atom, with the fine structure corrected. So the l equals 1 states and the l equals 0 states are split. Let's focus, for example, on E2. So these guys have a different energy. The l equals 1 guys have a different energy than the l equals 0 guys. What does this predict is going to happen if I look at my spectral line? So what this predicts is the following-- it says that if you have some spectrum that looks like this-- so this is as a function of wavelength and i'm looking literally at what comes off a prism. I'm looking at spectrum lines. And I'll have a line here. And these are going to be the l equals 0 states. This is n equals 2. And then there's the l equals 1 states. This is with zero magnetic field-- Bz equals 0. If we turn on a magnetic field, Bz not equal to zero, what should we see? Well, what happens to this state? Nothing-- it's the same thing, because the m is strictly 0, so it's the same value. And this guy, what happens to it? The m equals 0 state is the same. But then it breaks up-- as everyone said, it breaks up into three lines that are equally spaced. It breaks up into three lines, because one of them has m which is plus 1, one of them has m which is 0, and one is m is minus 1. So energies will increase or decrease. Let's say this is frequency. So we get this splitting. And another way to draw this is as a function of B, let's draw the energy levels. So there's the 1. There's the 0. And this stays constant. And these guys-- one is constant, one increases, and one decreases. When you start getting into strong magnetic fields, weird things happen, like these guys cross, and all sorts of subtle things can happen. But let's ignore the strong magnetic field case, and just focus on this. So we see that these guys split. So this is what the Schrodinger theory, or the full story would predict-- it came a little bit after this, but this experiment was done in '96, so there's actually kind of an entertaining story about people trying to explain this effect classically. Well, look you could have some angular momentum in the z direction, some angular momentum in the x direction, some angular momentum in the y direction. The part that has angular momentum in the x direction doesn't do anything, or in the z direction. The part with the x and part with the y can have different polars. There's a whole classical song and dance. It's crazy. It doesn't make any sense. But that's because they were trying to classical mechanics to do quantum mechanics, which doesn't work. So I'm not even going to bother explaining what they were thinking. But anyway, so people came up with a sort of classical crutch to say, well that's not totally insane. So this is what Zeeman got, for example, by looking at hydrogen. But he did more than just look at hydrogen. I can't remember exactly-- I think he did it with sodium. Matt, do you remember? I think he did sodium and iron. But anyway, here's the funny thing that he got with sodium and iron. So this is a different experiment. It's exactly the same set up, but he was looking at spectral lines in sodium. And spectral lines in sodium did a very similar thing. There were some where 1 went to 1. And there were others where 1 went to-- why did we have three here? Just as quick reminder, why do we have three states here? AUDIENCE: [INAUDIBLE]. PROFESSOR: Because we have three states for l equals 1. This one state went to one, two, three, four states, with none of them in the middle. And in fact, it's worse than that, because they didn't go to equally spaced states in general. In general, they did something slightly funny. And the beautiful plot in his paper looks like this. There are four states, and as you take the magnetic field to 0, these four states all coalesce exactly to this one state. So this one line is representing four different states. And this already included the fine structure, so we know that this is all a single value of l. But that apparently has four states. That's should worry you. What does that tell you about l? What does l have to be in order to have four states in your tower? AUDIENCE: [INAUDIBLE]. PROFESSOR: Half integer-- in particular, what value? 3/2, crap. We already know that these wave functions are described by the ylm's, right? We solved that problem. We explicitly solved it. Here's the exact answer. It's the ylm's. But we've observed experimentally that l has to be 3/2. But if l is 3/2, ylm is equal to 0, because it's equal to minus itself. This is bad. This is 0. So this effect is called the Zeeman effect, and this effect is called the anomalous Zeeman effect. [LAUGHTER] Which is strange for two reasons. The first reason it's strange is they showed up in the same paper. And the second thing is, OK, you call it the Zeeman effect, because there's some guy named Zeeman. Is there some guy named Anomalous Zeeman? It's a very strange name. Anyway so this was called the anomalous Zeeman effect, despite appearing in the same paper, because it's weird. It was deeply disconcerting to people. We now just call it the Zeeman effect, but we have the bad habit-- for entertainment value-- of referring to it still as the anamolous Zeeman effect. I'm not the only one responsible for that bad habit. I'm going to point out that at the end of this paper-- it's a totally awesome paper, by the way-- it's very readable and short-- he says-- I was looking at it last night-- it says, quote possibly the observed phenomenon will be regarded as nothing of any consequence. OK, so a few years later, Pauli says the following-- so Pauli says this, actually, and this is also totally lovely in Science from 1946 Pauli was in the US during the war. After the war, he was at the Institute for Advanced Study. At the Institute for Advanced Studies, while he was there, he got the Nobel Prize. And he gave-- for the exclusion principle, which we are about to get to-- and he gave a little spiel, and it's written up in this edition of Science. And he says, quote, a colleague who met me strolling rather aimlessly in the beautiful streets of Copenhagen said to me in a friendly manner, "You look very unhappy." Whereupon, I responded fiercely, "How can one look happy when he is thinking about the anamolous Zeeman effect?" [LAUGHTER] So this troubled people. Yeah. AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah he found all sorts of crazy states. So this isn't the only example. It's just the only one I'm going to draw. But indeed, you can find five half states you can find states with six. AUDIENCE: Why [INAUDIBLE]? PROFESSOR: Oh, I didn't draw all the states. The spectrum is complicated and messy, because it's sodium. And so there's all sorts of crap. So I'm just not drawing it. I'm just focusing on a specific set of spectral lines. Maybe the best way to draw that is-- OK. So this was really troubling. So let's put this on pause for the moment, and we'll come back to it in just a second. So meanwhile, there's something else that's sort of annoying about this system. We've got the Coulomb potential, and the Coulomb potential has the following beautiful property-- the degeneracy of any energy state is equal to n squared. Meanwhile, various people were fond of observing the fact that the number of states, which I will-- in an abuse of language-- call the degeneracy of the level n in the periodic table, is equal to-- well, the first one there's two. And then there's eight, and then there's 18. So two, 8, 8, 18, that's almost 2n squared. Two four 8, 9, 18. Cool, maybe there's a relationship, right? And meanwhile, it's very tempting to think there's a relationship, because imagine you have four electrons. If you have four electrons, what you do you do? Well, you put an electron here, and then you could put an electron here, or you could put an other electron here. You could also put another electron here and here. Or you could put them in any combination-- two here, put two here, two here, one, one-- you could do all sorts of stuff, right? But if, for the moment, you imagine that electrons have this funny property. Imagine for a moment that electrons have the property that they can't be in the same state at once. Then if you have four electrons-- I don't know why. I'm desperate. I'm thinking about the anamolous Zeeman effect, among other things. I have nothing better to do. So just for fun, let's imagine if I put an elector here, I can't put another one. So if I have a two electron atom, the second one goes with one of these. A three-electron atom, a four-electron atom, and these guys are all related to each other, because they live in the same energy level, but they have different l values, so they'll have slightly different properties. That suggests that the one l equals 0 state should be different from the 3 l equals 1 states, chemically. But wait-- there aren't groupings of one and three. They're groupings of two and six. Ah, two-- well, so Pauli staring at this in 1925 said the following-- he said, look, I'm going to conjecture two things. The first is that no two electrons can live in the same state-- inhabit the same quantum state. That doesn't give us the periodic table, though. It gives us half the periodic table. There are twice as many states in hydrogen as you think. And you might think that I'm being facetious, that he said something a little more sophisticated. But he really didn't. What he said is, I posit of the existence of an additional quantum number which the electron can have, which takes one of two values. Which translated, says, there are twice as many states as you think. So that's exactly what he said. So this is called the Pauli exclusion principle. And now this does a totally spectacular thing-- and you've probably all seen this in high school chemistry, or even college chemistry. Now, first electron goes here, second electron goes here, because there are twice as many states as you think, but then I can't put another state in there, because that's as many as you can have. One can go here, and you can put up to six in here. And that's it. So the E2 states, the n squared, which is 4 times 2 states, are somehow related to each other naturally. And this gives you the structure of the periodic table. In fact, an awful lot of chemistry follows directly from this. So this is the Pauli exclusion principle. However, it's a little bit disappointing, because while this is whatever-- some ridiculous rule, but we're doing quantum mechanics and they're all ridiculous rules. On the other hand, this one is just stupid. Right? This is just, like, look, it wouldn't it be nice? So in '25, a couple guys-- really interesting characters-- named Goudsmit and Uhlenbeck-- so Uhlenbeck-- wow, we're really low on time. OK, so I'll just tell you this, and then I'll get to the last of it later. So Goudsmit and Uhlenbeck-- who were young, kind of naive-- said, look we've got two effects here. One effect is the anamolous Zeeman effect, and there's this weird fact that states have the wrong total angular momentum. And we have this second thing, that there are twice as many states for an electron as you think in the Coulomb potential. How can these two things fit together naturally? Well here's a guess-- suppose that the angular momentum that we calculated when we did the estimate of the braking due to the Zeeman effect, suppose that angular momentum was not the right angular momentum. Maybe there's more angular momentum in the system. There's angular momentum in the system from the electron orbiting around, but maybe like a little, tiny Earth, the electron itself can have some intrinsic angular momentum. It turns out Pauli had had this idea before. In fact, Kramers had suggested this to him, who was a very young guy at the time. And Pauli said, you're a blithering idiot, because if you calculate how small an electron has to be in order to fit all the other things we know, and you figure out how fast it would have to rotate to explain the anamolous Zeeman effect, the surface would have to be moving faster the speed of light. That's ridiculous. Leave my office. So then Uhlenbeck and Goudsmit-- these two guys-- write a paper, and say aha! Well, we can explain this, but let's just not assume that the stupid thing is rotating. Let's just say an electron has some intrinsic angular momentum. I don't know, why not? Electrons have intrinsic angular momentum, and if you assume that electrons have an intrinsic angular momentum-- which I'll call S for the moment-- so then the total angular momentum, j, is equal to l, its orbital angular momentum, plus some intrinsic angular momentum, which-- I don't remember what symbol they used, but we'll call it for the moment S-- where this has the property that S squared, for an electron, sorry, the principle quantum number, which I will call little s is equal to 1/2, so m sub s is equal to plus or minus 1/2. So this is like the l equals 1/2 state, but it's not l. It's something intrinsic. It has nothing to do with anything rotating. It's just a fact about electrons. Suppose electrons have a little bit of spin. Then what you discover is if you have the l equals 1 state, and the electron has spin 1/2, what's the total angular momentum? 3/2, right? So what do you get? You get quadruplets, like that. So this turns out to explain-- if you include a small relativistic effect-- this turns out to explain the anomalous Zeeman effect bang on. Meanwhile, there was an experiment done in 1922, which is the Stern-Gerlach experiment, in which Stern and Gerlach discovered that nickel, when sent through a magnetic field gradient, bent into one of two different spots-- never three, never zero, always two. How can that be? Electron spin-- that wasn't realized until 1929, that the connection was there. So these guys came up with this ridiculous postulate. This was Uhlenbeck and Goudsmit. And let me just quickly-- U-H-L-E-N-B-E-C-K-- so Uhlenbeck-- amazing, amazing scientist, also the father of the mathematician Uhlenbeck, and she was a total badass, and has inspired an awful lot of physics-- Karen Uhlenbeck, who's at UT. So this is a pretty interesting and prolific guy. But he was also prolific in the following way-- he ran a program at the Rad Lab at MIT during World War II. And Goudsmit worked with him. And when they were finishing up, when the war was ending, Goudsmit became the scientific adviser to something called Project Alsos. Project Alsos was a project where the military went to the conquered territories in Germany to catch the German atomic scientists, and bring them to a place called Farm Hall in England, where they listened and eavesdropped on them-- Heisenberg, all the good guys. Well, bad guys-- it depends on-- all the great physicist in the German territory at the time were deeply complicated people. And they listened to them. Something called the Farm Hall Transcripts are the transcripts of those recordings. They were written up in a book called "The Epsilon Project," which is totally breathtakingly awesome. And Goudsmit wrote a book called "Alsos" about this process of hunting down the German scientists. So Goudsmit, G-O-U-D-S-M-I-T, I think, he wrote a book called "Alsos", which I heartily recommend to you, because it's like a combination adventure story and beautiful bit of physics history. So these guys both were at MIT during the World War. So there's a nice connection here. So these guys-- fascinating characters, and they came up with this idea of some intrinsic angular momentum. Pauli then calls it spin. He gives it the name. And he develops a mathematical theory of spin. And the mathematical theory of spin will lead to quantum field theory, relativistic quantum mechanics, and will eventually lead to quantum computation, which is going to be the topic of the last week of our course. So what we've done so far is we've explained the discreteness of atomic spectra, and we've explained the structure of the periodic table. What we haven't done, is we haven't explained why atoms form molecules or solids. And we also haven't explained what spin is at all. Those are the topics in the next three weeks. See you next time.
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_21_Periodic_Lattices_Part_2.txt
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The following content is provided under a Creative Commons License. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: OK, so, we're going to pick up on our study of periodic potentials and our search for the explanation of the physics of solids. And I want to quickly remind you of the logic we went through last time. This is one of the more subtle bits of logic in the semester, so I'm gonna do it again, a little faster. I'm mostly gonna do it in a slightly different form. I want to think for a moment about just a free particle, but I want to use all the language and formalism we used for the periodic potential last time-- translation operators, the block wave functions-- I want to use all of that for just a free particle. So, first let's just remind ourselves how the free particle story goes. So, for a free particle, by which I mean the energy operator is p squared upon 2m, and no potential, plus 0. We have a very important consequence of being free-- that the energy can be written as just p squared-- which is that the energy commutes with the momentum operator. And what this tells us is that we can find a basis of eigenfunctions which are common eigenfunctions of E and p, right? We can expand every wave function at any arbitrary-- fantastic. Thank you, sir. Oh! Brilliant! GUEST: Plan A, Plan B. PROFESSOR: I don't know. That's an awful huge straw. Pay no attention to the device beneath the table. So, 'cause you know, it's quantum mechanics, right? Bowl of water-- So, the energy and momentum commute. And as a consequence, we can find a basis of eigenfunctions exists which are common eigenfunction of E, E on phi E, and I'll label k, by the eigenvalue of k, is equal to E phi Ek. And p on phi Ek is equal to h bar k phi Ek. You'll notice, of course, that I don't label it by the eigenvalue, but I label it by something that's determined by the label, the eigenvalue's determined by the labels, h bar k. Now, here's the thing, can we allow E and k to be any values we want independently? Can I make E, 3, and k, 47? No. There's a consistency relation, which involves satisfying the energy eigenvalue equation. We actually have to find solutions of these equations. And once we find the solutions are those equations, we'll find relations between k and E. And in particular, for a free particle, we find that the relation that we need is E is equal to-- and I will write E sub k-- is equal to h bar squared k squared upon 2m. This is also known as omega is equal to h bar k squared upon 2m. Just to remind you, the [INAUDIBLE] E is h bar omega, omega sub k. That's not the thing I wanted to look at. This is. And when we work this way, we find that the energy eigenvalues, E, as a function of k, take a determined form. And that determined form is quite simple, it's a parabola. Everyone cool with that? So, if you know k, you know E. But if you know E, it turns out it's doubly degenerate. So, if you know E, you're one of two k's. You could be either the ikx-- oh, sorry, and I needed to say-- that the function, so, from solving the equation, gave us that the energy eigenfunctions, phi Ek are equal to some normalization 1 over 2 pi as conventional, e to the ikx. OK. So, if I tell you the energy, you don't know which momentum state I'm talking about. But if I tell you the momentum, you know the energy. So, there's a little degeneracy. That degeneracy is coming from the parity symmetry, the system, right? And note that what we've done here, did we have to write the wave functions in terms of these, the ikxes. Is there another basis we could've used for the energy eigenstates? Yeah. Because these are degenerate, we could've taken any linear combinations of E to the ikx and E to the minus ikx, and we could have got, for example, sine and cosine. That would have had the advantage that the energy eigenfunctions would have been pure real, which is one we proved was possible. You can always choose pure real energy eigenfunctions. However, it would have had the disadvantage that these states would not-- sine and cosine-- are not eigenstates of momentum, right? It's convenient to work with common eigenstates of momentum and energy, so we give up on having a real wave function and that's the cost of finding shared eigenfunctions of energy and momentum. Everyone cool? Any questions? OK. So, this is the usual story for the free particle. But we learn a couple of things about it immediately. Just by eyeballing the wave function we see some things to note. Note, all states-- all eigenfunctions-- phi sub E, which are of the form E to the ikxx-- and I'm gonna add the time dependence in now-- minus omega sum kt, are extended. They have equal probability density everywhere, and the probability distribution does not fall off to 0 as we go far away. And so, what this tells us we have to do is we have to build-- if we want to study real localized states that are normalizable-- we have to build wave packets of the form psi is equal to integral dk sum overall momentum modes f of k E to the ikx minus omega t sub k. Where this f we will generally take-- although we don't have to take, to be a real function-- but, is peaked at some k0. So, this is what we did when we studied the evolution of a wave packet of this form. But in particular, such wave packets have a couple of nice properties. The first property we've studied. And if you're not totally comfortable with the idea of group velocity, you really need to go over it again. It's be an excellent thing to show up in recitation, for example, stationary phase. So, the group velocity of such a wave packet-- as long as it's of this form and peaked around a single momentum value, k0-- the group velocity is d omega dk evaluated at k0, at the peak value of the distribution. And note, I wanna just quickly note that this is equal for the free particle. Omega is h bar k squared upon 2m, take a derivative with respect to k, the 2 cancels. We get h bar k upon m. Which notably, is equal to the expectation value of the momentum over the mass. Everyone cool with that? So, this gives us a nice way to, by the way, if you didn't know the mass, but you could measure the system, for example, you can compute on average the momentum, and you could compute the group velocity by watching wave packets move, this is a nifty way to compute the mass. The mass can be given by, well, put the mass up here, and divide by the group velocity. Measure the momentum and divide by the group velocity. OK, so, I can take this, and notice, that this just exactly gives me the mass. So, this is a way to, given a system with a momentum and a group velocity, I can compute an effective mass. Which for the free particle, is just the mass. Cool? Just a side observation. But this is all a consequence of the fact that we have wave packets of this form. Their plane waves, or at least things that have a phase, times what could be a real-- though it could but an overall phase on the whole thing-- a real function, which is peaked at a particular value of k0. Usual wave packet story. Any questions? Good. So, this should all be pretty familiar. Now, here's the thing, when we talk about a periodic potential, what I want to do next is, I want to talk about a periodic potential. So, now, we have a system that has maybe steps, and it's periodic, with period l, OK? And then, ditto. It's gonna be periodic. And I want to run through the same analysis. I want to find the energy eigenfunctions. I want to organize them in a nice way. And in particular, I want to know the following, if this is equal to v of x, and v of x, and I call the energy operator E sub g is equal to p squared upon 2m plus a coefficient, g, times-- let me call it g sub 0, just to make it unambiguous-- times v of x. Notice that when g0 is 0, this is just the free particle. I added a periodic potential times 0. That is stupid, but doable. But as I slowly turn on g0, what's going to happen? The energy eigenfunctions are going to slowly change. And the energy eigenvalues are going to slowly change. And the actual system we care about has some actual value of, you know, like 7, or e, or whatever. But this is a way to connect the free particle to the particle in a periodic potential. You know, and if you just turn on a little weak potential, you don't expect the system to radically change. So, one way to organize the question of periodic potentials is, what happens as you slowly turn on this periodic potential? But here's the first thing I want to notice, the first thing I want to notice is that it's no longer true that the energy commutes with the momentum. So, it would be silly to try to organize our wave functions in terms of energy eigenfunctions, which are also momentum eigenfunctions. Because that will only be a sensible thing to do when this interaction term-- when this potential-- is 0, right? So, instead, I want to organize them in terms of energy eigenvalues and any other operator that commutes with the energy eigenfunction. In particular, this is not invariant under arbitrarily translations, which it would have to be to be invariant to conserve momentum. But it is invariant under translations by L. And as a consequence, we have that the energy operator commutes with translations by L. Where translations by L. Takes a function and shifts it over by L. And so, because translations by L as we showed in a problem set long ago commutes with p squared, because this is just an exponential of Lp, with coefficients, with an i and an h bar. And because the potential is invariant under shifting by l, and so it commutes with TL, this is 0. So, what this tells us is we can find common eigenfunctions. In fact, a common eigenbasis of E and TL. And here's the thing that's nice about this, I could've made this argument whether the potential was 0 or not. For the free particle, it's also true that the system commutes with translations by l. It's free, there is no potential. It sort of silly, because I could've just used the momentum. But let's see what happens if we use the translate by l operator. Everyone cool with that? So, we want to find common eigenfunctions. So, as you guys showed on a problem set, the eigenfunctions of TL-- so, in order to do this, what are the eigenfunctions of TL?-- you showed that the eigenfunctions of TL must take the form-- sorry-- phi sub q of x is equal to E to the iqx times u of x, for some q where q is-- where u-- is a real periodic-- I don't really need the real, but it's going to simplify my life, so I'll take it to be real-- periodic function. Well, let's just drop real altogether, because we don't need that. We'll use a periodic function, and phi of q is this phase. It times the periodic function. So in particular, phi is not periodic. And as a consequence, TL on phi q is equal to E to the iqL phi q, because this guy shifts, and this is invariant periodic. So, this is the form-- general form-- of the eigenfunctions of translate by l. And periodic means that u of x plus l is equal to u of x, just be to explicit. OK. So, what this tells us is that the energy eigenfunctions can be put in a form-- can be organized-- as phi sub Eq, such that E acting on phi sub Eq is equal to E phi sub Eq. Just like this line, phi sub Eq. And TL phi Eq is equal to E to the iqL phi Eq. And it's just saying then we have common eigenfunctions, yeah, with energy E and TL eigenvalue, q. AUDIENCE: [INAUDIBLE]? PROFESSOR: No, because under translation by l, this x goes to x plus l. Oh, I've slightly changed notation from earlier in the course. You're right. I've slightly switched. So, ah! Good. So, yes. So, thank you. So, let's fix notation. TL takes f of x to f of x minus l, is what we've used previously, right? But I'm going to just switch notation, and call this x plus l. So, for all the notes in here, this is x plus L. Yeah, it doesn't really make a difference, you just have to be consistent. And you're right, I switched between lecture-- what was it?-- eight, and this one. Sorry about that. Yeah, good catch. Thank you for pointing that out. So, in which case, this shifts by x to x plus l. And we get E to the iqL. Good catch. Good. So, again, we have this property that we can find common eigenfunctions. However, can E and q be specified independently? We know on general grounds that it's possible to find eigenfunctions which are common eigenfunctions, because these guys commute-- E and TL commute-- so we can find common eigenfunctions. And here they are. And, meanwhile, we know that the form of these eigenfunctions is of this form. Because any eigenfunction of TL is of this form, a phase times a periodic functions. Does that, by itself, tell us a relationship between the eigenvalues E and q? No, right? Just like E and k were independent, until we solved the actual eigenvalue equations, right? And it required things like regularity, and no similarities. So, what we have to do at this point, is solve the differential equations for continuity, periodicity, and solving the energy eigenvalue equations. And when we solve the equations, we'll get a relationship between E and q. And in particular, if we do that, here-- do I want to do it this way? Yes. So, in particular for-- write this in the following way-- for g equals 0-- g0 equls 0, which is a free particle-- we find E is equal to h bar squared q squared E sub q is equal to h bar squared q squared upon 2m. And yeah, and the wave functions are of the form phi sub qE are equal to some normalization, 1 over 2 pi, times E to the iqx. And the coefficient function, u, is just constant, which is nice. So, times constant. And so, this is reassuring because it's exactly the same form of the wave functions, with exactly the same form of the energy. But where here, the label-- the value-- q is representing the possible value of-- it's controlling the eigenvalue of TL. So, there was a question? AUDIENCE: Yes. PROFESSOR: Yes? AUDIENCE: So, for the translation operator-- PROFESSOR: Yes? AUDIENCE: --that E iqx equal to x will work for any complex q in any u with period [INAUDIBLE]? Or any periodic u with [INAUDIBLE]? PROFESSOR: Yeah, although, if q is complex, then this is badly non-normalizable. Because if q has an imaginary part, then that function is gonna diverge in one direction or the other. So, q had better be real. But, that's correct. This works for-- this expression-- A function of this form, a periodic function-- periodic, with period L-- times an E to the iqx, for any real value q, and any periodic u, will be an eigenfunction of TL, translation. AUDIENCE: [INAUDIBLE]? PROFESSOR: No. It's not just non-hermitian. It's, in fact, unitary. I mean, what exactly is your question? Sorry, I'm misunderstanding. AUDIENCE: So, I mean, just the translation operator has a really, really large number of eigenfunctions? PROFESSOR: Indeed, it does. And that number is the same as the number of momentum eigenfunctions. Because they can be both used as a basis. They're both operators whose eigenfunctions form a basis. That's exactly right. So, I think the difficult thing is to think that-- we have a tendency to think that there aren't very many momentum functions, for example. But there's enough momentum eigenfunctions to be complete basis. So, if this seems like a vastly larger set, then yeah. Indeed. OK. So, when we do our analysis for the free particle of this form, we get exactly the same story. And I want to point out a couple of consequences for this. Working with this TL operation-- or working with TL eigenfunctions--- we have exactly the same properties. Our eigenfunctions are, again, extended. We have to use wave packets. And those wave packets move with a group velocity, which is given by d omega dk. On the other hand, here, when we write super positions, our general wave function-- our general wave pack, which I'm gonna write here-- psi, is gonna be of the form integral dq of E to the iqx u of x times some f of x-- or sorry, f of q-- u sub q. So, as long as this is a periodic function, then the-- oh, and I should say minus omega sub qt, to get the time dependence in there-- as long as this is a nice periodic function, especially for this case was, when it's real, and this is a sharply peaked function, which is sharply peaked around some q0, then we will find that the group velocity is, again, the omega, now dq, evaluated at the peak value, q0. And that's just a general consequence of this form of our wave packet. As I encourage the recitation searchers to review by stationary phase in recitation. And secondly, again, we can find this nice notion of a mass, which is the expectation value of the momentum divided by the group velocity. And, again, this, for our simple free particle, this is just a nice way of measuring the mass, if you happen to want to do so for your free particle. No one told you the mass. Everyone cool with that? OK. Questions? Yeah? AUDIENCE: Is that still the momentum when we have g0 is not equal to 0? PROFESSOR: Excellent question! So, now, let's study the case with g0 not equal to 0. OK? So, so far we looked at for g0 no equal to-- or g not equal to 0-- that was just this line. But everything else, was for general case of studying common eigenfunctions of E and TL. So, now, let's study the case for an actual periodic potential. So, in particular, repeat for g0 not equal to 0. And this is the analysis we did last time. We did exactly this analysis. And we looked for the common eigenfunctions of E and translate by L, right? And we found that the common eigenfunctions took the form Eq is equal to E to the iq of x times some u of x. And now, we need to find what's the relationship between E and q? Are any E and any q allowed? And do those give you eigenfunctions? From the free particle, we expect that that's certainly not the case. On the other hand, we have a guess as to what it should look like. The energy eigenfunction for-- let me, actually, write this here-- the energy eigenfunctions and energy eigenvalues, more importantly, of the free particle organized in terms of q, we just immediately see the energy is just h bar squared q squared upon 2m. So, the potential-- or sorry, the energy as a function of q-- is, again, a parabola. OK, so, here's q, and here's the energy. And this is for a free particle, right? Free particle. E equals h bar squared q squared upon 2m. Now, I want to ask what happens as we slowly turn on the potential? As we take g0 not equal 0. So, we did this last time. We turned on the potential. We turned on the a potential, which is, in fact, delta functions. So, now , in order to solve the differential equations, we have to actually know what the potential is. There's no getting around it. And so, we took a bunch of delta functions with dimensionless strength g0, and spaced by L. And what we found was an equation relating E and L, which took the following form-- and for simplicity, I'm gonna write E is equal to, this has nothing to do with momentum, although it looks like momentum, I'm just going to write E formally as h bar squared Scripty k squared upon 2m-- so, k, here, should just be understood as root 2m over h bar squared, which I hate writing out over, and over again. So, let me, actually, write that the other way around. Scripty k is equal to the square root of 2mE upon h bar squared. So, when you see k think E. When you see lions, think lions. OK. So, we found that the condition-- in order that we solve the differential equations-- the condition, in order that E and q label energy eigenstates which actually are energy eigenfunctions-- common energy eigenfunctions-- of the energy and the translation by operator, we found that there was a relation, which was cosine of qL-- AUDIENCE: [SNEEZE} PROFESSOR: --is equal to-- Bless you-- cosine of kL plus g0 upon 2kL sine kL, right? So, first off, let's make sure that this agrees with what we found when g0 was equal to 0. When g0 was equal to 0 we found that E was equal to h bar squared q squared for any q. And, well, when g0 is equal to 0, this term goes away. And we get cosine of qL is equal to cosine of kL. And an example of a solution for that is qL is equal to kL. Or q is equal to k. In which case, E is equal to h bar squared k squared upon 2m, by plugging in that expression. Cool? So, this reproduces the free particle result when we need it to. But in general, it's not the same. So, what does it look like? So, let's take a look at that. So, here, first off, what I'm plotting here is the plot that we draw on the board last time, only here I'm plotting it here horizontally is the energy, E, and vertically is cosine of qL. Now, for q a real number, cosine of qL goes between 1 and negative 1. And so, any value in here corresponds to a valid value of cosine-- or of q-- sorry. So, any value, any point, between 1 and minus 1 is an allowable value of q. So, for example, this point, right here, corresponds to this value is q, OK? Just horizontal lines. Meanwhile, this curve is the right hand side of that expression. It's cosine of kL plus-- or sorry-- in this case, it's cosine of root 2mE upon h bar squared L. This right-hand side plugged in here, and I'm plotting it as a function of energy. Just to be explicit, so this is energy-- or energy-- and the q. And here's that curve. So, for g equals 0-- which is what I've set it to here-- g is equal to 0-- that didn't work, there we go-- when g is equal to 0, we see that it goes between 0, and this is just cosine of root E. And there's cosine of root E. It gets spread out because it's square root of E, rather than E. Everyone cool with that? And so, here's my question, what pairs of E and q correspond to allowed energy eigenfunctions? How do you look at this diagram and decide which values of E and q are good energy eigenfunctions? AUDIENCE: Well, the fact that the boundary conditions of the q is such that cosine of qL is between negative 1 and 1? PROFESSOR: Good AUDIENCE: Or all values are allowed? PROFESSOR: Awesome. So, let me be explicit. So, how would I find? That's exactly right. Let me rephrase that and be a little more explicit. So, what we need is we need a solution of cosine qL is equal to the right hand side, in this case, cosine of root energy, L. So, that means we need to find a horizontal line corresponding to a particular value of q-- or a particular value of cosine qL-- and a point on the curve, cosine of root EL, which is this guy. So, at any point where we are inside these two extreme values, and this E, will give us a value. So, for example, for this point-- ah, I wish I could draw-- for this point, that corresponds to a particular value of q, right? Where the cosine of q is equal to that vertical value. And it corresponds to a particular value of E. So, each point, here, corresponds to a value of E-- an allowed value of E-- on the horizontal, and an allowed value of q, such that cosine qL is the vertical value. Everyone cool with that? So, what val-- Yeah? AUDIENCE: [INAUDIBLE] p degeneracy for values of q. Like for a specific q, we're going to find a lot of [INAUDIBLE]? PROFESSOR: For specific values of q, why does it-- AUDIENCE: [INAUDIBLE] horizontal line. You'll have a lot of intersections, right? PROFESSOR: Yeah, that's, excellent. OK. So, good. So, there are two ways of answering that. That's a very good question. So, the question is, look, this is slightly funny state of affairs. If we plot-- if we take a horizontal line-- corresponding to a particular value q, it hits at various different values of the energy. Excellent observation. So, let me make two points about that. The first point is that when we defined q, how did we define q? We said the translate be L on phi Eq was equal to E to the iqL phi eq. But now, suppose we take q to q plus 2pi upon L, OK? OK, so if we take q to q plus 2pi upon L, we don't do anything to the eigenvalue. So, that's not enough to specify which function we're talking about. Because we changed the value of q, and it doesn't change the eigenvalue. So, you can't just focus on the eigenvalue. There are two ways of thinking about that. One is, look, since q and q plus 2 pi upon L give you the same eigenvalue of TL, we should just think of them as equivalent. So, one, we should just-- look, any time you have q, and you add plus 2pi upon L, you just think of this as q. It's not different. There's just, you know, q now is valued periodic. If you shift q by 2 pi, you're actually back at the same point. So, q is now a periodic variable. And but, that's not enough. Different values of these q correspond to different values-- the same eigenvalue. So, you're getting multiple solutions as a consequence. But there's another way to think about this. Which is just in terms of this graph. In terms of this graph, if what we're specifying is cosine of qL, but we're not specifying the value for q, yeah? So, if we instead plotted this as a function of q, without assuming q is equivalent to 2pi q plus 2pi upon L-- which we can do, we're perfectly entitled to do that-- what do we get? What do you think you get? For the free particle, we know exactly what we get for the free particle. The parabola. And if you just take this, and I'm going to give you guys the Mathematica files for all these things. Yeah, if you just take this, and open it up, instead of plotting it in terms of cosine of qL, plot it in terms of q, that's exactly where you get. The satisfying thing. And I'm going to come back to that in just a minute, but it was a really good question and we'll come back to it. Other questions? OK, now, let's ask what happens as we turn on g0. As we turn on g0, the solutions to this equation are going to be different, because it's going to be a difference equation. So, let's do it. So, here it is. We have g0, we can tune it. And I'm going to slowly turn on g0. And watch what happens to these curves. So, now we see that the right hand side of the yellowish curve is now exceeding 1 and minus 1 in various places, right? And why is it doing that? Let's get some reasonable down here. So, why is it doing that's? It's doing that because this can now be greater than 1. Cosine is bounded between 1 and minus 1, but this is not. And so, we overshoot in some places. And what does this tell us? What does this really telling us? What its really telling us is that-- there we go-- how do we find allowed values of E and q such that we have a good energy eigenfunction which is simultaneously and eigenfunction of E and of TL? We need to find points where a horizontal line between 1 and minus 1 intersects the right hand side curve. But for the energies between here-- this value of the energy, and this value of the energy--- there is no such point. The energies in here-- any energy in here-- in order for those lines to intersect, line must be greater than 1, or less than minus 1 down here, or here. So, there's no allowed value of q, such that there's a solution with energy between these two points. Everyone cool with that? So, that's telling us that there are values of energy where there are no energy eigenvalues with that corresponding energy. Because there simply aren't solutions of this equation for any value of q with that energy. On the other hand, when the right hand side is between 1 and minus 1, we have allowed values of energy. We have a value of q, and a value of E that correspond to each other, and they correspond to a solution of the energy eigenvalue equation and the TL eigenvalue equation. So, let's see what that looks like in this presentation. Oh, before I do, I want to say one other thing about this. Let's take this plot, and let's use this periodicity. So, this periodicity is going to say that-- So, here's pi upon L, here's minus pi upon L-- let's see, yeah, good-- there's minus pi upon L, there's 2pi upon L, and 2pi upon L. If these guys are periodic, than I could've just said, look, at this point-- this is pi upon L, and minus pi upon L-- I could've said this point is the same as this point. So, I could've taken this whole bit of the curve, and I could've moved it over here. Everyone cool with that? Because this value of q, is the same as this value, q. This value of q, is the same as this value of q, right? So, I could have written this, and if we just take this over, you can see it's, you know, symmetric. As a consequence, we get-- And so, we can write-- we can draw-- this entire parabola folded up into one region, using the periodicity with q as 2pi upon L. And when you fold it up into one region, just using q as periodic with period 2pi over L, and now noting that there are several solutions for every value q-- corresponding to how far out on the parabola you went-- now we have a much simpler way of presenting this, where we don't need to see the whole parabola. This is a real advantage. When we plot things in this way, the structure of the energy bands and energy gaps becomes considerably more simple. So, let me do that. First let me set the g to 0. OK, so, here is our free particle. And let's look at that plot. So, here's the plot that I was just describing. What you see here, is the parabola. And I've use a periodicity to box it up into one fundamental period between pi and minus pi upon L, OK? And so, the blue line corresponds to the energy as a function of q-- the allowed energy as a function of q-- and what the green background represents is I'm going to put the horizontal section in green anytime there's an allowed energy with that. So, energy, here, is vertical, and q is on the horizontal, going between minus pi over L and pi over L. Yeah? AUDIENCE: I might be getting this just out, but is there a reason that as the energy increases the bands get broader and broader? Is that just an artifact of the math, or is there a good physical reason for that? PROFESSOR: Yeah, there're both. So, one way to think about it, so just purely mathematically, this is saying that cosine of qL is cosine of root EL. So, as we make the energy larger, and larger, some small variation in q is going to correspond to a quadratic variation in E. And that's what's making it stretched out. If we plotted this as a function of kL, instead of E, then it would have looked much more constant period, OK? So, that's the mathematical answer. The physical answer is this, if we have some potential, if I put you in a finite well-- So, there you are. You're in a finite well. And you look at the ceiling, you're like, damn it! And so, that's frustrating, right? And it's deeply frustrating if it's a deep well, but if it's a shallow well, it's not all that big a deal, you climb out. Being in a deep potential-- being at low energy, compared to the potential-- means that you're tightly bound, right? Takes a huge amount of energy to get out. And as a consequence, what you saw in your problem set is that the bands are quite thin. But as you get closer, and closer to the top, or as you go to higher energy, the thickness of that band grows as a measure of the fact that you're less tightly constrained by the potential. OK? Good. So, that's the physical intuition. Did that make sense? Good. OK, so, yeah? AUDIENCE: [INAUDIBLE] g0 different from 0, do you ever get to the point where the solution is always inside the [INAUDIBLE]. So, like the bumps [INAUDIBLE]. PROFESSOR: Ah! Good. Let's look. Let's look. So, let's answer that by looking. So, the question is, basically, look, as you crank up g0, do you ever lose the bands entirely, or do the bands just disappear? Is that the question? AUDIENCE: As you get to higher energies. PROFESSOR: Yeah, as you go to higher energies. Good. So, we can answer that. Let me do that in two ways. So, the first thing I want to do is I want to ask the first of that question, which is what happens as we turn on the potential? Here we have the free particle, let's turn on the potential. We know what happens down here, I want to ask what happens to the picture of the allowed values of q and the allowed value-- sorry, the allowed values of energy-- and their corresponding values of q. So, here, it's easy. This particular point on the blue line corresponds to a value of the energy, and a value of q. What happens to those allowed pairs as we increase the potential? And as we increase the potential, you can see on the left that the right hand side is exceeding 1 and minus 1. And correspondingly, on the right, here the red dot is what was the free particle-- parabola-- and the blue line is the actual solutions. What you find is when you turn on the potential, the actual solution moves away from the free particle line in a very particular way. In particular, for this value of energy-- around 100-- there are absolutely no allowed energy eigenstates. There are no solutions. And so as correspondingly, that area is not shaded green. There are only solutions where you have this blue line. Everyone see that? So, we get these gaps. And we get bands of continuously allowed energies. And then gaps between those allowed bands. And I'm going to post this-- the guy-- to play with on Stellar. So, now, let's go up to very, very large values of the interaction. OK, and let's see what happens when you go to large of interaction. Well, one thing that happens it that Mathematica sort of panics down here, so it lost an entire band, which shouldn't happen. Let's try this, there we go. That's better There's a wonderful book on spectral methods in solving differential equations by a guy named Boyd. And he's pithy, if somewhat degenerate, and he says at one point, the definition of an idiot is-- or he says, idiot definition, someone who doesn't reproduce their numerics twice with different parameter values. And so, we could have just lost this band and written a paper, oh, band's disappear! This is great! Never believe Mathematica until you've checked it. So, anyway, here you see a band, and another band, and another. But what's happened is the bands become very thin. But that makes sense from the earlier intuition. We're making the potential much stronger. We're more tightly binding particles into wells. And as a consequence, the bands are very tightly restricted. And that fits very much with what we saw on your phet simulations. OK, but-- hold on for one sec-- coming back to the question over here-- yours-- coming back to that question, if you look at higher and higher energy, what you're going to find is that the bands get wider and wider, and the gaps get thinner and thinner, as you go to high-- well, it's a little more complicated than that, it depends on exactly what you're doing, but-- the bands get wider, the gaps get wider, but both of them don't get wider at the same rate. So, this is actually something you'll look at on your problem set. But the important thing you will find is that the bands never disappear. Like a band in 1D doesn't just close and disappear altogether, and the bands don't overlap. And here's an easy way to understand why the bands don't overlap. Let's look at what these states are. So in particular, I want to look at the states at the bottom of each loud energy band. So, let's go back down to small value of g0. So, we have these big thick bands with a weak little potential. We have big, thick bands, little, tiny gaps, and I want to look at the states at the bottom of the gap. Let's make it maybe a little bit bigger. So, we'll look at the states at the bottom of the gap. And there's an optional problem in your problem set that works through the mechanics of this. But I want to look through it in some detail. So, what we're plotting here-- Did that work? Crap. Unfortunately, this is-- OK, good. So, here what we have is the ground state of the lowest band at zero interaction. So, here, is g0. And it's 0.001, it's basically 0. As I turn on the delta function potentials, what's going to happen to the lowest energy state, which is just the constant, has momentum 0. What's going to happen to this guy? Well, we saw from this guy, that the lowest energy state got a little bit of a gap there. There is no state at 0 energy anymore. There's now a state only at slightly positive energy. A wave from what would have been the zero energy, zero momentum state. Everyone see that? There's this little gap between 0, and the bottom. And if we make the interaction bigger, we'll see that a little more obviously. Here's the red is the zero energy, and it goes up to finite energy. But has that lowest state developed any nodes? Does the lowest state in a 1D potential have nodes? No, right? There's nodes here. 0 nodes, then one, then two. And do they ever switch orders? Does it ever go zero, two, one? No. There's the node theory. So, what happens to these wave functions is all the wave functions continuously change as we turn on the potential, but they don't switch order. Their ordering is completely fixed by the number of nodes. So, let's watch what happens to the actual wave functions. So, here's that lowest state, and I'm going to turn on the potential. What's going to happen? Well, we're going to see the effect of the delta function. And there, you see the effect of the delta function turning on. And as we crank up the potential, you see that the wave function generates a kink where there is a delta function. And that state has zero nodes. That's the ground state. Everyone cool with that? OK. Similarly, let's go to 0. Whoops. Oh, shoot. Don't give me infinite expressions. You're not allowed to divide by 0. Oh, shoot! OK, here's the second excited state, this guy. And let's look at what happens as we turn up the potential now. Exactly the same thing. We see a little kink. We don't get new nodes. But what we do get are kinks at the potential. This is for the special case of the delta function potential, but it's illustrative of everything else. So, going back to the bands, what's going on here is that these states are getting compressed-- they're getting pushed up together-- but they're not getting swapped in their order. OK? And this was in the service of a question that was asked. What was the last question? AUDIENCE: [INAUDIBLE]. PROFESSOR: OK, I don't remember exactly which questions I was answering, I'm sorry, but-- AUDIENCE: [INAUDIBLE]. PROFESSOR: Sorry? AUDIENCE: I think it was if you lose the bands. PROFESSOR: Ah, yeah. If you lose the bands. So, you know by the node theorem that you never lose states. States with three nodes can't disappear. That same with three nodes always has to be between the state with two-- thank you-- between the state with two nodes, and the state with four nodes. And they can't just disappear. So, those gaps never disappear. And they never overlap, because they never cross each other, again, by the node theorem. All you get is things separating into bands and squishing together, or spreading back apart and becoming the free particle. AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah? AUDIENCE: [INAUDIBLE] strange periodic potential and you'll get the [INAUDIBLE]. PROFESSOR: Indeed. Indeed. So, that's exactly right. So, the observation is, look, this was for the delta function potential, but we didn't have to use delta function barriers. We could have use little square barriers, I could have used little, you know, goat-shaped barriers. I could have used whatever. And there's going to be some answer. And how's it going to change? Well, the way it's going to change is where the bands are, how wide they are, and how wide the gaps are. But the rest of the story goes over in exactly the same way. The energy eigenfunctions take the form E to the iqx times the periodic function. Cool? OK, and on your problem set, where you're gonna show-- one of the things you're gonna show-- and I love this result, because it encapsulates so much physics, you're going to show that if you have a potential-- a single potential barrier-- and you know the reflection amplitude, and the transmission amplitude-- or really, you just need to know one-- if you know the reflection amplitude-- if it's parity symmetric-- then you can write down the band structure entirely in terms of-- you can write down an equation for the allowed energies as a function of q-- entirely in terms of the reflection amplitude and the phase shift. Which is an amazing fact. So, you know about reflection off of one barrier, you make an infinite lattice of those barriers, and you deduce the structure of the allowed energy bands and gaps, which is cool. So, the scattering information-- as I kept promising-- contains a huge amount of the physics of your system. So, we saw that it contains bound energies, but it also contains these band gaps. OK, questions. Yeah? AUDIENCE: [INAUDIBLE] If we make the g0 go to infinity, would that approach a single [INAUDIBLE]? PROFESSOR: Yeah, exactly. Excellent. So, the question was, what happens as g0 goes to infinity, is it like getting single wells? So, let's look at that. It's a very good observation. So, here's a band between-- I really should put a line here-- but between here, and here, we have this band. And that corresponds to between here, and here. As we make g0 stronger, the derivation from simple cosine gets larger and larger. The amplitude gets larger and larger, and those bands get steeper, and steeper, and steeper. And the allowed energy bands get thinner, and thinner, and thinner. And, in fact, they get so thin, that Mathematica loses track of some of them, which is the annoyance. So, here we have very, very thin bands. What happens is we take g0 very, very large. So, let's make it 400. And, you can see, Mathematica doesn't like that. It's totally lost the band. So, let's try something a little less extreme. Oh, no, still too big for Mathematica. Eight, you can do it, little buddy. Ah, there we go! We can keep tack of one. So, what's happening is these bands are getting extremely thin. And what's happening, you're erecting infinitely high barriers between each period in the lattice. And if you have infinitely high barriers, what are the allowed eigenvalues? It's the same as the energy eigenvalues of the infinite well, because it has to be 0 at those infinite barriers, right? So, all of those states-- all of the states in the band-- become degenerate, and have the same energy, the energy of the periodic well, or of the infinite well. Very good question. OK. So, let's let Mathematica relax a little bit again. Well done. OK. And so, I want to quickly ask why do we have bands in the first place? Why do we have gaps? So, what's going on? Why do we have gaps in the first place? AUDIENCE: Because our math told us we had to? PROFESSOR: OK. Good. This is an excellent answer. Not the one I was looking for. The answer is a good answer because we did the calculation and that's what the calculation tells us. And that is a completely valid answer for a theorist, right? At this stage of writing down what your theory is and making a prediction. But before you can write that paper, publish it, and say, aha! I've discovered bands! You must have an explanation to the following question, why? How do you know your calculation was correct, before you've done the experiment? So, one answer is, of course, you do the experiment, and it fits like a champ. But that's not a very illuminating experiment as you know you're going to find departures from your theory, you want to improve the situation. You want some intuition about why this works. And you actually all know the physics behind this. And we did it on the first problem set. Remember the experiment that we studied on a problem set, the Davis and Germer Experiment, where you send plane waves onto a crystal, and due to interference effects, the transmission amplitude changes-- the transition probability changes-- as a function of the angle. And the reason it changes as a function of the angle was you had a crystal plane, right? So, atom, atom, atom, atom, atom, atom, atom, atom, and a plane wave sent in could scatter off of any of these scatterers, right? Or it could transmit and scatter off a further one down. And if you look at the scattering off different scatterers, if the effect of those scatterings-- so one down, and then one much deeper down-- if those interfere destructively, then that's going to suppress the reflection. And if they interfere constructively, it's gonna enhance the reflection. So, usually we think about that Davis and Germer Experiment, we think about it as a function of the angle of incidence, either straight at, or as a function of angle. And we ask, how does the transmission depend on those angles? But there's another way to phrase that question, you can ask, look, on average, how deep does that wave go? Does the wave propagate through the crystal? Or if it doesn't propagate through the crystal, it's gonna exponentially decay with some wavelength, with some decay-length. And what you find is that whether the wave propagates in, or whether it decays exponentially, depends on the energy and the lattice spacing of that crystal. And this is exactly the same thing, only we have a crystal in both directions. We don't have empty air and we're sending in a plane wave. Now we have a plane wave, can it propagate through? Well, for some energies, it'll propagate with continuous energy, or it'll propagate continuously. And for some energies, it will damp out exponentially. But if it damps out exponentially, and it damps out exponentially in the other direction too, then you don't get any solution at all. It just completely dies. So, what's really going on here is the fact that all of these scatterers in this lattice are scattering centers. And if you send a little wave packet in-- which you should think of as a superposition of energy eigenstates-- if you send a wave packet in, the probably to get-- the probability amplitude to get from here to here-- is the probability amplitude to go from here to here, plus to scatter through-- to transmit through and reflect back-- plus to transmit, transmit, reflect, reflect-- or sorry, transmit, transmit, reflect, transmit-- plus all the other crazy things you could do. And you have to sum up over all those terms. You have to sum the effect of how do you get from here to here? Every possible way to get from here to there, a la the two slit experiment. Sum of over all possible paths where you bounce around scattering off each of these scattering centers. And that's why the band structure's encoded in the reflection amplitudes from an individual scattering center. Because now, it's just a common [INAUDIBLE] problems of how many different paths, and summing up the phase that you get. OK? So, it's just about coherent and incoherent scattering. Yeah? AUDIENCE: [INAUDIBLE] crystal and I have an electron. PROFESSOR: Yeah. AUDIENCE: If I sent the electron through the crystal in just the right way, with just the right energy, won't it just go through in some states, without seeing it? PROFESSOR: Excellent. Excellent. So, the question is under what conditions can you send an electron and it'll just cruise right on through the potential? So, let's answer that question. That's exactly the right question. So, let's pick up from here. And I want to ask, what have we learned? So, what are the lessons for g0 equal to 0? So, the first lesson is that the energy eigenvalues, E eigenvalues, are restricted to lie within a band-- this bands-- and the bands are separated by gaps. Two, the energy eigenstates-- so this was eigenvalues-- the energy eigenstates are all extended. And we didn't even need to find the full solution of this equation to know that. All we needed to know is that it was a solution. That the energy eigenfunction are eigenfunctions of the translation operator as well. And that tells us the probability of that form. And that tells us what the probability distribution is, which is the norm squared of that guy, is independent, it's periodic. And if it's periodic, it can't fall off exponentially one direction or the other. So, we've seen that the eigenfunctions are all extended, and so as a consequence, we have to build wave packets. So, therefore, we must build wave packets. And, again, the form of the wave packets, psi is equal to the integral dq f of q-- sorry, I'll say fq0-- sharply peaked around the point q0 of q. And then the wave function's E to the iqz minus omega t times ru of x sub q. OK. So, we have to build wave packets. So, three, this tells us that those wave packets move with group velocity, by the same logic as for the free particle, V group is equal to d omega dk-- or dq-- evaluated at q0. OK, where q0 is where our wave packet is sharply peaked. So, that's already telling you something really cool. Look at this lowest band. Let's make this a little more reasonable. So, let's look at this lowest band. And in particular, I'm going to make this big. So, focus in on the lowest energy band, here. So, there's the lowest energy band. It goes from the top of the band to the bottom of the band. It's just a little bit away from the free particle, because we're looking at a relatively weakly, perturbed system. I chose a small value of q0. Actually, let's make it a little more exaggerated. OK, so there's the free particle curve, the red one, and the blue one is our actual band. So, that's the energy as a function of q. And let's make it even bigger. This one goes to 11. Let's see. OK. And so what you can see is the following, it looks, again, it's looks sort of like a parabola at the bottom. And at the top, it curves over, and, actually, goes to zero derivative. And it has to, because it's got to be periodic. Because q is periodic with period 2pi over L. So, what does that tell you about the velocity? Well, let's look at the velocity. V group velocity, of wave packet with-- whoops, say, here's 0-- as a function of q between minus 2pi over L, and 2pi over L. What is the group velocity? And here's what I mean. What I mean is let's build a wave packet which has a reasonably well-localized momentum, say, at this value of q. So, I built a little wave packet it has a peak here, and what does that wave function do? Well, it has a group velocity. And the group velocity is given by d omega dq. And now, we can't solve that equation analytically, because it's [INAUDIBLE]. But we can see, just by eyeball, what this does. When q is equal to 0, what's the group velocity? Yeah. It's the slope, right? D omega dq. So, what's d omega dq at the origin? 0, right? 'Cause it's just horizontal. So, it's 0. Good. What about a little bit to the right of 0? What about if I give it a small positive q-- wave packet localized with a small positive q? It's got a little bit of a slope now, right? Little bit of a positive slope. So, it increases. But then, it reaches a maximum slope, halfway through the domain. So, here, it hits a maximum value, and then the slope comes back down. And if we'd went the other way, if we'd made q a little bit negative, the slope becomes negative. And that's the velocity. So, that's a slightly strange thing. As we increase q, it increases the velocity. Increasing q increases the velocity for a while. That's just like k and momentum. Momentum is h bar k. So, that would be increasing the group velocity for a free particle. But now, as we increase the q to a reasonably large value, to 2pi upon L-- oh, sorry, this is pi upon L, and pi upon L-- so, if we do this to pi upon 2L, or roughly, to the middle, we get to a maximum value. And if we increase q further, what happens? The velocity goes down, right? If we increase q further, the velocity goes. This is strange. For a free particle, if you had momentum h bar k, and you increased k, what happens to the expectation-? What happens to the group velocity? If you increase the momentum, what happens to the group velocity for a free particle? AUDIENCE: [MURMURS] PROFESSOR: It's gonna increase, right? But not for a particle in the periodic potential. The wave packet, as you increase the crystal momentum, q, past this maximum, as you increase the crystal momentum, you actually decrease the velocity. And, in fact, if you keep increasing q, eventually the velocity goes to 0. And you increase q a little more, and it actually goes negative. So, this is a funny thing, it's not behaving like normally you'd think of as a momentum. And, indeed, is there momentum conservation in this system? No. We have a periodic potential. There are forces at work. Is momentum conserved? No. This is not a system that conserves a momentum. Momentum can be exchanged between the particle and the potential. Q is a good quantity. It's the eigenfunction of TL, and that does commute with the energy operator. So, it's a perfectly reasonable quantity. But the momentum itself, p, is not conserved. So, this crystal momentum, q, is not the momentum, for all these reasons. But it plays a lot of the same role, especially near the bottom of the band. Near the top of the band, it's a little bit funnier, because if you increase q, you get a negative velocity. If you decrease q, you get a positive velocity. It's exactly the opposite of what you'd normally expect. But let's hold on to that thought for a second. So, there's another sense in which the crystal momentum is like a momentum. And you're gonna show this on your problem set. But it's very important in giving you some understanding-- some intuition-- about what the crystal momentum is. On your problem set, you'll show the following. Suppose I induce a constant force. For example, imagine this particle in my periodic potential had a charge, and I turn on a constant, uniform electric field. Then that particle experiences a constant, uniform driving force, which I can write as a linearly increasing potential, right? So, what you showed on the problem set is that the force-- the expectation value of the force-- is equal to-- for a wave packet-- sharply localized around some value, q0, the force, it gives the time rate of change, d dt, of the expectation value of q. So, if you turn on a force, what happens is q increases linearly. Thank you. It's so hard. It's just so obviously 1. So, if the force is constant, then h bar q increases constantly. So, q just increases linearly in time, yeah? But that's funny. Well, we'll come back to that, just how funny that is in just a-- yeah? Question? No, OK. Well, let's think about how funny that it. Let's tackle this, right now. So, imagine we start with a wave packet, which is well localized around zero crystal momentum, around q equals 0. Then what is that wave function doing in time? How does that wave packet move? What's its group velocity? 0. So, it's just sitting there. It's just little wave packets slowly dispersing, quantum mechanically, yeah? But if we don't make it too tightly constrained, and p and 2 too tightly constrained. And the dispersion could be made quite slow. So, it's just a little wave packet, localized around a particular position. And localized around zero momentum, and it's just sitting there. Cool? Now, let's apply a force. What happens? As we apply a force, a constant, force with a positive sign. Q is gonna increase. So, the central value-- I should say, yeah-- so, the central value of our wave packet is gonna go from 0 to something slightly positive. Because q is increasing. What that means is that the velocity is increasing at first. So, the velocity is gonna increase, and in equal units of time it's gonna march linearly along this way. So, the particle is going faster, and faster, and faster, and faster. And that makes sense. You're driving the system by putting on a constant force. Of course it accelerates. The velocity accelerates linearly, so the position accelerates quadratically. So, let's plot the position-- expectation value of the position-- as a function of time. At zero time-- it's at the origin-- it's at some trivial position, x0. As we increase time, velocity's increasing linearly for awhile, and the position is gonna increase quadratically. Everyone cool with that? But eventually, at some point in time, the velocity-- or the q-- is gonna get to the point where we're at the maximum allowed velocity. And at that point, what is the curve for x gonna look like? It's gonna have an inflection point. And as we increase-- as we wait longer-- q is gonna continue linearly increasing, which means the velocity is going to slow down. So, the thing slows down, so its slope decreases, until finally it's got zero velocity again. There we go-- zero velocity-- dx dt. And then what happens? We keep forcing it. It's a constant driving force. But this is this, by periodicity of q, and the thing is, we increase q further linearly in time. As we increased q further, the velocity becomes negative. So, instead of continuing to accelerate-- instead of continuing to move-- in a positive x direction, it starts going backwards! That's weird. You're putting a force in this direction, and it's accelerating in that direction. That's a strange effect. And as we continue, eventually we get to maximum negative velocity. Again, an inflection point. And, at which point, the velocity starts getting less, and less negative, or closer, and closer to 0. And we return. And so, in some period, capital T, which is determined by how big this constant driving force is, the particle returns to its original position. Everyone see that? Yeah? AUDIENCE: Is it starting at q equals 0? PROFESSOR: Well, yeah. At q0 equals 0. Exactly. So, at first point, so this is q equals 0, this is-- sorry, q equals 0-- is the initial point. And x is equal to x0. So, under this process, the particle goes through an oscillation, called a Bloch Oscillation. It's named after the guy who invented the wave functions, the Bloch Wave Functions. And this is a deeply weird thing. Normally, you think, well, look, if I take a charged particle, and I just put it in an empty space, or in a box, or whatever, and I turn on an electric field, what happens to it? It accelerates. But if you put that charged particle in a periodic lattice, like copper-- idealized copper, here, perfect uniform lattice-- and you turn on an electric field-- or you put a capacitor plate across the thing-- that electron, at first, wants to behave like an electron in free space. It wants to accelerate. But then it finds out it's in a lattice. And it goes backwards. And it just oscillates back and forth, [INAUDIBLE]-like, you know? I really want to get there, oh crap. I really want to get there. Oh no, I'm in a periodic potential. And it just oscillates back and forth. Is there any conductions in this system due to the electromagnetic field that we've induced? None. That is a strange little beasty. That is a strange, strange property. And yet, copper conducts. Copper, which is a uniform lattice of ions-- of potentials-- to which electrons are stuck in bands. Copper conducts! The electrons don't do this Bloch Oscillation. How come? Yeah. It's not perfect. And it's not perfect for a whole bunch of reasons. So, one reason it's not perfect is that, so, first off, in a lattice of, say, copper, or whatever, the ions have a finite mass. So, what that means is they're not stuck in place, they also wiggle. When the electron moves, one of the ions move. So, the whole thing you should think of as a little wiggling piece of jello. And as a consequence, they're not perfectly periodic. This does a bunch of things. One is these are no longer the exact energy eigenfunctions. You now have to deal with the oscillations and wiggling of the lattice. But more importantly, here's something that can happen-- and electron can move along, and it can kick one of these ions-- it can bounce off one of the ions-- and scatter some of its momentum into the ion. Not into the rigid lattice, but it could just make one of those ions wiggle. So, it's changed the structure of the lattice. OK? And what you can do then, is you can have an electron that is accelerating, and it hits something, and it stops. And it accelerates. And it stops. Accelerates, and it hits something, then it stops. OK? So, when you have lots of disorder in your system-- or when you allow the electron to bounce off things in the lattice-- you get this effective conductivity. Things start their Bloch Oscillation, then they collide, and scatter off their momentum, and go back to zero momentum, but they've moved over a little bit in their oscillation already. They move up, and they move. So, what that picture looks like is, you accelerate, and then, boom, you scatter, and you fall back down to zero momentum by scattering off your momentum into something else. Same thing, you move up, and then, now, you have zero momentum again. You move up, you have zero momentum again. You move up, you have zero momentum. You move up, you have zero momentum, again. And so, you get this effective drifting of the electrons, pointed out by a guy named Drude. That gives you, effectively, a conductivity and a solid. Now, there are a bunch of other effects that are important for conductivity. Normally, we think of disorder as something that, you know, if you make things messy, they're probably gonna transmit less well, right? But disorder is absolutely essential for conductivity in real solids. Both for this reason, because you scatter off of fluctuations, but also, when you have a chunk of copper, it's not a chunk of copper. It's a chunk of copper, but it's got, you know, here and there, there's a little carbon atom that got stuck, and maybe there's a bit of nickel, and, you know, palladium, or, you know, Berkelium-- pretty unlikely, but could be there-- so, you have all sorts of schmutz sort of distributed around. And, again, those are things that make the potential not periodic. And, so, will change the conservation of this q. OK? Yeah? AUDIENCE: [INAUDIBLE] but if you actually managed to put a alternating current-- or an alternating voltage-- at the right frequency, you could actually drive the electrons from one side to the other in this fashion? PROFESSOR: Yeah, so, I-- AUDIENCE: --even though the conventional way of thinking about alternating currents is that the electrons really don't move at all. PROFESSOR: That's a very good observation. It's a little subtle, so come to my office hours and ask me about that. But there's an interesting-- For a very specific reason, because it's easy and unambiguous, I chose look the DC. But you're right, looking at the AC, is really a very entertaining example. So, yeah, I'm not gonna get into it. There's a cool story with parametric resonance. And yeah, there's a very nice story there. Come to my office hours, because that is a particularly fun story. OK, but I want to do one last thing on this before we move on. So, in this process, something very strange happened. When we got to the top of the potential, what we found was as we increase-- as we continue constant, positive force-- the velocity decreased. Yeah? That is weird. Normally, when I take a force, and I take some object with some mass, and I take a force and I apply it to the mass, and then the acceleration is the force divided by the mass. In particular, mass is always a positive thing, so if I take my force, and I divide it by the positive mass, I get an acceleration in the same direction as the force. But here, we have an applied force, and we get an acceleration in the opposite direction. I thought our particle had a positive mass. Didn't it have a positive mass? We started off with a positive mass. We should still have a positive mass. So, what's going on here? Well, we have a rule for calculating what the mass is. We take the expectation value of the momentum and divide by the group velocity. So, let's find out what the mass is for this guy. And I'll do that here. So, this is a slightly surreal moment with a nice little codon. So, I'm going to write that expression for the mass slightly differently. And you're gonna show in your problem set-- PSet-- you're gonna show that that definition of the mass-- M star is p upon the group velocity-- leads to the following expression, you can write that as-- you can use that to derive-- that the mass can be written as 1 upon h bar squared d squared E dq squared. So, the group velocity is the first derivative of the energy with respect to q. The mass-- or really q over the mass-- is proportional to the second derivative, the curvature. Everyone see that? So, this should not be an obvious equation, unless you're breathtakingly good at calculus in your head. It's just chain rule, but it does take a little bit of careful thinking, because it's a physical argument, it's not a rigorous mathematical argument. You're gonna do it on your problem set. So, let's take this expression and let's see what the effective mass is. What is the mass of our particle? Now, here, when I see what is the mass of our particle? I mean a very precise thing. What is the mass of the object moving in this periodic potential? That's what I mean. What is the mass of this thing? And we have a definition for it. Here it is. And let's plot this 1 over mass. So, let's plot this 1 over mass. Oh, that's not the way I wanna draw it. Yeah. Yeah, OK, good. So, here-- oh, shoot-- so, let's do it this way. So, I want to plot 1 over mass, in the vertical, as a function of q. And here is q is equal to 0, and here is 1 upon mass is equal to 0. So, at the bottom of the band-- and this is pi upon L, and minus pi upon L. At the bottom of at the band, at q equals 0, what is the mass? Well, here's the velocity, and there's the band. So, what's the mass at the bottom of the band? Tell me properties about it. Is it 0? Yeah. There's still some curvature. It's approximated by some parabola, right? It's even. And, in fact, you can take that equation and derive properties about it. But it's some curvy thing. You can see that. It's more obvious if we look at a higher energy band. So, here's the minimum. Here's what it would have been if we didn't have a potential, right? But here's the minimum, and it's nice and curvy, and at the top, again, it's nice and curvy. But it's got a second derivative there, that's non-zero. Everyone agree with that? Now, here's the thing I want to ask, is that second derivative the same as it would have been if we had g arbitrarily small? AUDIENCE: [INAUDIBLE]? PROFESSOR: Yeah. Not so much. No, if g was much lower, the second derivative would be a bit different. OK. So, in particular, this is some funny value. And I'm just gonna give it a name. It's gonna have some value at 0. And, in particular, it is positive. Everyone agree with that? But there's a point here-- there's an inflection point-- where this thing has zero second derivative. And that inflection point is the same point where the velocity becomes a maximum, because the velocity's a first derivative. So, when the velocity's at a maximum, the second derivative has a 0. It's the derivative of the first velocity, zero slope. And so, at that inflection point, the velocity is 0-- or sorry, the velocity is a maximum-- and 1 upon the mass, goes to 0. And then, if we increase q further, what happens? 1 over m star goes negative. And it reaches a minimum at pi over L, come back, and, again, repeats. So, it's periodic. I'm just artistically challenged. So, how can this possibly be? Right? So, first off, what this is telling us is that we have a point where the mass is 0. It's positive around the origin, where things normally behave intuitively. It goes to 0-- 1 over the mass-- goes to 0, which tells me that the mass, m star, is going to infinity. 1 over the mass going to 0? Mass must be going to infinity. What does that mean? Well, what does it mean for something to be infinitely massive? It mean f equals ma, if it's infinitely massive-- then let's write that as acceleration is force over m star-- then that tells you that if you apply a force, you get no acceleration. But that's exactly what we saw. We get to this point, we keep applying our force, and the velocity stays constant. There's no acceleration. Yeah? AUDIENCE: [INAUDIBLE] in this equation, that's an external force, right? PROFESSOR: That's an external force. AUDIENCE: So, only-- PROFESSOR: Exactly. Yeah. Purely external force. I put a capacitor plate across my piece of metal. AUDIENCE: So, my next question would be is it all that insane then? Because, like, you haven't calculated the net force on the object, so to speak, and so there is the potential. And so, yeah, the net force is zero, sure, you're applying force, but it's not going anywhere. PROFESSOR: Yeah, it's not a complete answer to what's going on here, but it's a very good observation. So, let me rephrase that slightly. So, look, I'm doing a slightly strange thing. I'm applying an external force, and I'm not treating it quantum mechanically, I'm treating this particle quantum mechanically. That's clearly stupid. We should treat everything quantum mechanically, and derive the results in that fashion. It turns out, in this case, that the negative effective mass doesn't go away if we treat the electromagnetic field quantum mechanically. So, you're absolutely correct, but it's not enough to sort of make us comfortable with the result. It doesn't change this result. But you're absolutely correct that that's an important thing to do if we want to be honest. But this is a perfectly good approximation for our purposes. So, the mass goes infinite, and then it goes a negative! 1 over the mass goes negative, so that's ridiculous, but that's exactly what we wanted. The acceleration is the force divided by the mass, but we found that when we apply a force in this direction, we get an acceleration in this direction. That's a negative coefficient, so, in particular, here's what I want to think about. I want to ask the following question, this value of the mass-- even around the bottom, this should already be disturbing observation-- because this value of the mass is not the same as the mass of our original particle. And the way you can see that is if you go to various strong coupling, then this band shrinks down. It becomes very thin, and the second derivative becomes arbitrarily small, which means the mass becomes arbitrarily large. So, the mass is getting larger and larger as we crank up the potential-- the effective mass-- of whatever this thing is that's moving. But this is strange. The thing that's moving is our particle. It has mass, m. Everyone agree with that? What is going on here? And here, I want to do an experiment. And my experiment-- AUDIENCE: [MURMURING] PROFESSOR: So, yeah, the seamless pong ball company, in recognition of my contributions to physics and $3, has given me a series of ping pong balls, with which, I'm going to do the following experiment. OK. Here's the calculation I want to do. The lesson I want you to take away from this, is that you've got to be very careful what you mean by, "the mass" of an object. Take one of these ping pong balls, I'm not gonna do the whole experiment for you, but I'll tell you the set up. Take a ping pong ball, and the first thing I want to do is, I want to measure it's mass. Well, what do you mean by measure its mass? Well, I'm gonna take it, and I'm gonna put it in a vacuum, you know, we could take it to France to were they do the-- And take the ping pong ball-- ANNOUNCEMENT: One, two, three, four. One, two, three, four. One, two. AUDIENCE: [LAUGHTER] PROFESSOR: We're just gonna turn that down. OK. ANNOUNCEMENT: One, two, three, four. One, two, three, four. PROFESSOR: Can we kill that? OK, so. So, we take our ping pong ball, and we put some mass over here, with a known mass, and we put it on a scale, and we wait until this goes to the vertical, and we declare the mass. And what we find is that the mass of our ping pong ball-- this is, you know, not the greatest ping pong balls-- but by legislation, this is supposed to be about 2.7 grams, or people get in fistfights. So, the mass of the ping pong ball needs to be about 2.7 grams. And, meanwhile, the radius of this guy is about 20 millimeters, so 2 centimeters. Yeah, that's about right. And yeah. So, this is supposed to-- morally-- it's supposed to be 2 centimeters. And so, if you go through this and you compute the density from these guys-- so, this gives you a volume-- and if you compute the density, the density is about 12.4 by regulation. Now, I've done this experiment, and I've done this measurement. And I must've used very cheap balls, because I got a very, very low density. I got-- oh sorry, this is the wrong way to do it-- the density is equal to 1, in usual CGS units, is 1 upon 12.4. And I can write that as 1 upon 12.4 times the density of water. OK? Now, when I did this experiment, I got 1 over 20. So, I was probably using very cheap ping pong balls. But if you look at the regulations on Wikipedia, this is what you're supposed to get. Let's assume these are perfect. So, you do this. There's the mass. That is the mass. That's what I mean by the mass. You take a scale, you weigh the thing! Right? Does anyone object to my definition of the mass? It's an observational, empirical process for determining the mass. If I take my mass, OK? And now I ask the following question, if I take some water, and I pull my ping pong ball, and I put it under water, and I put a spring scale here, and they measure the force on this ping pong ball, what will the force be? Well, the force will be-- and let's just call this ten, because the calculation's going to be horrible otherwise-- well, whatever, let's call it 12. So, the force-- and another way to write this is that rho water is equal to 12 rho ping pong ball-- I did math there. So, the force-- we know this from Archimedes-- the force is a combination of the weight of the ping pong ball down, so, it's going to be the density times the volume. So, the volume times rho ping pong ball down, so, minus. Plus the volume times rho water up, because that's the weight of the water displaced, right? So, your weight down, minus the weight of the water displaced. Everyone cool with that? Archimedes. It's pretty well known at this point. So, this is equal to the volume times rho water minus rho ping pong ball. And I can write this as the volume times rho water is 12 rho ping pong balls, so that's 12 minus 1 rho ping pong ball. That's 11 rho ping pong ball. But this is equal to 11 times the weight of the ping pong ball-- oh sorry-- 11 times the mass of the ping pong ball. The volume times the density. So, mass ping pong ball. Now, there's an important thing that I forgot to add here, what did I forget to add? G. Right? It's the force is the mass times g. OK, g, g, and g. So, we get that the force is equal to 11 m ping pong ball g. Everyone agree with that? I haven't done anything sneaky. And so, as a consequence, what this predicts is that if I take the force and I divide it by the mass of the ping pong ball, I get 11 times the acceleration of gravity. And here's what that predicts, that predicts that when I let go of this thing, what should happen? It should accelerate upward through the water at 11 times the acceleration of gravity, which is roughly 10 meters per second squared. So, 110 meters per second squared. Does that sound right? Just intuitively, does that sound right? If you pull something like a ping pong ball, or a beach ball or something underwater, what happens when you let go of it? Well, at this point, I'm going to remind you that physics is an empirical science. So, let's see if we can do this. OK, now, what's supposed to happen is this thing is supposed to shoot out at nine times-- or 11 times-- the acceleration of gravity. So, it just shoots through this thing much faster than equivalently. So which one's going to hit first? AUDIENCE: [INAUDIBLE]. PROFESSOR: Should it come out, or should it hit the ground first? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah, the ping pong ball. Great, that's helpful. Uh-huh. So, what's gonna happen first? Is the ping pong ball gonna hit the surface, or is it gonna come flying out of the water? This is accelerating with the acceleration of gravity. And this it's accelerating up with 11 times the acceleration of gravity. So, let's do the experiment. And this is what they pay me for. Oh. This would be much more satisfying if we had a nice long-- how do I want to phrase this-- cylinder. Oh, shoot! Stop. So, it's best done with a nice, long cylinder, but let's do this. So, which one wins? AUDIENCE: [MURMURS] PROFESSOR: Gravity wins, right? And why? Intuitively, why does this work? AUDIENCE: [MURMURING] PROFESSOR: I am neglecting the drag effect, but I'm not really heating up the water all that much. I mean, it's not like the dissipation is all that significant. The dissipation of the momentum, which is drag-- the friction-- is not the most important thing for our purposes here. That water hasn't heated up at all. So, what's going on? AUDIENCE: [INAUDIBLE]. PROFESSOR: OK, so here's the question. Has a mass of the ping pong ball, my empty little ping pong ball changed? So, something like that has to be true. But what's the right answer? The right answer is this, the thing that's moving, is not the ping pong ball. The ping pong ball is interacting with the fluid. And as a ping pong ball accelerates, it drags the fluid along with it. And as it goes faster, and faster, it drags more and more of the fluid along with it. And as it's dragging more and more of that fluid, the effective mass of the entire object that's moving-- which is now the ping pong ball, and all of its associated water-- is increasing. And now, if you take that force, which was constant, and divide by this growing effective mass, you find that the acceleration rapidly falls off. And you get a terminal velocity. And you know this works, because if you blow a bubble under water, it doesn't shoot upwards with an infinite acceleration. It goes glurg, glurg, glurg. And this idea, this idea of the effective mass of a particle changing with its state of motion, is an idea called renormalization. And it plays an essential role, not just in particle physics, but in all of modern condensed matter physics. And we'll pick up at this point next time. AUDIENCE: [APPLAUSE]
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MIT_804_Quantum_Physics_I_Spring_2013_2013
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Lecture_6_Time_Evolution_and_the_Schrödinger_Equation.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So, finally, before I get started on the new stuff, questions from the previous lectures? No questions? Yeah. AUDIENCE: I have a question. You might have said this last time, but when is the first exam? PROFESSOR: Ah, excellent. Those will be posted on the Stellar page later today. Yeah. AUDIENCE: OK, so we're associating operators with observables, right? PROFESSOR: Yes. AUDIENCE: And Professor [? Zugoff ?] mentioned that whenever we have done a wave function with an operator, it collapses. PROFESSOR: OK, so let me rephrase the question. This is a very valuable question to talk through. So, thanks for asking it. So, we've previously observed that observables are associated with operators-- and we'll review that in more detail in a second-- and the statement was then made, does that mean that acting on a wave function with an operator is like measuring the observable? And it's absolutely essential that you understand that acting on a wave function with an operator has nothing whatsoever to do with measuring that associated observable. Nothing. OK? And we'll talk about the relationship and what those things mean. But here's a very tempting thing to think. I have a wave function. I want to know the momentum. I will thus operate with the momentum operator. Completely wrong. So, before I even tell you what the right statement is, let me just get that out of your head, and then we'll talk through that in much more detail over the next lecture. Yeah. AUDIENCE: Why doesn't it collapse by special relativity? PROFESSOR: We're doing everything non-relativistically. Quantum Mechanics for 804 is going to be a universe in which there is no relativity. If you ask me that more precisely in my office hours, I will tell you a relativistic story. But it doesn't violate anything relativistic. At all. We'll talk about that-- just to be a little more detailed-- that will be a very important question that we'll deal with in the last two lectures of the course, when we come back to Bell's inequality and locality. Other questions? OK. So, let's get started. So, just to review where we are. In Quantum Mechanics according to 804, our first pass at the definition of quantum mechanics is that the configuration of any system-- and in particular, think about a single point particle-- the configuration of our particle is specified by giving a wave function, which is a function which may depend on time, but a function of position. Observables-- and this is a complete specification of the state of the system. If I know the wave function, I neither needed nor have access to any further information about the system. All the information specifying the configuration system is completely contained in the wave function. Secondly, observables in quantum mechanics are associated with operators. Something you can build an experiment to observe or to measure is associated with an operator. And by an operator, I mean a rule or a map, something that tells you if you give me a function, I will give you a different function back. OK? An operator is just a thing which eats a function and spits out another function. Now, operators-- which I will denote with a hat, as long as I can remember to do so-- operators come-- and in particular, the kinds of operators we're going to care about, linear operators, which you talked about in detail last lecture-- linear operators come endowed with a natural set of special functions called Eigenfunctions with the following property. Your operator, acting on its Eigenfunction, gives you that same function back times a constant. So, that's a very special generically. An operator will take a function and give you some other random function that doesn't look all like the original function. It's a very special thing to give you the same function back times a constant. So, a useful thing to think about here is just in the case of vector spaces. So, I'm going to consider the operation corresponding to rotation around the z-axis by a small angle. OK? So, under rotation around the z-axis by a small angle, I take an arbitrary vector to some other stupid vector. Which vector is completely determined by the rule? I rotate by a small amount, right? I take this vector and it gives me this one. I take that vector, it gives me this one. Everyone agree with that? What are the Eigenvectors of the rotation by a small angle around the z-axis? AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah, it's got to be a vector that doesn't change its direction. It just changes by magnitude. So there's one, right? I rotate. And what's its Eigenvalue? AUDIENCE: One. PROFESSOR: One, because nothing changed, right? Now, let's consider the following operation. Rotate by small angle and double its length. OK, that's a different operator. I rotate and I double the length. I rotate and I double the length. I rotate and I double the length. Yeah, so what's the Eigenvalue under that operator? AUDIENCE: Two. PROFESSOR: Two. Right, exactly. So these are a very special set of functions. This is the same idea, but instead of having vectors, we have functions. Questions? I thought I saw a hand pop up. No? OK, cool. Third, superposition. Given any two viable wave functions that could describe our system, that could specify states or configurations of our system, an arbitrary superposition of them-- arbitrary linear sum-- could also be a valid physical configuration. There is also a state corresponding to being in an arbitrary sum. For example, if we know that the electron could be black and it could be white, it could also be in an arbitrary superposition of being black and white. And that is a statement in which the electron is not black. The electron is not white. It is in the superposition of the two. It does not have a definite color. And that is exactly the configuration we found inside our apparatus in the first lecture. Yeah. AUDIENCE: Are those Phi-A arbitrary functions, or are they supposed to be Eigenfunctions? PROFESSOR: Excellent. So, in general the superposition thank you. It's an excellent question. The question was are these Phi-As arbitrary functions, or are they specific Eigenfunctions of some operator? So, the superposition principle actually says a very general thing. It says, given any two viable wave functions, an arbitrary sum, an arbitrary linear combination, is also a viable wave function. But here I want to mark something slightly different. And this is why I chose the notation I did. Given an operator A, it comes endowed with a special set of functions, its Eigenfunctions, right? We saw the last time. And I claimed the following. Beyond just the usual superposition principle, the set of Eigenfunctions of operators corresponding to physical observables-- so, pick your observable, like momentum. That corresponds to an operator. Consider the Eigenfunctions of momentum. Those we know what those are. They're plane waves with definite wavelength, right? u to the ikx. Any function can be expressed as a superposition of those Eigenfunctions of your physical observable. We'll go over this in more detail in a minute. But here I want to emphasize that the Eigenfunctions have a special property that-- for observables, for operators corresponding to observables-- the Eigenfunctions form a basis. Any function can be expanded as some linear combination of these basis functions, the classic example being the Fourier expansion. Any function, any periodic function, can be expanded as a sum of sines and cosines, and any function on the real line can be expanded as a sum of exponentials, e to the ikx. This is the same statement. The Eigenfunctions of momentum are what? e to the ikx. So, this is the same that an arbitrary function-- when the observable is the momentum, this is the statement that an arbitrary function can be expanded as a superposition, or a sum of exponentials, and that's the Fourier theorem. Cool? Was there a question? AUDIENCE: [INAUDIBLE] PROFESSOR: OK, good. Other questions on these points? So, these should not yet be trivial and obvious to you. If they are, then that's great, but if they're not, we're going to be working through examples for the next several lectures and problem sets. The point now is to give you a grounding on which to stand. Fourth postulate. What these expansion coefficients mean. And this is also an interpretation of the meaning of the wave function. What these expansion coefficients mean is that the probability that I measure the observable to be a particular Eigenvalue is the norm squared of the expansion coefficient. OK? So, I tell you that any function can be expanded as a superposition of plane waves-- waves with definite momentum-- with some coefficients. And those coefficients depend on which function I'm talking about. What these coefficients tell me is the probability that I will measure the momentum to be the associated value, the Eigenvalue. OK? Take that coefficient, take its norm squared, that gives me the probability. How do we compute these expansion coefficients? I think Barton didn't introduce to you this notation, but he certainly told you this. So let me introduce to you this notation which I particularly like. We can extract the expansion coefficient if we know the wave function by taking this integral, taking the wave function, multiplying by the complex conjugate of the associated Eigenfunction, doing the integral. And that notation is this round brackets with Phi A and Psi is my notation for this integral. And again, we'll still see this in more detail later on. And finally we have collapse, the statement that, if we go about measuring some observable A, then we will always, always observe precisely one of the Eigenvalues of that operator. We will never measure anything else. If the Eigenvalues are one, two, three, four, and five, you will never measure half, 13 halves. You will always measure an Eigenvalue. And upon measuring that Eigenvalue, you can be confident that that's the actual value of the system. I observe that it's a white electron, then it will remain white if I subsequently measure its color. What that's telling you is it's no longer a superposition of white and black, but it's wave function is that corresponding to a definite value of the observable. So, somehow the process of measurement-- and this is a disturbing statement, to which we'll return-- somehow the process of measuring the observable changes the wave function from our arbitrary superposition to a specific Eigenfunction, one particular Eigenfunction of the operator we're measuring. And this is called the collapse of the wave function. It collapses from being a superposition over possible states to being in a definite state upon measurement. And the definite state is that state corresponding to the value we observed or measured. Yeah. AUDIENCE: So, when the wave function collapses, does it instantaneously not become a function of time anymore? Because originally we had Psi of (x,t). PROFESSOR: Yeah, that's a really good question. So I wrote this only in terms of position, but I should more precisely write. So, the question was, does this happen instantaneously, or more precisely, does it cease to be a function of time? Thank you. It's very good question. So, no, it doesn't cease to be a function of time. It just says that Psi at x-- what you know upon doing this measurement is that Psi, as a function of x, at the time which I'll call T star, at what you've done the measurement is equal to this wave function. And so that leaves us with the following question, which is another way of asking the question you just asked. What happens next? How does the system evolve subsequently? And at the very end of the last lecture, we answered that-- or rather, Barton answered that-- by introducing the Schrodinger equation. And the Schrodinger equation, we don't derive, we just posit. Much like Newton posits f equals ma. You can motivate it, but you can't derive it. It's just what we mean by the quantum mechanical model. And Schrodinger's equation says, given a wave function, I can determine the time derivative, the time rate of changes of that wave function, and determine its time evolution, and its time derivative, its slope-- its velocity, if you will-- is one upon I h bar, the energy operator acting on that wave function. So, suppose we measure that our observable capital A takes the value of little a, one of the Eigenvalues of the associated operators. Suppose we measure that A equals little a at some particular moment T start. Then we know that the wave function is Psi of x at that moment in time. We can then compute the time derivative of the wave function at that moment in time by acting on this wave function with the operator e hat, the energy operator. And we can then integrate that differential equation forward in time and determine how the wave function evolves. The point of today's lecture is going to be to study how time evolution works in quantum mechanics, and to look at some basic examples and basic strategies for solving the time evolution problem in quantum mechanics. One of the great surprises in quantum mechanics-- hold on just one sec-- one of the real surprises in quantum mechanics is that time evolution is in a very specific sense trivial in quantum mechanics. It's preposterously simple. In particular, time evolution is governed by a linear equation. How many of you have studied a classical mechanical system where the time evolution is governed by a linear equation? Right. OK, all of you. The harmonic oscillator. But otherwise, not at all. Otherwise, the equations in classical mechanics are generically highly nonlinear. The time rate of change of position of a particle is the gradient of the force, and the force is generally some complicated function of position. You've got some capacitors over here, and maybe some magnetic field. It's very nonlinear. Evolution in quantum mechanics is linear, and this is going to be surprising. It's going to lead to some surprising simplifications. And we'll turn back to that, but I want to put that your mind like a little hook, that that's something you should mark on to as different from classical mechanics. And we'll come back to that. Yeah. AUDIENCE: If a particle is continuously observed as a not evolving particle? PROFESSOR: That's an awesome question. The question is, look, imagine I observe-- I'm going to paraphrase-- imagine I observe a particle and I observe that it's here. OK? Subsequently, its wave function will evolve in some way-- and we'll actually study that later today-- its wave function will evolve in some way, and it'll change. It won't necessarily be definitely here anymore. But if I just keep measuring it over and over and over again, I just keep measure it to be right there. It can't possibly evolve. And that's actually true, and it's called the Quantum Zeno problem. So, it's the observation that if you continuously measure a thing, you can't possibly have its wave function evolve significantly. And not only is it a cute idea, but it's something people do in the laboratory. So, Martin-- well, OK. People do it in a laboratory and it's cool. Come ask me and I'll tell you about the experiments. Other questions? There were a bunch. Yeah. AUDIENCE: So after you measure, the Schrodinger equation also gives you the evolution backwards in time? PROFESSOR: Oh, crap! Yes. That's such a good question. OK. I hate it when people ask that at this point, because I had to then say more words. That's a very good question. So the question goes like this. So this was going to be a punchline later on in the in the lecture but you stole my thunder, so that's awesome. So, here's the deal. We have a rule for time evolution of a wave function, and it has some lovely properties. In particular-- let me talk through this-- in particular, this equation is linear. So what properties does it have? Let me just-- I'm going to come back to your question in just a second, but first I want to set it up so we have a little more meat to answer your question precisely. So we note some properties of this equation, this time evolution equation. The first is that it's a linear equation. The derivative of a sum of function is a sum of the derivatives. The energy operator's a linear operator, meaning the energy operator acting on a sum of functions is a sum of the energy operator acting on each function. You guys studied linear operators in your problem set, right? So, these are linear. What that tells you is if Psi 1 of x and t solves the Schrodinger equation, and Psi 2 of x and t-- two different functions of position in time-- both solve the Schrodinger equation, then any combination of them-- alpha Psi 1 plus Beta Psi 2-- also solves-- which I will call Psi, and I'll make it a capital Psi for fun-- solves the Schrodinger equation automatically. Given two solutions of the Schrodinger equation, a superposition of them-- an arbitrary superposition-- also solves the Schrodinger equation. This is linearity. Cool? Next property. It's unitary. What I mean by unitary is this. It concerns probability. And you'll give a precise derivation of what I mean by unitary and you'll demonstrate that, in fact, Schrodinger evolution is unitary on your next problem set. It's not on the current one. But what I mean by unitary is that conserves probability. Whoops, that's an o. Conserves probability. IE, if there's an electron here, or if we have an object, a piece of chalk-- which I'm treating as a quantum mechanical point particle-- it's described by the wave function. The integral, the probability distribution over all the places it could possibly be had better be one, because it had better be somewhere with probability one. That had better not change in time. If I solve the Schrodinger equation evolve the system forward for half an hour, it had better not be the case that the total probability of finding the particle is one half. That means things disappear in the universe. And much as my socks would seem to be a counter example of that, things don't disappear, right? It just doesn't happen. So, quantum mechanics is demonstrably-- well, quantum mechanics is unitary, and this is a demonstrably good description of the real world. It fits all the observations we've ever made. No one's ever discovered an experimental violation of unitarity of quantum mechanics. I will note that there is a theoretical violation of unitarity in quantum mechanics, which is dear to my heart. It's called the Hawking Effect, and it's an observation that, due quantum mechanics, black holes in general relativity-- places from which light cannot escape-- evaporate. So you throw stuff and you form a black hole. It's got a horizon. If you fall through that horizon, we never see you again. Surprisingly, a black hole's a hot object like an iron, and it sends off radiation. As it sends off radiation, it's losing its energy. It's shrinking. And eventually it will, like the classical atom, collapse to nothing. There's a quibble going on right now over whether it really collapses to nothing, or whether there's a little granule nugget of quantum goodness. [LAUGHTER] We argue about this. We get paid to argue about this. [LAUGHTER] So, but here's the funny thing. If you threw in a dictionary and then the black hole evaporates, where did the information about what made the black hole go if it's just thermal radiation coming out? So, this is a classic calculation, which to a theorist says, ah ha! Maybe unitarity isn't conserved. But, look. Black holes, theorists. There's no experimental violation of unitarity anywhere. And if anyone ever did find such a violation, it would shatter the basic tenets of quantum mechanics, in particular the Schrodinger equation. So that's something we would love to see but never have. It depends on your point of view. You might hate to see it. And the third-- and this is, I think, the most important-- is that the Schrodinger evolution, this is a time derivative. It's a differential equation. If you know the initial condition, and you know the derivative, you can integrate it forward in time. And they're existence and uniqueness theorems for this. The system is deterministic. What that means is that if I have complete knowledge of the system at some moment in time, if I know the wave function at some moment in time, I can determine unambiguously the wave function in all subsequent moments of time. Unambiguously. There's no probability, there's no likelihood, it's determined. Completely determined. Given full knowledge now, I will have full knowledge later. Does everyone agree that this equation is a deterministic equation in that sense? Question. AUDIENCE: It's also local? PROFESSOR: It's all-- well, OK. This one happens to be-- you need to give me a better definition of local. So give me a definition of local that you want. AUDIENCE: The time evolution of the wave function happens only at a point that depends only on the value of the derivatives of the wave function and its potential energy at that point. PROFESSOR: No. Unfortunately, that's not the case. We'll see counter examples of that. The wave function-- the energy operator. So let's think about what this equation says. What this says is the time rate of change of the value of the wave function at some position and some moment in time is the energy operator acting on Psi at x of t. But I didn't tell you what the energy operator is. The energy operator just has to be linear. But it doesn't have to be-- it could know about the wave function everywhere. The energy operator's a map that takes the wave function and tells you what it should be later. And so, at this level there's nothing about locality built in to the energy operator, and we'll see just how bad that can be. So, this is related to your question about special relativity, and so those are deeply intertwined. We don't have that property here yet. But keep that in your mind, and ask questions when it seems to come up. Because it's a very, very, very important question when we talk about relativity. Yeah. AUDIENCE: Are postulates six and three redundant if the Schrodinger equation has superposition in it? PROFESSOR: No. Excellent question. That's a very good question. The question is, look, there's postulate three, which says, given any two wave functions that are viable wave functions of the system, then there's another state which is a viable wave function at some moment in time, which is also a viable wave function. But number six, the Schrodinger equation-- or sorry, really the linearity property of the Schrodinger equation-- so it needs to be the case for the Schrodinger question, but it says something slightly different. It doesn't just say that any any plausible or viable wave function and another can be superposed. It says that, specifically, any solution of the Schrodinger equation plus any other solution of the Schrodinger equation is again the Schrodinger operation. So, it's a slightly more specific thing than postulate three. However, your question is excellent because could it have been that the Schrodinger evolution didn't respect superposition? Well, you could imagine something, sure. We could've done a differ equation, right? It might not have been linear. We could have had that Schrodinger equation was equal to dt Psi. So imagine this equation. How do we have blown linearity while preserving determinism? So we could have added plus, I don't know, PSI squared of x. So that would now be a nonlinear equation. It's actually refer to as the nonlinear Schrodinger equation. Well, people mean many different things by the nonlinear Schrodinger equation, but that's a nonlinear Schrodinger equation. So you could certainly write this down. It's not linear. Does it violate the statement three that any two states of the system could be superposed to give another viable state at a moment in time? No, right? It doesn't directly violate. It violates the spirit of it. And as we'll see later, it actually would cause dramatic problems. It's something we don't usually emphasize-- something I don't usually emphasize in lectures of 804, but I will make a specific effort to mark the places where this would cause disasters. But, so this is actually a logically independent, although morally-- and in some sense is a technically related point to the superposition principle number three. Yeah. AUDIENCE: For postulate three, can that sum be infinite sum? PROFESSOR: Absolutely. AUDIENCE: Can you do bad things, then, like creating discontinuous wave functions? PROFESSOR: Oh yes. Oh, yes you can. So here's the thing. Look, if you have two functions and you add them together-- like two smooth continuous functions, you add them together-- what do you get? You get another smooth continuous function, right? Take seven. You get another. But if you take an infinite number-- look, mathematicians are sneaky. There's a reason we keep them down that hall, far away from us. [LAUGHTER] They're very sneaky. And if you give them an infinite number of continuous functions, they'll build for you a discontinuous function, right? Sneaky. Does that seem terribly physical? No. It's what happens when you give a mathematician too much paper and time, right? So, I mean this less flippantly than I'm saying it, but it's worth being a little flippant here. In a physical setting, we will often find that there are effectively an infinite number of possible things that could happen. So, for example in this room, where is this piece of chalk? It's described by a continuous variable. That's an uncountable infinite number of positions. Now, in practice, you can't really build an experiment that does that, but it is in principle an uncountable infinity of possible positions, right? You will never get a discontinuous wave function for this guy, because it would correspond to divergent amounts of momentum, as you showed on the previous problem set. So, in general, we will often be in a situation as physicists where there's the possibility of using the machinery-- the mathematical machinery-- to create pathological examples. And yes, that is a risk. But physically it never happens. Physically it's extraordinarily rare that such infinite divergences could matter. Now, I'm not saying that they never do. But we're going to be very carefree and casual in 804 and just assume that when problems can arise from, say, superposing an infinite number of smooth functions, leading potentially to discontinuities or singularities, that they will either not happen for us-- not be relevant-- or they will happen because they're forced too, so for physical reasons we'll be able to identify. So, this is a very important point. We're not proving mathematical theorems. We're not trying to be rigorous. To prove a mathematical theorem you have to look at all the exceptional cases and say, those exceptional cases, we can deal with them mathematically. To a physicist, exceptional cases are exceptional. They're irrelevant. They don't happen. It doesn't matter. OK? And it doesn't mean that we don't care about the mathematical precision, right? I mean, I publish papers in math journals, so I have a deep love for these questions. But they're not salient for most of the physical questions we care about. So, do your best to try not to let those special cases get in the way of your understanding of the general case. I don't want you to not think about them, I just want you not let them stop you, OK? Yeah. AUDIENCE: So, in postulate five, you mentioned that [? functions ?] in effect was a experiment that more or less proves this collapse [INAUDIBLE] But, so I read that it is not [? complicit. ?] PROFESSOR: Yeah, so as with many things in quantum mechanics-- that's a fair question. So, let me make a slightly more general statement than answering that question directly. Many things will-- how to say-- so, we will not prove-- and experimentally you almost never prove a positive thing. You can show that a prediction is violated by experiment. So there's always going to be some uncertainty in your measurements, there's always going to be some uncertainty in your arguments. However, in the absence of a compelling alternate theoretical description, you cling on to what you've got it as long as it fits your data, and this fits the data like a champ. Right? So, does it prove? No. It fits pretty well, and nothing else comes even within the ballpark. And there's no explicit violation that's better than our experimental uncertainties. So, I don't know if I'd say, well, we could prove such a thing, but it fits. And I'm a physicist. I'm looking for things that fit. I'm not a metaphysicist. I'm not trying to give you some ontological commitment about what things are true and exist in the world, right? That's not my job. OK. So much for our review. But let me finally come back to-- now that we've observed that it's determinist, let me come back to the question that was asked a few minutes ago, which is, look, suppose we take our superposition. We evolve it forward for some time using the Schrodinger evolution. Notice that it's time reversal. If we know it's time reverted, we could run it backwards just as well as we could run it forwards, right? We could integrate that in time back, or we could integrate that in time forward. So, if we know the wave function at some moment in time, we can integrate it forward, and we can integrate it back in time. But, If at some point we measure, then the wave function collapses. And subsequently, the system evolves according to the Schrodinger equation, but with this new initial condition. So now we seem to have a problem. We seem to have-- and I believe this was the question that was asked. I don't remember who asked it. Who asked it? So someone asked it. It was a good question. We have this problem that there seem to be two definitions of time evolution in quantum mechanics. One is the Schrodinger equation, which says that things deterministically evolve forward in time. And the second is collapse, that if you do a measurement, things non-deterministically by probabilities collapse to some possible state. Yeah? And the probability is determined by which wave function you have. How can these things both be true? How can you have two different definitions of time evolution? So, this sort of frustration lies at the heart of much of the sort of spiel about the interpretation of quantum mechanics. On the one hand, we want to say, well, the world is inescapably probabilistic. Measurement comes with probabilistic outcomes and leads to collapse of the wave function. On the other hand, when you're not looking, the system evolves deterministically. And this sounds horrible. It sounds horrible to a classical physicist. It sounds horrible to me. It just sounds awful. It sounds arbitrary. Meanwhile, it makes it sound like the world cares. It evolves differently depending on whether you're looking or not. And that-- come on. I mean, I think we can all agree that that's just crazy. So what's going on? So for a long time, physicists in practice-- and still in practice-- for a long time physicists almost exclusively looked at this problem and said, look, don't worry about. It fits the data. It makes good predictions. Work with me here. Right? And it's really hard to argue against that attitude. You have a set of rules. It allows you to compute things. You compute them. They fit the data. Done. That is triumph. But it's deeply disconcerting. So, over the last, I don't know, in the second or the last quarter, roughly, the last third of the 20th century, various people started getting more upset about this. So, this notion of just shut up and calculate, which has been enshrined in the physics literature, goes under the name of the Copenhagen interpretation, which roughly says, look, just do this. Don't ask. Compute the numbers, and get what you will. And people have questioned the sanity or wisdom of doing that. And in particular, there's an idea-- so I refer to the Copenhagen interpretation with my students as the cop out, because it's basically disavowal of responsibility. Look, it doesn't make sense, but I'm not responsible for making sense. I'm just responsible for making predictions. Come on. So, more recently has come the theory of decoherence. And we're not going to talk about it in any detail until the last couple lectures of 804. Decoherence. I can't spell to save my life. So, the theory of decoherence. And here's roughly what the theory says. The theory says, look, the reason you have this problem between on the one hand, Schrodinger evolution of a quantum system, and on the other hand, measurement leading to collapse, is that in the case of measurement meaning to collapse, you're not really studying the evolution of a quantum system. You're studying the evolution of a quantum system-- ie a little thing that you're measuring-- interacting with your experimental apparatus, which is made up of 10 to the 27th particles, and you made up of 10 to the 28 particles. Whatever. It's a large number. OK, a lot more than that. You, a macroscopic object, where classical dynamics are a good description. In particular, what that means is that the quantum effects are being washed out. You're washing out the interference of fringes, which is why I can catch this thing and not have it split into many different possible wave functions and where it went. So, dealing with that is hard, because now if you really want to treat the system with Schrodinger evolution, you have to study the trajectory and the motion, the dynamics, of every particle in the system, every degree of freedom in the system. So here's the question that decoherence is trying to ask. If you take a system where you have one little quantum subsystem that you're trying to measure, and then again a gagillion other degrees of freedom, some of which you care about-- they're made of you-- some of which you don't, like the particles of gas in the room, the environment. If you take that whole system, does Schrodinger evolution in the end boil down to collapse for that single quantum microsystem? And the answer is yes. Showing that take some work, and we'll touch on it at the end of 804. But I want to mark right here that this is one of the most deeply unsatisfying points in the basic story of quantum mechanics, and that it's deeply unsatisfying because of the way that we're presenting it. And there's a much more satisfying-- although still you never escape the fact that quantum mechanics violates your intuition. That's inescapable. But at least it's not illogical. it doesn't directly contradict itself. So that story is the story of decoherence. And if we're very lucky, I think we'll try to get one of my friends who's a quantum computing guy to talk about it. Yeah. AUDIENCE: [INAUDIBLE] Is it possible that we get two different results? PROFESSOR: No. No. No. There's never any ambiguity about what result you got. You never end up in a state of-- and this is also something that decoherence is supposed to explain. You never end up in a situation where you go like, wait, wait. I don't know. Maybe it was here, maybe it was there. I'm really confused. I mean, you can get up in that situation because you did a bad job, but you don't end up in that situation because you're in a superposition state. You always end up when you're a classical beast doing a classical measurement, you always end up in some definite state. Now, what wave function describes you doesn't necessarily correspond to you being in a simple state. You might be in a superposition of thinking this and thinking that. But, when you think this, that's in fact what happened. And when you think that, that's in fact what happened. OK. So I'm going to leave this alone for the moment, but I just wanted to mark that as an important part of the quantum mechanical story. OK. So let's go on to solving the Schrodinger equation. So what I want to do for the rest of today is talk about solving the Schrodinger equation. So when we set about solving the Schrodinger equation, the first thing we should realize is that at the end of the day, the Schrodinger equation is just some differential equation. And in fact, it's a particularly easy differential equation. It's a first order linear differential equation. Right? We know how to solve those. But, while it's first order in time, we have to think about what this energy operator is. So, just like the Newton equation f equals ma, we have to specify the energy operative before we can actually solve the dynamics of the system. In f equals ma, we have to tell you what the force is before we can solve for p, from p is equal to f. So, for example. So one strategy to solve the Schrodinger equation is to say, look, it's just a differential equation, and I'll solve it using differential equation techniques. So let me specify, for example, the energy operator. What's an easy energy operator? Well, imagine you had a harmonic oscillator, which, you know, physicists, that's your go-to. So, harmonic oscillator has energy p squared over 2m plus M Omega squared upon 2x squared. But we're going quantum mechanics, so we replace these guys by operators. So that's an energy operator. It's a perfectly viable operator. And what is the differential equation that this leads to? What's the Schrodinger equation leads to? Well, I'm going to put the ih bar on the other side. ih bar derivative with respect to time of Psi of x and t is equal to p squared. Well, we remember that p is equal to h bar upon i, derivative with respect to x. So p squared is minus h bar squared derivative with respect to x squared upon 2m, or minus h bar squared upon 2m. Psi prime prime. Let me write this as dx squared. Two spatial derivatives acting on Psi of x and t plus m omega squared upon 2x squared Psi of x and t. So here's a differential equation. And if we want to know how does a system evolve in time, ie given some initial wave function, how does it evolve in time, we just take this differential equation and we solve it. And there are many tools to solve this partial differential equation. For example, you could put it on Mathematica and just use NDSolve, right? This wasn't available, of course, to the physicists at the turn of the century, but they were less timid about differential equations than we are, because they didn't have Mathematica. So, this is a very straightforward differential equation to solve, and we're going to solve it in a couple of lectures. We're going to study the harmonic oscillator in detail. What I want to emphasize for you is that any system has have some specified energy operator, just like any classical system, has some definite force function. And given that energy operator, that's going to lead to a differential equation. So one way to solve the differential equation is just to go ahead and brute force solve it. But, at the end of the day, solving the Schrodinger equation is always, always going to boil down to some version morally of solve this differential equation. Questions about that? OK. But when we actually look at a differential equation like this-- so, say we have this differential equation. It's got a derivative with respect to time, so we have to specify some initial condition. There are many ways to solve it. So given E, given some specific E, given some specific energy operator, there are many ways to solve. The resulting differential equation. And I'm just going to mark that, in general, it's a PDE, because it's got derivatives with respect to time and derivatives with respect to space. And roughly speaking, all these techniques fall into three camps. The first is just brute force. That means some analog of throw it on Mathematica, go to the closet and pull out your mathematician and tie them to the chalkboard until they're done, and then put them back. But some version of a brute force, which is just use, by hook or by crook, some technique that allows you to solve the differential equation. OK. The second is extreme cleverness. And you'd be amazed how often this comes in handy. So, extreme cleverness-- which we'll see both of these techniques used for the harmonic oscillator. That's what we'll do next week. First, the brute force, and secondly, the clever way of solving the harmonic oscillator. When I say extreme cleverness, what I really mean is a more elegant use of your mathematician. You know something about the structure, the mathematical structure of your differential equation. And you're going to use that structure to figure out a good way to organize the differential equation, the good way to organize the problem. And that will teach you physics. And the reason I distinguish brute force from cleverness in this sense is that brute force, you just get a list of numbers. Cleverness, you learn something about the way the physics of the system operates. We'll see this at work in the next two lectures. And see, I really should separate this out numerically. And here I don't just mean sticking it into MATLAB. Numerically, it can be enormously valuable for a bunch of reasons. First off, there are often situations where no classic technique in differential equations or no simple mathematical structure that would just leap to the imagination comes to use. And you have some horrible differential you just have to solve, and you can solve it numerically. Very useful lesson, and a reason to not even-- how many of y'all are thinking about being theorists of some stripe or other? OK. And how many of y'all are thinking about being experimentalists of some stripe or another? OK, cool. So, look, there's this deep, deep prejudice in theory against numerical solutions of problems. It's myopia. It's a terrible attitude, and here's the reason. Computers are stupid. Computers are breathtakingly dumb. They will do whatever you tell them to do, but they will not tell you that was a dumb thing to do. They have no idea. So, in order to solve an interesting physical problem, you have to first extract all the physics and organize the problem in such a way that a stupid computer can do the solution. As a consequence, you learn the physics about the problem. It's extremely valuable to learn how to solve problems numerically, and we're going to have problem sets later in the course in which you're going to be required to numerically solve some of these differential equations. But it's useful because you get numbers, and you can check against data, but also it lets you in the process of understanding how to solve the problem. You learn things about the problem. So I want to mark that as a separate logical way to do it. So today, I want to start our analysis by looking at a couple of examples of solving the Schrodinger equation. And I want to start by looking at energy Eigenfunctions. And then once we understand how a single energy Eigenfunction evolves in time, once we understand that solution to the Schrodinger equation, we're going to use the linearity of the Schrodinger equation to write down a general solution of the Schrodinger equation. OK. So, first. What happens if we have a single energy Eigenfunction? So, suppose our wave function as a function of x at time t equals zero is in a known configuration, which is an energy Eigenfunction Phi sub E of x. What I mean by Phi sub E of x is if I take the energy operator, and I act on Phi sub E of x, this gives me back the number E Phi sub E of x. OK? So it's an Eigenfunction of the energy operator, the Eigenvalue E. So, suppose our initial condition is that our system began life at time t equals zero in this state with definite energy E. Everyone cool with that? First off, question. Suppose I immediately at time zero measure the energy of this system. What will I get? AUDIENCE: E. PROFESSOR: With what probability? AUDIENCE: 100% PROFESSOR: 100%, because this is, in fact, of this form, it's a superposition of energy Eigenstates, except there's only one term. And the coefficient of that one term is one, and the probability that I measure the energy to be equal to that value is the coefficient norm squared, and that's one norm squared. Everyone cool with that? Consider on the other hand, if I had taken this wave function and I had multiplied it by phase E to the i Alpha. What now is the probability where alpha is just a number? What now is the probability that I measured the state to have energy E? AUDIENCE: One. PROFESSOR: It's still one, because the norm squared of a phase is one. Right? OK. The overall phase does not matter. So, suppose I have this as my initial condition. Let's take away the overall phase because my life will be easier. So here's the wave function. What is the Schrodinger equation? Well, the Schrodinger equation says that ih bar time derivative of Psi is equal to the energy operator acting on Psi. And I should be specific. This is Psi at x at time t, Eigenvalued at this time zero is equal to the energy operator acting on this wave function. But what's the energy operator acting on this wave function? AUDIENCE: E. PROFESSOR: E. E on Psi is equal to E on Phi sub E, which is just E the number. This is the number E, the Eigenvalue E times Psi at x zero. And now, instead of having an operator on the right hand side, we just have a number. So, I'm going to write this differential equation slightly differently, ie time derivative of Psi is equal to E upon ih bar, or minus ie over h bar Psi. Yeah? Everyone cool with that? This is the easiest differential equation in the world to solve. So, the time derivative is a constant. Well, times itself. That means that therefore Psi at x and t is equal to E to the minus i ET over h bar Psi at x zero. Where I've imposed the initial condition that at time t equals zero, the wave function is just equal to Psi of x at zero. And in particular, I know what Psi of x and zero is. It's Phi E of x. So I can simply write this as Phi E of x. Are we cool with that? So, what this tells me is that under time evolution, a state which is initially in an energy Eigenstate remains in an energy Eigenstate with the same energy Eigenvalue. The only thing that changes about the wave function is that its phase changes, and its phase changes by rotating with a constant velocity. E to the minus i, the energy Eigenvalue, times time upon h bar. Now, first off, before we do anything else as usual, we should first check the dimensions of our result to make sure we didn't make a goof. So, does this make sense dimensionally? Let's quickly check. Yeah, it does. Let's just quickly check. So we have that the exponent there is Et over h bar. OK? And this should have dimensions of what in order to make sense? AUDIENCE: Nothing. PROFESSOR: Nothing, exactly. It should be dimensionless. So what are the dimensions of h bar? AUDIENCE: [INAUDIBLE] PROFESSOR: Oh, no, the dimensions, guys, not the units. What are the dimensions? AUDIENCE: [INAUDIBLE] PROFESSOR: It's an action, which is energy of time. So the units of the dimensions of h are an energy times a time, also known as a momentum times a position. OK? So, this has dimensions of action or energy times time, and then upstairs we have dimensions of energy times time. So that's consistent. So this in fact is dimensionally sensible, which is good. Now, this tells you a very important thing. In fact, we just answered this equation. At time t equals zero, what will we get if we measure the energy? E. At time t prime-- some subsequent time-- what energy will we measure? AUDIENCE: E. PROFESSOR: Yeah. Does the energy change over time? No. When I say that, what I mean is, does the energy that you expect to measure change over time? No. Does the probability that you measure energy E change? No, because it's just a phase, and the norm squared of a phase is one. Yeah? Everyone cool with that? Questions at this point. This is very simple example, but it's going to have a lot of power. Oh, yeah, question. Thank you. AUDIENCE: Are we going to deal with energy operators that change over time? PROFESSOR: Excellent question. We will later, but not in 804. In 805, you'll discuss it in more detail. Nothing dramatic happens, but you just have to add more symbols. There's nothing deep about it. It's a very good question. The question was, are we going to deal with energy operators that change in time? My answer was no, not in 804, but you will in 805. And what you'll find is that it's not a big deal. Nothing particularly dramatic happens. We will deal with systems where the energy operator changes instantaneously. So not a continuous function, but we're at some of them you turn on the electric field, or something like that. So we'll deal with that later on. But we won't develop a theory of energy operators that depend on time. But you could do it, and you will do in 805. There's nothing mysterious about it. Other questions? OK. So, these states-- a state Psi of x and t, which is of the form e to the minus i Omega t, where Omega is equal to E over h bar. This should look familiar. It's the de Broglie relation, [INAUDIBLE] relation, whatever. Times some Phi E of x, where this is an energy Eigenfunction. These states are called stationary states. And what's the reason for that? Why are they called stationary states? I'm going to erase this. Well, suppose this is my wave function as a function of time. What is the probability that at time t I will measure the particle to be at position x, or the probability density? Well, the probability density we know from our postulates, it's just the norm squared of the wave function. This is Psi at x t norm squared. But this is equal to the norm squared of e to the minus Psi Omega t Phi E by the Schrodinger equation. But when we take the norm squared, this phase cancels out, as we already saw. So this is just equal to Phi E of x norm squared, the energy Eigenfunction norm squared independent of time. So, if we happen to know that our state is in an energy Eigenfunction, then the probability density for finding the particle at any given position does not change in time. It remains invariant. The wave function rotates by an overall phase, but the probability density is the norm squared. It's insensitive to that overall phase, and so the probability density just remains constant in whatever shape it is. Hence it's called a stationary state. Notice its consequence. What can you say about the expectation value of the position as a function of time? Well, this is equal to the integral dx in the state Psi of x and t. And I'll call this Psi sub E just to emphasize. It's the integral of the x, integral over all possible positions of the probability distribution, probability of x at time t times x. But this is equal to the integral dx of Phi E of x squared x. But that's equal to expectation value of x at any time, or time zero. t equals zero. And maybe the best way to write this is as a function of time. So, the expectation value of x doesn't change. In a stationary state, expected positions, energy-- these things don't change. Everyone cool with that? And it's because of this basic fact that the wave function only rotates by a phase under time evolution when the system is an energy Eigenstate. Questions? OK. So, here's a couple of questions for you guys. Are all systems always in energy Eigenstates? Am I in an energy Eigenstate? AUDIENCE: No. PROFESSOR: No, right? OK, expected position of my hand is changing in time. I am not in-- so obviously, things change in time. Energies change in time. Positions-- expected typical positions-- change in time. We are not in energy Eigenstates. That's a highly non-generic state. So here's another question. Are any states ever truly in energy Eigenstates? Can you imagine an object in the world that is truly described precisely by an energy Eigenstate in the real world? AUDIENCE: No. PROFESSOR: Ok, there have been a few nos. Why? Why not? Does anything really remain invariant in time? No, right? Everything is getting buffeted around by the rest of the universe. So, not only are these not typical states, not only are stationary states not typical, but they actually never exist in the real world. So why am I talking about them at all? So here's why. And actually I'm going to do this here. So here's why. The reason is this guy, the superposition principle, which tells me that if I have possible states, I can build superpositions of them. And this statement-- and in particular, linearity-- which says that given any two solutions of the Schrodinger equation, I can take a superposition and build a new solution of the Schrodinger equation. So, let me build it. So, in particular, I want to exploit the linearity of the Schrodinger equation to do the following. Suppose Psi. And I'm going to label these by n. Psi n of x and t is equal to e to the minus i Omega nt Phi sub En of x, where En is equal to h bar Omega n. n labels the various different energy Eigenfunctions. So, consider all the energy Eigenfunctions Phi sub En. n is a number which labels them. And this is the solution to the Schrodinger equation, which at time zero is just equal to the energy Eigenfunction of interest. Cool? So, consider these guys. So, suppose we have these guys such that they solve the Schrodinger equation. Solve the Schrodinger equation. Suppose these guys solve the Schrodinger equation. Then, by linearity, we can take Psi of x and t to be an arbitrary superposition sum over n, c sub n, Psi sub n of x and t. And this will automatically solve the Schrodinger equation by linearity of the Schrodinger equation. Yeah. AUDIENCE: But can't we just get n as the sum of the energy Eigenstate by just applying that and by just measuring that? PROFESSOR: Excellent. So, here's the question. The question is, look, a minute ago you said no system is truly in an energy Eigenstate, right? But can't we put a system in an energy Eigenstate by just measuring the energy? Right? Isn't that exactly what the collapse postulate says? So here's my question. How confident are you that you actually measure the energy precisely? With what accuracy can we measure the energy? So here's the unfortunate truth, the unfortunate practical truth. And I'm not talking about in principle things. I'm talking about it in practice things in the real universe. When you measure the energy of something, you've got a box, and the box has a dial, and the dial has a needle, it has a finite width, and your current meter has a finite sensitivity to the current. So you never truly measure the energy exactly. You measure it to within some tolerance. And In fact, there's a fundamental bound-- there's a fundamental bound on the accuracy with which you can make a measurement, which is just the following. And this is the analog of the uncertainty equation. We'll talk about this more later, but let me just jump ahead a little bit. Suppose I want to measure frequency. So I have some signal, and I look at that signal for 10 minutes. OK? Can I be absolutely confident that this signal is in fact a plane wave with the given frequency that I just did? No, because it could change outside that. But more to the point, there might have been small variations inside. There could've been a wavelength that could change on a time scale longer than the time that I measured. So, to know that the system doesn't change on a arbitrary-- that it's strictly fixed Omega, I have to wait a very long time. And in particular, how confident you can be of the frequency is bounded by the time over which-- so your confidence, your uncertainty in the frequency, is bounded in the following fashion. Delta Omega, Delta t is always greater than or equal to one, approximately. What this says is that if you want to be absolute confident of the frequency, you have to wait an arbitrarily long time. Now if I multiply this whole thing by h bar, I get the following. Delta E-- so this is a classic equation that signals analysis-- Delta E, Delta t is greater than or approximately equal to h bar. This is a hallowed time- energy uncertainty relation, which we haven't talked about. So, in fact, it is possible to make an arbitrarily precise measurement of the energy. What do I have to do? I have to wait forever. How patient are you, right? So, that's the issue. In the real world, we can't make arbitrarily long measurements, and we can't isolate systems for an arbitrarily long amount of time. So, we can't put things in a definite energy Eigenstate by measurement. That answer your question? AUDIENCE: Yes. PROFESSOR: Great. How many people have seen signals in this expression, the bound on the frequency? Oh, good. So we'll talk about that later in the course. OK, so coming back to this. So, we have our solutions of the Schrodinger equation that are initially energy Eigenstates. I claim I can take an arbitrary superposition of them, and by linearity derive that this is also a solution to the Schrodinger equation. And in particular, what that tells me is-- well, another way to say this is that if I know that Psi of x times zero is equal to sum over n-- so if sum Psi of x-- if the wave function at some particular moment in time can be expanded as sum over n Cn Phi E of x, if this is my initial condition, my initial wave function is some superposition, then I know what the wave function is at subsequent times. The wave function by superposition Psi of x and t is equal to sum over n Cn e to the minus i Omega nt Phi n-- sorry, this should've been Phi sub n-- Phi n of x. And I know this has to be true because this is a solution to the Schrodinger equation by construction, and at time t equals zero, it reduces to this. So, this is a solution to the Schrodinger equation, satisfying this condition at the initial time t equals zero. Don't even have to do a calculation. So, having solved the Schrodinger equation once for energy, Eigenstates allows me to solve it for general superposition. However, what I just said isn't quite enough. I need one more argument. And that one more argument is really the stronger version of three that we talked about before, which is that, given an energy operator E, we find the set of wave functions Phi sub E, the Eigenfunctions of the energy operator, with Eigenvalue E. So, given the energy operator, we find its Eigenfunctions. Any wave function Psi at x-- we'll say at time zero-- any function of x can be expanded as a sum. Specific superposition sum over n Cn Phi E sub n of x. And if any function can be expanded as a superposition of energy Eigenfunctions, and we know how to take a superposition, an arbitrary superposition of energy Eigenfunctions, and find the corresponding solution to the Schrodinger equation. What this means is, we can take an arbitrary initial condition and compute the full solution of the Schrodinger equation. All we have to do is figure out what these coefficients Cn are. Everyone cool with that? So, we have thus, using superposition and energy Eigenvalues, totally solved the Schrodinger equation, and reduced it to the problem of finding these expansion coefficients. Meanwhile, these expansion coefficients have a meaning. They correspond to the probability that we measure the energy to be equal to the corresponding energy E sub n. And it's just the norm squared of that coefficient. So those coefficients mean something. And they allow us to solve the problem. Cool? So this is fairly abstract. So let's make it concrete by looking at some examples. So, just as a quick aside. This should sound an awful lot like the Fourier theorem. And let me comment on that. This statement originally was about a general observable and general operator. Here I'm talking about the energy. But let's think about a slightly more special example, or more familiar example. Let's consider the momentum. Given the momentum, we can find a set of Eigenstates. What are the set of good, properly normalized Eigenfunctions of momentum? What are the Eigenfunctions of the momentum operator? AUDIENCE: E to the ikx. PROFESSOR: E to the ikx. Exactly. In particular, one over 2 pi e to the ikx. So I claim that, for every different value of k, I get a different value of p, and the Eigenvalue associated to this guy is p is equal to h bar k. That's the Eigenvalue. And we get that by acting with the momentum, which is h bar upon i, h bar times derivative with respect to x. Derivative with respect to x pulls down an ik times the same thing. H bar multiplies the k over i, kills the i, and leaves us with an overall coefficient of h bar k. This is an Eigenfunction of the momentum operator with Eigenvalue h bar k. And that statement three is the statement that an arbitrary function f of x can be expanded as a superposition of all possible energy Eigenvalues. But k is continuously valued and the momentum, so that's an integral dk one over 2 pi, e to the ikx times some coefficients. And those coefficients are labeled by k, but since k is continuous, I'm going to call it a function. And just to give it a name, instead of calling C sub k, I'll call it f tilde of k. This is of exactly the same form. Here is the expansion-- there's the Eigenfunction, here is the Eigenfunction, here is the expansion coefficient, here is expansion coefficient. And this has a familiar name. It's the Fourier theorem. So, we see that the Fourier theorem is this statement, statement three, the superposition principal, for the momentum operator. We also see that it's true for the energy operator. And what we're claiming here is that it's true for any observable. Given any observable, you can find its Eigenfunctions, and they form a basis on the space of all good functions, and an arbitrary function can be expanded in that basis. So, as a last example, consider the following. We've done energy. We've done momentum. What's another operator we care about? What about position? What are the Eigenfunctions of position? Well, x hat on Delta of x minus y is equal to y Delta x minus y. So, these are the states with definite value of position x is equal to y. And the reason this is true is that when x is equal to y, x is the operator that multiplies by the variable x. But it's zero, except at x is equal to y, so we might as well replace x by y. So, there are the Eigenfunctions. And this statement is a statement that we can represent an arbitrary function f of x in a superposition of these states of definite x. f of x is equal to the integral over all possible expansion coefficients dy delta x minus y times some expansion coefficient. And what's the expansion coefficient? It's got to be a function of y. And what function of y must it be? Just f of y. Because this integral against this delta function had better give me f of x. And that will only be true if this is f of x. So here we see, in some sense, the definition of the delta function. But really, this is a statement of the superposition principle, the statement that any function can be expanded as a superposition of Eigenfunctions of the position operator. Any function can be expanded as a superposition of Eigenfunctions of momentum. Any function can be expanded as a superposition of Eigenfunctions of energy. Any function can be expanded as a superposition of Eigenfunctions of any operator of your choice. OK? The special cases-- the Fourier theorem, the general cases, the superposition postulate. Cool? Powerful tool. And we've used this powerful tool to write down a general expression for a solution to the Schrodinger equation. That's good. That's progress. So let's look at some examples of this. I can leave this up. So, our first example is going to be for the free particle. So, a particle whose energy operator has no potential whatsoever. So the energy operator is going to be just equal to p squared upon 2m. Kinetic energy. Yeah. AUDIENCE: When you say any wave function can be expanded in terms of-- PROFESSOR: Energy Eigenfunctions, position Eigenfunctions, momentum Eigenfunctions-- AUDIENCE: Eigenbasis, does the Eigenbasis have to come from an operator corresponding to an observable? PROFESSOR: Yes. Absolutely. I'm starting with that assumption. AUDIENCE: OK. PROFESSOR: So, again, this is a first pass of the axioms of quantum mechanics. We'll make this more precise, and we'll make it more general, later on in the course, as we go through a second iteration of this. And there we'll talk about exactly what we need, and what operators are appropriate operators. But for the moment, the sufficient and physically correct answer is, operators correspond to each observable values. Yeah. AUDIENCE: So are the set of all reasonable wave functions in the vector space that is the same as the one with the Eigenfunctions? PROFESSOR: That's an excellent question. In general, no. So here's the question. The question is, look, if this is true, shouldn't it be that the Eigenfunctions, since they're our basis for the good functions, are inside the space of reasonable functions, they should also be reasonable functions, right? Because if you're going to expand-- for example, consider two dimensional vector space. And you want to say any vector can be expanded in a basis of pairs of vectors in two dimensions, like x and y. You really want to make sure that those vectors are inside your vector space. But if you say this vector in this space can be expanded in terms of two vectors, this vector and that vector, you're in trouble, right? That's not going to work so well. So you want to make sure that your vectors, your basis vectors, are in the space. For position, the basis vector's a delta function. Is that a smooth, continuous normalizable function? No. For momentum, the basis functions are plane waves that extend off to infinity and have support everywhere. Is that a normalizable reasonable function? No. So, both of these sets are really bad. So, at that point you might say, look, this is clearly nonsense. But here's an important thing. So this is a totally mathematical aside, and for those of you who don't care about the math, don't worry about it. Well, these guys don't technically live in the space of non-stupid functions-- reasonable, smooth, normalizable functions. What you can show is that they exist in the closure, in the completion of that space. OK? So, you can find a sequence of wave functions that are good wave functions, an infinite sequence, that eventually that infinite sequence converges to these guys, even though these are silly. So, for example, for the position Eigenstates, the delta function is not a continuous smooth function. It's not even a function. Really, it's some god- awful thing called a distribution. It's some horrible thing. It's the thing that tells you, give it an integral, it'll give you a number. Or a function. But how do we build this as a limit of totally reasonable functions? We've already done that. Take this function with area one, and if you want, you can round this out by making it hyperbolic tangents. OK? We did it on one of the problem sets. And then just make it more narrow and more tall. And keep making it more narrow and more tall, and more narrow and more tall, keeping its area to be one. And I claim that eventually that series, that sequence of functions, converges to the delta function. So, while this function is not technically in our space, it's in the completion of our space, in the sense that we take a series and they converge to it. And that's what you need for this theorem to work out. That's what you need for the Fourier theorem. And in some sense, that observation was really the genius of Fourier, understanding that that could be done. That was totally mathematical aside. But that answer your question? AUDIENCE: Yes. PROFESSOR: OK. Every once in a while I can't resist talking about these sort of details, because I really like them. But it's good to know that stupid things like this can't matter for us, and they don't. But it's a very good question. If you're confused about some mathematical detail, no matter how elementary, ask. If you're confused, someone else the room is also confused. So please don't hesitate. OK, so our first example's going to be the free particle. And this operator can be written in a nice way. We can write it as minus-- so p is h bar upon iddx, so this line is minus h bar squared upon 2m to the derivative with respect to x. There's the energy operator. So, we want to solve for the wave functions. So let's solve it using an expansion in terms of energy Eigenfunctions. So what are the energy Eigenfunctions? We want to find the functions E on Phi sub E such that this is equal to-- whoops. That's not a vector. That's a hat-- such as this is equal to a number E Phi sub E. But given this energy operator, this says that minus h bar squared over 2m-- whoops, that's a 2. 2m-- Phi prime prime of x is equal to E Phi of x. Or equivalently, Phi prime prime of x plus 2me over h bar squared Phi of x is equal to zero. So I'm just going to call 2me-- because it's annoying to write it over and over again-- over h bar squared. Well, first off, what are its units? What are the units of this coefficient? Well, you could do it two ways. You could either do dimensional analysis of each thing here, or you could just know that we started with a dimensionally sensible equation, and this has units of this divided by length twice. So this must have to whatever length squared. So I'm going to call this something like k squared, something that has units of one over length squared. And the general solution of this is that phi E E of x-- well, this is a second order differential equation that will have two solutions with two expansion coefficients-- A e to the ikx plus B e to the minus iks. A state with definite momentum and definite negative momentum where such that E is equal to h bar squared k squared upon 2m. And we get that just from this. So, this is the solution of the energy Eigenfunction equation. Just a note of terminology. People sometimes call the equation determining an energy Eigenfunction-- the energy Eigenfunction equation-- sometimes that's refer to as the Schrodinger equation. That's a sort of cruel thing to do to language, because the Schrodinger's equation is about time evolution, and this equation is about energy Eigenfunctions. Now, it's true that energy Eigenfunctions evolve in a particularly simple way under time evolution, but it's a different equation. This is telling you about energy Eigenstates, OK? And then more discussion of this is done in the notes, which I will leave aside for the moment. But I want to do one more example before we take off. Wow. We got through a lot less than expected today. And the one last example is the following. It's a particle in a box And this is going to be important for your problem sets, so I'm going to go ahead and get this one out of the way as quickly as possible. So, example two. Particle in a box. So, what I mean by particle in a box. I'm going to take a system that has a deep well. So what I'm drawing here is the potential energy U of x, where this is some energy E naught, and this is the energy zero, and this is the position x equals zero, and this is position x equals l. And I'm going to idealize this by saying look, I'm going to be interested in low energy physics, so I'm going to just treat this as infinitely deep. And meanwhile, my life is easier if I don't think about curvy bottoms but I just think about things as being constant. So, my idealization is going to be that the well is infinitely high and square. So out here the potential is infinite, and in here the potential is zero. U equals inside, between zero and l for x. So that's my system particle in a box. So, let's find the energy Eigenfunctions. And again, it's the same differential equations as before. So, first off, before we even solve anything, what's the probability that I find x less than zero, or find the particle at x greater than l? AUDIENCE: Zero. PROFESSOR: Right, because the potential is infinitely large out there. It's just not going to happen. If you found it there, that would correspond to a particle of infinite energy, and that's not going to happen. So, our this tells us effectively the boundary condition Psi of x is equal to zero outside the box. So all we have to do is figure out what the wave function is inside the box between zero and l. And meanwhile, what must be true of the wave function at zero and at l? It's got to actually vanish at the boundaries. So this gives us boundary conditions outside the box and at the boundaries x equals zero, x equals l. But, what's our differential equation inside the box? Inside the box, well, the potential is zero. So the equation is the same as the equation for a free particle. It's just this guy. And we know what the solutions are. So the solutions can be written in the following form. Therefore inside the wave function-- whoops. Let me write this as Phi sub E-- Phi sub E is a superposition of two. And instead of writing it as exponentials, I'm going to write it as sines and cosines, because you can express them in terms of each other. Alpha cosine of kx plus Beta sine of kx, where again Alpha and Beta are general complex numbers. But, we must satisfy the boundary conditions imposed by our potential at x equals zero and x equals l. So from x equals zero, we find that Phi must vanish when x equals zero. When x equals zero, this is automatically zero. Sine of zero is zero. Cosine of zero is one. So that tells us that Alpha is equal to zero. Meanwhile, the condition that at x equals l-- the wave function must also vanish-- tells us that-- so this term is gone, set off with zero-- this term, when x is equal to l, had better also be zero. We can solve that by setting Beta equal to zero, but then our wave function is just zero. And that's a really stupid wave function. So we don't want to do that. We don't want to set Beta to zero. Instead, what must we do? Well, we've got a sine, and depending on what k is, it starts at zero and it ends somewhere else. But we need it hit zero. So only for a very special value of k will it actually hit zero at the end of l. We need kl equals zero. Or really, kl is a multiple of pi. Kl is equal-- and we want it to not be zero, so I'll call it n plus 1, an integer, times pi. Or equivalently, k is equal to sub n is equal to n plus 1, where n goes from zero to any large positive integer, pi over l. So the energy Eigenfunction's here. The energy Eigenfunction is some normalization-- whoops-- a sub n sine of k and x. And where kn is equal to this-- and as a consequence, E is equal to h bar squared kn can squared E sub n is h bar squared kn squared over 2m, which is equal to h bar squared-- just plugging in-- pi squared n plus 1 squared over 2ml squared. And what we found is something really interesting. What we found is, first off, that the wave functions look like-- well, the ground state, the lowest possible energy there is n equals zero. For n equals zero, this is just a single half a sine wave. It does this. This is the n equals zero state. And it has some energy, which is E zero. And in particular, E zero is not equal to zero. E zero is equal to h bar squared pi squared over 2ml squared. It is impossible for a particle in a box to have an energy lower than some minimal value E naught, which is not zero. You cannot have less energy than this. Everyone agree with that? There is no such Eigenstate with energy less than this. Meanwhile, it's worse. The next energy is when n is equal to 1, because if we decrease the wavelength or increase k a little bit, we get something that looks like this, and that doesn't satisfy our boundary condition. In order to satisfy our boundary condition, we're going to have to eventually have it cross over and get to zero again. And if I could only draw-- I'll draw it up here-- it looks like this. And this has an energy E one, which you can get by plugging one in here. And that differs by one, two, four, a factor of four from this guy. E one is four E zero. And so on and so forth. The energies are gaped. They're spread away from each other. The energies are discrete. And they get further and further away from each other as we go to higher and higher energies. So this is already a peculiar fact, and we'll explore some of its consequences later on. But here's that I want to emphasize for you. Already in the first most trivial example of solving a Schrodinger equation, or actually even before that, just finding the energy Eigenvalues and the energy of Eigenfunctions of the simplest system you possibly could, either a free particular, or a particle in a box, a particle trapped inside a potential well, what we discovered is that the energy Eigenvalues, the allowed values of the energy, are discrete, and that they're greater than zero. You can never have zero energy. And if that doesn't sound familiar, let me remind you of something. The spectrum of light coming off of a gas of hot hydrogen is discrete. And no one's ever found a zero energy beam of light coming out of it. And we're going to make contact with that experimental data. That's going to be part of the job for the rest. See you next time. [APPLAUSE]
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MIT_508J_Biological_Chemistry_II_Spring_2016
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R2_PreSteady_State_and_SteadyState_Kinetic_Methods_Applied_to_Translation.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or view additional materials from hundreds of MIT courses, visit MIT open courseware at ocw.mit.edu. JOANNE STUBBE: Recitation 2 and recitation 3 are on the same paper. You only have to read one paper that Liz has been discussing in class, the Rodnina paper. The paper was published in 1999, OK? And it's still, I would say, a seminal paper. And what they propose or what you read about their model is still the working hypothesis in the field. But if you go and Google the ribosome in elongation, you will find out that in the last 10 years there are hundreds of papers now taking pot shots at this model using modern technological mechanisms, like single-molecule spectroscopy, Cryo-EM. So they're flushing things out, but so still the basic model holds. So we continue to go through this because, in my opinion, all the machines that you're going to be talking about-- and this is part of the course-- have complex behavior like this with numerous substrates and many, many steps. And so hopefully one thing you got out of this paper is that kinetics are important. OK, so today what I want to do, I'm going to ask you questions. I'm going to put some things on the board. We get you at talking points. I'm going to ask you some questions. And then the discussion will continue into the next recitation on the same exact same topic. But kinetics are important. But to do kinetics, what do you have to have? What's required to do kinetics if you look at this model? So this is the model out of your paper. If you want to do kinetics, what do you need? Not only that, you have to speak loud because I'm deaf anyhow. What do you need? AUDIENCE: To do kinetic studies? JOANNE STUBBE: Yeah, to do kinetic studies. AUDIENCE: [INAUDIBLE] a detection method in very controlled conditions. JOANNE STUBBE: Yeah, so you have to have an assay, OK? And you'll see that everything you're doing over the course of the semester requires development of an assay. And I would say the more complex you get the more complex these machines. And that's what people are studying now as opposed to if you looked at it Liz's lecture on tRNA synthetases, you saw a simple reaction, OK? That assay was developed decades ago. But when you get into these more complicated machines, you have to be really pretty creative to develop an assay. And you need to have substrates. You need to get them from somewhere. And then you need to do kinetics. And so today, what I want to do is go through the kinetics part of this, asking you questions as we go along. And I'm going to start. So kinetics, in my opinion, is a key tool. So we're using kinetics as a tool to study machines. And the machine we're studying is-- and have been studying is, is the ribosome. OK, so how many of you have had an introductory last lab course where you did kinetics? Only one? Two? OK, because steady state kinetics is where you start for everything, OK? And I find when I-- I've been teaching for many years-- that there are certain things about steady state kinetics that people don't seem to get. And furthermore, were steady state kinetics important in this paper you had to read? Can anybody tell me? Did you get anything about steady state kinetics? Did you think about it? This will tell me how closely you read the paper. No? No one? OK, so this paper is hard and this is a paper that, even though I read it probably 20 times, I still learn stuff every time I read it. So you can't read a paper once. There's huge amounts of information in this paper. And if you go back and look at it three weeks from now, you'll probably get a lot more because we're continually filling in pieces of information from you in this complex system. Yeah? AUDIENCE: The part, I think, related to the steady state kinetics they measure Kcat, and Km, and their ratio. JOANNE STUBBE: Right, so that that's where the steady state kinetics is. And so if it goes to the question of what can you learn from steady state kinetics, OK? So let me just put down a simple system, which you've all seen if you take an introductory biochemistry course. People use this system because you don't have very many rate constants. So when I write down rate constants, I don't put K's. I just put 1, 2, 3 because it becomes hard to read anything, OK? So this is a simple system for any catalyst, OK, where some substrate could be EFTU, and tRNA, and GTP binding to the ribosome, OK? You do some chemistry to form some product. OK, and then the product dissociates. So if you look at the rate of the reaction-- so this involves the assay. You have to develop an assay where you can monitor something as easily as possible. That's the key thing. So I think here is where your chemistry background plays an incredibly important role because you can be creative about your assays. And so and you look at this as a function of the concentration of your substrate. What does the spectrum look like? What does the graph look like? How would you describe the graph? This is something you've seen in 507. We're just going back. What does it look like? Right, exactly-- rectangular hyperbole. OK, and so I think what's important is that this kind of behavior has been observed over and over and over again since 1940s when this curve was first described by Michaelis and Menton with many variations on the theme. And so what you need to think about is you have two parts of the curve. What's happening up here? What is the dependence on the reaction on substrate? So we have an enzyme that's a catalyst. It doesn't matter whether you're an organic chemist, an inorganic chemist, a biochemist. All of these things can be described by this simple, simple cartoon. So what's happening up here? What's happening in this part of your graph? AUDIENCE: It's saturated. JOANNE STUBBE: Yeah, see, you're saturated. So you're zero water and substrate, OK? And then what's happening over here? Your first order N substrate, OK? So from those observations, people derived equations, a general equation. So the rate of product formation, whatever you're assay is that you're using, is equal to Vmax times a concentration of substrate over Km plus the concentration of substrate. OK, so you've all seen this before. And if you look at this one simple case, and you look at what is Vmax equal to-- can anybody tell me what are the rate constants within Vmax? So Kcat times the concentration of enzyme. OK, so Vmax, and what does that mean? Kcat we'll see in a minute, is the turnover number times the concentration of enzyme. That means all your catalysts have stuff on it. It can't go any faster. It doesn't matter if you add more, more, and more substrate. You have no catalysts. So that's what's limiting the reaction. So if you derive this equation using steady state assumptions, what are the four sets of equations you need to be able to derive this expression? Can anybody tell me? What are the conditions you need to do? So what's the goal of deriving this equation, first of all? And then what are the assumptions you make? OK, so you want to be able to describe what you see experimentally, OK? So the first thing you have to do is be able to measure it experimentally, OK? So you have to have something in terms of an experimentally measurable parameter. And if you look at e, es, ep, which one of these are going to be measurable? AUDIENCE: Going to be the substrate and the product. JOANNE STUBBE: OK, so substrate and product. Yeah, you can measure substrate. You can measure product. But I'm talking about e. OK, so we have the e, we have an es, in this case we have an ep, most of the times you have 20 more e, equilibria. So which one can you measure experimentally? AUDIENCE: [INAUDIBLE]. JOANNE STUBBE: Pardon me? AUDIENCE: [INAUDIBLE] enzyme. JOANNE STUBBE: Yeah, so you can measure the total enzyme. OK, so that's this the enzyme conservation equation. So you have-- I'm not going to draw this all out. This is a review that you've already seen, presumably. So that's the conservation equation. How do you measure the concentration of an enzyme? AUDIENCE: UV vis. JOANNE STUBBE: UV vis. What amino acids absorb in the visible? AUDIENCE: Tryptophan. JOANNE STUBBE: In the visible. AUDIENCE: Oh, in the visible? None. JOANNE STUBBE: None. So don't say UV vis. Say UV, OK? So this is key to being able to sort things out. So what are the amino acid side chains that absorb in the UV? This comes back to-- you need to-- AUDIENCE: Tryptophan, tyrosine. JOANNE STUBBE: Right. So tryptophan and tyrocine are the major ones. Then phenylalanine is much smaller. So you can measure this. But, in general, you can't measure all the other forms. OK, so you know this, and that's required to be able to get this expression that describes this rectangular hyperbole. What about the substrate concentration? Under normal assays, if you've done an assay in the lab, how is the reaction set up? How much enzyme do you have in there? How much substrate do you have in there? AUDIENCE: A lot of substrate. JOANNE STUBBE: A lot of substrate. And what conditions are you under, perhaps, if you have a lot of substrate, over here in this graph? AUDIENCE: You have to saturate your-- JOANNE STUBBE: Yeah, well, you don't have to, but you would be saturated if you had a lot of substrate. How much enzyme do you have in there? A lot or a little? AUDIENCE: A little. JOANNE STUBBE: A little, OK? So the enzyme, the concentration of the substrate is much, much greater than the concentration of the enzyme, OK? So that's a typical steady state assay when you go to determine the rate of your reaction. So because the concentration of the enzyme is much, much greater than the concentration-- the concentration of the substrate is much, much greater than the concentration of the enzyme, you don't have to worry about substrate being bound in these forms. So the second equation you routinely use is called the substrate conservation equation, because it doesn't change, because the amount of this es on the enzyme, which is tiny, you don't need to measure it. So this is the second. So these are both conservation equations. OK, so we just said we were doing steady state kinetics, OK? So now you need to be able to make the steady state assumption, which hopefully all of you know. So the rate of change of some intermediate with respect to time is equal to 0, that is we're under a set of conditions where the rate of formation is equal to the rate of disappearance of whatever this species is. And what is the fourth equation we need to be able to set up this? What is the fourth thing, which is probably the most straightforward? And, again, it needs to be in terms of experimentally measurable parameters. What are we measuring in our reaction? AUDIENCE: You said the position from vs to [INAUDIBLE] is irreversible. JOANNE STUBBE: No, you I could have done this. And what would that have done to my equation? It just would have put in more rate constants. I'm going to show you what the rate constants are in a minute. There's nothing-- in fact, almost no enzyme reactions are irreversible. If you look, you can find reversibility in almost all reactions. This is-- so why do people write equations like this? They like irreversible reactions because it makes the kinetic derivation simpler. You don't have as many rate constants, OK? So, but what do you need now? We're monitoring the reaction? What are you going to monitor? So we know how much enzyme we have. We can measure that. We know how much substrate we can have. We can measure that. We know what the steady state assumption is, and we have an equation. So we can describe that. What's the other thing you need? It's the standard thing. How do you describe the rate of product formation? That's this guy over here. So what do you need? You need some kind of an equation that expresses just appearance of substrate, formation of products. So you need a way of measuring this. And you can do this many ways, even from a simple equation we've shown over here because a description of the rate of product formation is simply the net flux through any step in the pathway. And so what you see people writing is they immediately go to an irreversible step because it makes the algebra simpler. So k3 times the concentration of ep would be the net flux through this step. But I could write the net flux through this step and I would get the same answer. So it's the net flux through any step in the pathway. OK, so why am I going through all of this? OK, and the reason I'm going through this is because of this Kcat over km, which I just described to you. So one of the questions I asked you to think about when you're thinking about steady state kinetics is what are the two important parameters you get out of Michaelis Menton analysis? And the reason I ask this is because, in my opinion, it's not correct in most textbooks. So what are the two important parameters you get that you learned about that you probably even evaluated if you did something in the lab? AUDIENCE: Kcat. JOANNE STUBBE: Kcat is one of them. OK, and what's the other one? AUDIENCE: Km. JOANNE STUBBE: OK, so this is what everybody says, is km. And that's not correct, OK? So let me put down what the-- what did I do with it-- the values for the kinetic constants are here. So in this particular simple equation, it's Kcat is 2 times 3 over 2 plus 3. OK, so this is Kcat. And Km out of this analysis is 3. The numbers really aren't important. What I want you to see is that there are a huge number of first order rate constants in each of these parameters Km and Kcat, OK? Can you measure these? Can you measure these rate constants? That's what you want to know if you want to understand how this works, you would like to understand the reaction coordinate and what the rate constants are for every step in the pathway. That's what the whole Rodnina paper is about with the long range goal of understanding fidelity. Can we come up with a model for fidelity in the translational process that's contributed by EFTU? So can we measure these guys from an assay the concentration of the enzyme, The concentration of the substrate, the steady state assumption? What do you think? Don't be afraid. This is a discussion. What do you think? Can we measure? AUDIENCE: Does it depend how fast it is? JOANNE STUBBE: No. AUDIENCE: No? JOANNE STUBBE: No. It is dependent on how fast it is, but it doesn't matter how fast it is to answer this question, OK? Anybody else got another guess? What? Your name? AUDIENCE: Rebecca. JOANNE STUBBE: Rebecca. What's your name? AUDIENCE: Nicole. JOANNE STUBBE: Nicole. OK, yeah? AUDIENCE: Yeah. [INAUDIBLE] measure them [INAUDIBLE] we measure the initial rate, [INAUDIBLE] take that [INAUDIBLE]. JOANNE STUBBE: So you get Kcat and you get Km. That's not the question I asked. I asked, can you measure all the first order rate constants that make up Kcat and Km? No. So the problem with steady state kinetics is you can't really learn very much, OK? So what can you learn from steady state kinetics, and why do we keep looking at it? OK, why is it the first thing you've seen this with the tRNA synthetases? You saw Kcat over Km values charging with valine or isoleucine, right? In this paper, if you go back and look carefully at the discussion at the end of the paper-- so hopefully after this class you'll go back and you'll read that-- a lot of the discussion is about mechanisms of fidelity where they are thinking about these initial steps. And so these initial steps are really the selection steps of these things binding, OK? And if you go back and you look at the equation that they derive, it's amazingly complicated. Why? Because we have many more equilibria in our equation, but what you can get out of all this is Kcat and Kcat over Km. So Km really is not very informative at all because it's composed of a whole bunch of first order rate constants. It's always never equal to the dissociation constant, OK? So you can't-- so what it is mathematically, it's the concentration required to reach half maximum of velocity. So it doesn't really tell you anything. It's just half maximum of velocity. OK, so the two parameters that you need to think about-- and this goes back to the way you do experiments in the steady state versus the pre-steady state, which is what we're focusing on in this paper, is that you have two extremes when you do kinetics. And kinetics is something-- how do you learn how to do kinetics? You do them yourself. And you think about-- you think about what you think is going on. And then you make guesses about what's going on. And these are one of the types of experiments, when you're doing them you change your experimental design in the middle of your experiment. So it takes a lot of practice to get good at kinetics. But what you do with all kinetics, look at the extremes, the limits. So one extreme is the concentration of s goes to infinity. OK, so if you look at that extreme, what do you have? If s goes to infinity, what happens to b? What happens to this equation? AUDIENCE: [INAUDIBLE]. JOANNE STUBBE: Yeah, so it goes to Vmax, or Kcat times the concentration of e. So you're up here, OK? And so what is Kcat? So you can get out of this Kcat. Why is Kcat an important parameter? Why do people care about Kcat? Hey, what's your name? AUDIENCE: Alex. JOANNE STUBBE: Alex. My nephew's name is Alex. I'll remember that, OK? You're stuck. What's Kcat? AUDIENCE: It's like how quickly the enzyme turns over-- JOANNE STUBBE: Per active site. So it's called the turnover number. OK, so what does it tell you. It tells you how good your catalyst is, OK? So that's pretty important. So this is the turnover number. And I would also say it's pretty-- in the age of recombinant production or proteins, where we never isolate proteins from the normal source-- we isolate them all from bacteria or from yeast-- Kcat becomes really important to know, OK? So how do you know what the real Kcat should be if you isolate your enzyme from a protein that's expressed in E. coli? Do you think you get the real Kcat? Have you ever thought about that? Most people haven't, OK? So you're not alone. What could happen if you expressed your protein in a bacteria, or in another model organism like yeast? AUDIENCE: [INAUDIBLE]. JOANNE STUBBE: Yeah, it might not have the appropriate-- it's probably not-- it could be post-translational modification. It could be co-factors. So there are examples in the literature of very, very smart scientists who have spent 25 years of their life studying an enzyme that was only 1% active. So this is, in this course-- and I think in general in biochemistry-- you've got to go back and forth between the cell and what you see in the test tube. So this Kcat, if the number is 0.00001 per second, you have to have some intuition that tells you, oh, my god. That's so slow. Something-- something is wrong. So this number of turnover is incredibly important. It gives you a feeling for how good your catalyst is. But the number we're really after is the second example and the other limit. And what happens is s goes to 0, what happens to this equation. So those are the two extremes, OK? So as s goes to 0, OK, that's the other part of this equation. What happens to the equation? The rate of product formation is equal to-- and I'll write Vmax as KT times the concentration of total enzyme, OK? I didn't write it down. Hopefully you all know that. So what you now get is Kcat over Km times the concentration of e times the concentration of s. So what is this guy, if you look at this equation? What's Kcat over Km? What are the units? Kinetics isn't that hard. These are pretty-- if you think this is hard, wait till you start getting-- we're not going to go into derivation of steady state, pre-steady state analysis. But this is pretty simple compared to pre-steady state analysis. So what's Kcat over Km? What are the units? AUDIENCE: [INAUDIBLE]. JOANNE STUBBE: Yeah. Yeah. So it's second order rate constant. So that's the key thing. So what are you looking at? You can look at that by this equation. What you're looking at is the enzyme combining with the substrate, OK? And that's what we care about. That's the specificity, specificity, or efficiency of your reaction. So if you have a tRNA loosing and an tRNA phenylalanine, they're both competing for binding to the substrate. So the important parameter to think about that selection-- and that's why that's important at the end of this paper-- relates to Kcat over Km. It's the specificity or efficiency number. And if any of you ever work in a pharmaceutical industry, you'll find out that, of course, you never-- and you're looking for inhibitors, you never look at Kcat. Why don't you look at Kcat? You always look at Kcat over Km. Why is that true? Can anybody tell me? If you were looking for a drug, if you were looking for an antibiotic, fusidic acid that we talked about today that inhibits the EFG that Liz talked about, how would you set up the experiment to look for inhibition? What would you do with your concentration of substrate? Do you want it high or do you want it low? AUDIENCE: You want it low. JOANNE STUBBE: You want it low. Why do you want it low? AUDIENCE: Because [INAUDIBLE]. JOANNE STUBBE: Yeah, so if you're inhibitor is binding to the same site, and you have a huge amount of this, no matter what you do, even if this was a great inhibitor, if you had 10,000 times the amount of this, you're never going to see any inhibition. So understanding these simple principles-- which I can tell you there are people that don't get this in drug companies-- are pretty important, OK? So Kcat and Kcat over Km, boring. But it's not really so boring. It's sort of central to everything that you'll be thinking about over the course of the semester and almost all the modules in some form, although we won't highlight it like we're highlighting it here. OK, so, again, the reason we care about Kcat over Km is this question of selectivity. And I urge you to go back and look in the methods section of your paper. Now, this paper is packed full of stuff, OK? so as I said, I read it 20 times. Every time I read it, I find out something new. And furthermore, I think the paper-- how many of you found it a tough slog to go through this paper? This is probably the hardest paper you're going to look at in my opinion? Did you think it was well written? Did you get the ideas? OK, Did you all get the ideas or not, or where you completely confused, or you didn't spend enough time on it? How much time did you have to spend on it? AUDIENCE: Probably about an hour. JOANNE STUBBE: OK, an hour. OK, so I would say-- I read the paper 45 times, and it takes me two hours to read a paper like this. OK, so again, it's a question of what level you want to look at things. And I think part of what this course is about is looking at experimental details. You're want to see that. And the problems set, you're going to see that in lecture. You're going to see that probably next time when we continue to look at the primary data that they collected, how important it is to look at the axes, and not just looking at it rapidly. You really have to think about what the data is telling you. So this paper is complicated from my point of view because it's based on-- it's based on 15 other papers. OK, so for you to really believe what they say, which is what you need to do as a scientist, how to critically evaluate somebody else's data, you need to really go back-- and we didn't ask you to do that-- and really critically evaluate the earlier experiments they've done, because some of the conclusions they've done, when we look at the primary data, I could have drawn-- without knowing all that primary data, I could have drawn a conclusion completely different. So you see something and you've got to explain it, OK? And so when you start out, you have no idea. You have a very simple model. And in general, the model's almost always get more and more complex. That's what you're going to see over and over again. You start out as simple as possible, and then things get more complex. OK, so what we want to do now is ask the question. And I've just told you, you can't evaluate these individual rate constants. We just don't have enough variables, OK? We don't have enough that we can measure, that we can change the substrate concentration we can change, which changes the rate of product formation. So those are the two variables. But we have many more unknowns. We have k1, k2, k minus 2, k2, et cetera. So we can't evaluate these things. So the question is, is there any way you can start getting the primary rate constant, the numbers to the primary rate constants, OK? And so one way that people do this nowadays-- and when this paper was done, this was not an easy task. OK, now because of molecular biology where you can get large amounts of protein, it has become much more of an easy task-- you can get a large amount of protein-- you want to turn to the pre-steady state. So what I want to do very briefly is discuss the pre-steady state. I asked you to think about-- I asked you draw this out. This is one of the talking points in the questions I handed out. But in the steady state, we're over here. And the pre-steady state is before we get to the steady state. And does anybody have any idea what timescale you are on in that region of the curve? Is it hours? AUDIENCE: Milliseconds? JOANNE STUBBE: Yes, so it's milliseconds. So, fortunately, this didn't necessarily have to be true-- most enzymatic reactions occur. the catalysis occurs in that time regime, or maybe 0.1 milliseconds to milliseconds, allowing you to be able to use this method in an effort to try to understand what these-- evaluate what the rate constants are. And when you look at the table in the Rodnina paper, we're going to talk about where all those three constants came from, OK? Are they good or are they not good? But that's what you'd like to know for every system to really understand the question of fidelity, whether it's translation fidelity, DNA fidelity in replication, transcriptional fidelity. And you'll even see in Liz's section on polyketide syntases, which make natural products, you also have fidelity issues almost everywhere. So you'd like to be able to evaluate these things. And you can get a handle on this if you're a chemist and really care about the molecular details using potentially kinetics. So this is why kinetics is one of the first places that you actually start to think about what's going on in any reaction. OK, so let's say a few things about pre-steady state. I'm going to ask you a few questions, if I can remember what I'm going to ask you. OK, so OK, so let's suppose in this simple case, which I just covered up, this step is rate limiting, OK? So what is that step? Do you think it's common that a step like this-- so we have e plus s. And eventually, the enzyme gets recycled. I'm saying this is the rate-limiting step. Where is the chemical steps? Where are the chemical steps in this reaction? AUDIENCE: 2, 2. JOANNE STUBBE: Yeah, so k2. What are these steps over here, k1, and k minus 1, and k3? AUDIENCE: It's like association of the-- JOANNE STUBBE: Yeah, so the physical steps, OK? So as a chemist, and you're trying to understand what's going on, isn't it a problem if the rate limiting step is physical? It masks all the chemistry, OK? So what you see in this paper is they have to figure out clever ways to get around these kinds of issues. And you see that over and over again when you're studying enzyme systems because enzymes have have had billions of years to evolve. They are evolved. Their catalytic transformations are amazing. They go 10 to the 15th per second, right? That's totally mind boggling. Chemists can't come close. And so what happens then issued a limited by physical steps. So what we want to do is try and then look at the first part of this transformation. And basically, what we're then doing is using the enzyme sort of as a reagent. There are numbers of ways you can do this so that you can have a way of not looking at multiple turnovers because you can only look at one turnover if this is blocked in terms of product release. OK, so I think this product release is quite often the rate-limiting step in biological transformations. And what have you seen from reading the Rodnina paper? Have you seen conformational changes in thinking about the kinetic model we had up there before, and Liz had on the slide? Have you seen conformational changes? Is that part of the mechanism? Are they fast or are they slow? AUDIENCE: Wasn't that part of their reasoning that the difference between if you have a cognitive versus if you have a mismatched? That influences the rate of the reaction based on how it can affect the conformational change? JOANNE STUBBE: Wait, so that's exactly what's going to happen. So there are multiple places. How are you going to discriminate between two amino acids? Cognate and near cognate, whatever they are, will get to the data. The question is, how do you do that? And one of the steps is-- we talked about today GTP hydrolysis. But GTP hydrolysis is limited by a conformational change. And then once that go, the hydrolysis is very fast. And so it looks like the rate constant for GTP hydrolysis is the same as the rate constant for the conformational change. Where else have we seen a confirmation change the accommodation? AUDIENCE: Peptide. JOANNE STUBBE: Yeah, so peptide confirmation. This is shown here is this little cartoon where this red ball is the amino acid. It needs to reorient itself so it can form a peptide bond. So confirmation changes are all over the place in entomology. And if you look at the ribosome, do you think it's easy to tell with those conformational changes are from a molecular point of view? What do you think? Do you think it's easy or hard? AUDIENCE: Hard. JOANNE STUBBE: Very hard. OK, so here-- one of the most amazing things about the ribosome-- you've got to think this is amazing. You have this called the anti-codon loop way down here on the [INAUDIBLE]. And the GTPase is 80 angstroms away. And somehow, twiddling-- you saw this in class today-- these guys to form the right confirmation is transferred 80 Angstroms. And that triggers the reaction, rapid and irreversible. And the reaction goes to the right. You see this over and over and over again in these machines. OK, so this is a really important observation. How does that happen? Well, I think what's mindboggling about the ribosome-- again if you Google ribosome and elongation, you'll see we have another 150 papers published where people are trying to sort out-- because we have cryoem structures, we have stagnant crystallographic structures, we have single molecule stuff now. On top of all this model we have from Rodnina, people are trying to sort all those out because they care about how it works in some detail. So who ever would have thought we could get to the stage where we-- you've seen the pictures you saw in class today. Those pictures-- when I was your age, do you know how many structures there were? Maybe three. OK, we had hemoglobin. We had chymotrypsin. There were no structures. And why was that true? Because we had no molecular biology. So I used to spend three-- I'm digressing. This happens to me all the time. You're going to hate me for this. I'm going to get hammered for this. But I used to spend three months in the cold room isolating a microgram or protein, OK? And then molecular biology came in. And it's still not easy. And Liz will tell you what the issues are with purification of protein. But you can get grams of protein now in a day, OK? So there's been a revolution. And that allowed these crystallographic-- a pure material that crystallized more readily. And then the technology on top of that has really revolutionized what you can do. I think it's a very exciting time. And I think any of you who want to be biochemists, or are thinking about drug design, you really need to learn how to look at structures. So that was the first module. It takes a lot of practice. You need to figure that all out. OK, so pre-steady state-- so we're going to look at pre-steady state. And the goal is to evaluate the individual rate constants. OK, so that's the goal. And you may or may not be able to achieve this goal. But this happens, we just said, on the millisecond timescale. And so one of the questions-- and we're doing this under single turnover. OK, so let's look at a simple-- and we've just talked about it in the steady state. The concentration of the substrate has to be much, much greater than the concentration of the enzyme. And the enzyme concentrations are really low. So let's say we have an enzyme concentration of 0.01 micromolar, OK? So that's our enzyme concentration. And that would be typical if you would be using in a steady state assay if you have done those. And let's say that we're going to monitor this reaction by some kind of absorption change, a unique absorption change. So we're looking at absorption at some wavelength, OK? And let's say the extinction coefficient for this is 10 to the 4 per molar per centimeter. It would be ATP or coA. Then you can ask yourself the question, under these conditions, the change in absorption at whatever this wavelength is, is equal to the path length of light in centimeters times 10 to the minus 8th molar times 10 to the fourth molar per centimeter. OK, so what would your change in absorption be if you were measuring this in a single turnover? It would be really, really small, 0.0001. Can you measure that? Maybe you could measure this if you took hundreds of samples and you did a statistical analysis on it. But it's really low. So what do you want to do then to do pretty steady state? What's the thing to change so that you will be able to see something? AUDIENCE: [INAUDIBLE]. JOANNE STUBBE: Yeah, so you increase. So when you have this, and you can't see something-- and, obviously, it depends on what this extinction coefficient is-- but this is a pretty high extinction coefficient. So what you do is you increase the concentration of enzyme. And if we increase it, say, 1,000 fold, then then so we're now at 10 to the minus 5. Then now what is the change in absorption? The change in absorption is 0.1, which you can measure fairly easily in any kind of current instrumentation. So the thing is you have to be able to see. So the key thing with pre-steady state, and the reason you need to have large amounts of enzyme, is you need to be able to see what you're monitoring. So it's all about sensitivity. You need to see. And this usually implies increasing the concentration of the enzyme. OK, so what's the problem if you increase the concentration of the enzyme? Say we normally are at 0.01 micromolar steady state. We now are at 1,000 times higher. What's going to happen that makes the analysis complicated? If you increase the concentration of the enzyme, what does that do? AUDIENCE: You're going to burn through-- [INTERPOSING VOICES] JOANNE STUBBE: You're going to-- yeah, it increases the rate of the reaction because the rate of the reaction is proportional to the concentration of your catalyst. If you don't remember anything else out of this course, or anything in chemistry, that's pretty important no matter what area of chemistry you're in. So now what happens is the reaction is going like a bat out of hell instead of pipetting by hand. By the time you pipetted and put this into however you're observing it in the spectrophotometer, reaction's over, OK? So that's what the issue is, OK? So the sensitivity is key. And then the second thing you need to think about-- so sensitivity is one thing. And the other thing you need to think about is, what are the limitations of this method? How fast can the rate come-- on the millisecond timescale, what are the limitations in terms of the rate constants you can actually measure? So when you're looking at these reactions, you're looking at, in general, first order reactions. So all of these take place on the enzyme. So everything is stuck on the enzyme. So it's all first order. So the half life of the reaction is, if you go back and you look at your introductory kinetics, is 0.693 divided by k observed. And so if you had, say, a rate constant of 500 per second, then that gives you a half life of 1.5 milliseconds, OK? So that means you have to be able to make your measurement faster than that, OK? So the instrumentation we're going to be using can't do that. So the instrumentation we're doing-- so this would give you a half life if you calculate this. I don't even remember what the number is. But the dead time of the instruments that you would be using to make pre-steady state kinetics is approximately 2 milliseconds. So by the time you could stop looking at the reaction in some form, you know more than 50% of the reaction is gone, OK? So the rate constant then limits also what you can measure. So we asked this question before-- how could you modify this rate constant? What could you actually do? How could you make it so you might be able to say your rate consent was 500 per second-- you missed more than half your reaction. What could you potentially do so that you could monitor more of the reaction? What's the parameter that you would change? Kinetics. Think about kinetics. What do you always control in kinetics? AUDIENCE: Substrate concentration. JOANNE STUBBE: OK, you can control substrate concentration, but that's not the one I'm looking for. AUDIENCE: Temperature? JOANNE STUBBE: Temperature. Yeah. So in our body, we're at 37 degrees. That really is where you want to be making all of your measurements. In reality, many of the measurements are right on the edge. And so if you read the papers carefully, you'll see that people do lower the temperature, and that the rate of the reaction is related to the temperature. What's the problem with lowering the temperature? These are all things just you got to think about in the back of your mind. They're all playoffs in terms of how bad you want your experimental data and what the issues are with interpretation of data. Yeah? Rebecca. AUDIENCE: It makes it difficult to compare the different values. And you might not know exactly what the relationship between temperature and rate is, like if scales linearly. JOANNE STUBBE: OK, so that's true. You could have a huge conformational change that doesn't have erroneous behavior. I think quite frequently, most enzymes, they're designed to work at 37 degrees. And when you start cooling them down, they do weird things. So you could make the measurements, but then you have this issue-- I think, which is what you were saying, of how do you extrapolate that? So a lot of times you will change the temperature because that's the only way you could make the measurement. But the caveat is, like with everything, that you need to think about what the consequences of that actually are. OK, so the methods that we're going to be using Liz already introduced you to in class, not today, but the previous time. And so what you want to do, since you can't pipette on the millisecond time scale by hand, you want to have an instrument that allows you to control the rate of reaction by-- you put two different things in your syringes. And then you have an instrument-- push the two syringes into a chamber where they're mixed. Do you know what the rate-limiting step in this process is? It's the mixing process, OK? So the mixing processes is 2-millisecond dead time. I don't know if any of you have ever mixed something viscus with something not so viscus? What do you see? Have you ever made up a solution of glycerol? No? They probably give you all this in a kit nowadays. You don't have to make up your solutions of glycerol aluminum anymore. So this goes back to experimental design. And I'm not here to teach you how to do experimental design. But if you had very high concentration of the enzyme-- because we need a lot to be able to see something-- and you're mixing it against substrate-- you have something very viscous and something not viscous-- and when you mix them it takes a lot longer to remove all the lines from the mixing process. So experimental design, you really need to do some thinking about that. Once it gets into the mixer, the liquid in the mixer pushes a third syringe back. It fills the syringe up with liquid. It hits some kind of a stop position, which then triggers detection, OK? So that's what you're looking at. And the beauty of this method is it's continuous. And so what do you have to do to be able to look at this? What did they do in the case of the Rodnina paper? What kind of method did they use? Did you think about that? They described it, but you might not have thought about it in terms of experimental design. How did they monitor their reaction, one of their reactions? AUDIENCE: [INAUDIBLE]. JOANNE STUBBE: Yeah, so they are going to be able to somehow tag-- and this is a key thing, is how do you tag something in the right place so you can see a unique fluorescent change? That's not so easy, OK? So you mix this. You can monitor this continuously by change in fluorescence. If you had something that has a visible absorption, could you use that? What would limit that? Say, if you looked at tRNA synthetases that you talked about in class two times ago, where you were looking at ATP that isolates the amino acid to form the adenylate, which then reacts with the tRNA, could you use-- ATP as an absorption at 260. Could you use that absorption change? Do you remember what that equation is? Do you remember installation of amino acid? You're going to see this again in polyketide syntases. It's used quite frequently in biology. Nobody has any idea? Nebraska, how about you? OK, so here we have amino acid plus ATP. I'm digressing. But if you learn this part, you've learned something out of all of this, that forms the acyl adenylate. OK, how did they monitor this reaction? You discussed this in class. How do they monitor this reaction? AUDIENCE: [INAUDIBLE]. JOANNE STUBBE: You need to talk louder. You need to be assertive, OK? AUDIENCE: [INAUDIBLE]. JOANNE STUBBE: Yeah. So we're going to talk about radioactivity next time. This will be one of the methods we're going to be introduced to. Why couldn't they use ATP? AUDIENCE: The absorbent's different between [INAUDIBLE].. JOANNE STUBBE: Yeah, they're the same. Yeah. So you have to have a difference in absorption to be able to measure the visible. So the total absorption is due to the adenosine moiety which is the same in both molecules. OK, so you can't do that, OK? So that's one. And then let me just do one more thing, and then we'll quit. I still have another minute according to my watch. OK, so the second method, which they also used in the Rodnina paper is rapid chemical quench, OK? So, again, you have two things. You mix them. There's some plunger that allows the mixing. And then this is a discontinuous method. So what happens is you mix. And then you have to stop the reaction, OK? So you have a third syringe where you mix in something to stop the reaction. And then you have to analyze what comes out the other side. And this is where they're going to use radioactivity. And so this is rapid chemical quench. And how can you monitor a rapid chemical quench experiment? What's the best way to stop the reaction? What did they do in this paper? How else could you stop the reaction? Anybody got any ideas? So what what's the first criteria if you're going to stop the reaction? What does it have to be able to do? You just can't throw in anything, right? What is the key criteria for successful stopping? AUDIENCE: Something that will turn off the catalytic activity? JOANNE STUBBE: Yeah. And it has to be able to do it on a millisecond timescale. So you need millisecond stopping. OK, how could you millisec? How could you stop something on the millisecond? What would you use? Anybody got any ideas? So this is not trivial, actually. AUDIENCE: You could change the pH? JOANNE STUBBE: Yeah, so changing the pH. But so you could go acid or base. Acid works. In general, base doesn't work. So if you read the handout that I've given you, it does work. But it's much, much, much slower. And every base is different. Acid works all the time. There's another thing that you can use that is quite frequently used, especially with polymerases that work on nucleic acids. And that's EDTA. So this is a chelator and EDTA chelates the metal, which is essential for viability. So that also-- the chelation can occur in the millisecond timescale. So what we're going to do next time, I've sort of introduced you to the pre-steady state. The next time we'll come in. And we're going to look at the actual experiments. We'll look at fluorescence. We'll look at radioactivity and how you measure radioactivity. We're going to look at antibiotics, like you talked about today. We're going to look at non-hydrolyzable GTP analogs. If you look carefully at this paper, it's amazing how many methods they used to come up with this model. And that's one of the take home messages that you have to use many, many methods. And then you never get an exact solution to your equations. It's numerical integration of all the data that leads you to the model that they've actually used, OK?
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MIT_508J_Biological_Chemistry_II_Spring_2016
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R4_Purification_of_Native_and_Mutant_Ribosomes_Protein_Purification.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ELIZABETH NOLAN: Today's recitation will all stem from this reading by Youngman and Green on ribosome purification. But also, we posted an optional-- well, not really optional but an additional reading that's a review that talks about purifying macromolecular machines from the source. And I guess one thing I'd just like to say about this review, something I like is figure three because it indicates many, many different types of methods that can be used to purify some biomolecule. And I think often it's easy just to think all the time, well, let's just use a tag. And tags have enabled many things, but there's also many possibilities out there and decades worth of work before tags for how to purify proteins. And in some instances, you might not want to use a tag and that's discussed to some degree here. So I guess I'm curious. What did you all think about this week's paper? Did you like it? Did you not like it? Was it easy to hard read? Did you read the paper? AUDIENCE: It wasn't too bad. I've purified-- JOANNE STUBBE: You need to speak louder because I can't hear you. AUDIENCE: It wasn't too bad. I've purified proteins before, so I felt like I could follow along. ELIZABETH NOLAN: So not too bad means it was easy to understand and follow the text there. Yeah, so compared to the reading for recitations two and three, this one was probably easier to work with, but there's a lot of details in this one too there. One thing, I guess, I like this paper from the standpoint of it being a methods paper, is the amount of detail they give with how they did this purification and things that didn't work, right? So often, sharing those details with your readers can make such difference for an experimentalist in another lab in another part of the world, right? And I think, from this, one with biochemistry training could go into the lab and reproduce their purification there. So it's a good example in terms of what details to include and what types of pitfalls to include. So how many of you have done a protein purification, either in a lab class or in your research? OK. And so what type of purification did you use? AUDIENCE: We used a histamine-tag with nickel. ELIZABETH NOLAN: OK. So you used a His6-tag, probably a polyhistamine tag in a nickel NTA column. For everyone else, has it been that type of methodology or another methodology? AUDIENCE: I used the same one. ELIZABETH NOLAN: OK. OK. So would someone like to kind of comment on the basics of affinity tag purification? So how does this work? And why did you do it? So what are the advantages? AUDIENCE: So you have a light chain of histamines on your protein, and you use nickel, which is a metal that chelates to histamine very well. Is that right? ELIZABETH NOLAN: Yeah. The nickel's bound to something though. You have the NTA ligand on the resin. AUDIENCE: I did this over a year ago. Sorry. AUDIENCE: So if you have whatever the type is, the [INAUDIBLE] bound to some solid substrate. And then you can elute everything that's not bond to that. Elute what it does bind. So you just tag the protein. It's bound to a high concentration of the free ligand. ELIZABETH NOLAN: Or right, to push it off. So the idea is that you have your biomolecule of interest, whether that be a protein, which is what we're all most familiar with, or the ribosome, and then you attach a tag. And that tag can be any number of things, right? So in this paper, we saw they used a stem loop structure incorporated into the 23s rRNA. And the idea is that you're going to use that tag to separate your biomolecule of interest from the complexity of the cellular environment there, so all of the other proteins. And so you have some bead or resin that this tag can bind to. So in the case of the nickel column, the His6-tag will bind to the nickel NTA on the resin. And then you can wash away other components. And then you devise some method to elute the protein you hope to have trapped there. So what are some advantages of using an affinity tag, just thinking about this generally before we delve into the paper? AUDIENCE: Easy to install. ELIZABETH NOLAN: OK. So what do you mean by easy to install? AUDIENCE: If you wanted to just encode a 6 His-tag at the terminus of your target protein, I don't know if you can say it's trivial. It's easy. ELIZABETH NOLAN: I'd agree it's easy. There's many plasmids available that you can insert your gene of interest into in order to have this tag genetically encoded. So when you express the protein, the tag's there. So beyond that, from the standpoint of purification-- AUDIENCE: It's more specific. ELIZABETH NOLAN: Pardon? AUDIENCE: It's more specific. ELIZABETH NOLAN: More specific-- AUDIENCE: Pure proteins as opposed to various charges. AUDIENCE: It simplifies the purification. Instead of doing size exclusion and ion exchange that is much different. ELIZABETH NOLAN: So the hope is it simplifies the purification because you have some way initially to pull your protein of interest out of your cell lysate there. So that can be a big help. What are some potential disadvantages of using a tag? So have any of you run into trouble with a tag in the lab? AUDIENCE: Having a tag can highly deform your protein and change it. ELIZABETH NOLAN: Yeah. So it might change your protein or deform it. What do you mean by "deform?" AUDIENCE: It could just cause a conformation change or the tag could make it localize somewhere else based on size. ELIZABETH NOLAN: Yeah. So that's an example, say if you were doing a study in cells, say, rather than a purification. But maybe you tag a protein and it goes somewhere other than it would go untagged right? And that will affect your observations and your data there. And it might alter the conformation. So it might affect the folding, right? It might affect the oligomerization. His-tags bind metal ions. So is that a factor to consider there? If you have an enzyme, will the tag affect activity, right? And these things can be a positive or a negative. Sometimes the tag is helpful in these regards. You can't get soluble protein without the tag, right? And sometimes the reverse. You decide to express your protein or biomolecule with a tag and you find out you get an aggregate, something that the protein shouldn't be. So these are just things to keep in mind when designing a fusion protein and thinking about how you're going to use an affinity tag to express your protein there and purify your protein. So there's pluses and minuses, right? And you can always make the choice not to use a tag. So you saw some of that in the review article, right? And I guess one other thing, too, there's this idea of using the affinity tag in the affinity column, which we'll talk about more in the context of the ribosome, but is that always enough? So is the affinity column alone always enough to purify your protein of interest? So in lab class, that's where it will end, because that's an exercise made for lab class, and it's your first adventure into protein purification for most people, right? Oftentimes, it's not enough. That you do enrich what you've purified with what you want, but often there's contaminants. So you actually do move forward with doing some other type of purification. Like Rebecca mentioned, ion exchange or size exclusion, those are things you can use after the affinity purification there as needed, right? So contaminations are something to look out for here. JOANNE STUBBE: What other types of steps do you use for purification besides columns and [INAUDIBLE]. Because people have forgotten all of this. Everybody uses the tag and that's the end of it, and it can be the kiss of death. What other kind of fractionation steps? Do you do any other fractionation steps? What is it, 535 or something? AUDIENCE: [INAUDIBLE] salt. JOANNE STUBBE: The what? AUDIENCE: The [INAUDIBLE] salt. So some proteins will precipitate. Some will not. JOANNE STUBBE: Yeah. So that's what you use to precipitate. So that's a mild method. It's fast. It gives you separation on a fair amount of proteins. Anybody know what you use? AUDIENCE: Ammonium sulfate. JOANNE STUBBE: Yeah, ammonium sulfate. And then what's the other thing that really is important? What is the other thing you want to remove from your protein when you're using this tag that oftentimes people miss in the literature? ELIZABETH NOLAN: Yeah, we were getting there. JOANNE STUBBE: There's another component inside the cell that you need to get rid of that you've been talking about. AUDIENCE: Where one component has His-tag on the cell, you remove it. JOANNE STUBBE: So you have the protein. What are the other components in the cell that you need to remove? You can go back and look at your cartoon of the inside of the cell. ELIZABETH NOLAN: Yeah. So we're getting a little ahead, but that's totally fine. So if you're going to lyse your cell, and then, imagine your protein's soluble, right? So you do a centrifugation to remove the insoluble components. So you have the membrane and all that debris. And then you take your soluble fraction, and you incubate that on your column, right? And then you elute, you wash, you elute your protein. What might come out with your protein? AUDIENCE: Nucleic acid. ELIZABETH NOLAN: Yeah, right. So how do you know if you have nucleic acid contaminating? AUDIENCE: The A260. ELIZABETH NOLAN: Right. So A260 will give you a readout of nucleic acids. A280 is what people typically look at for their protein concentration, but you should look at both so you know what's in your protein. You need to look at both. JOANNE STUBBE: Do that and adaptively repeat stuff out of the literature. Very frequently, you take an absorption spectrum, there's some nucleic acid. ELIZABETH NOLAN: You have contamination. So a lot of ribosome, DNA. JOANNE STUBBE: If you don't remember anything else, it's important. ELIZABETH NOLAN: Yeah. So sure, just something to think about. So Alex noted, it's easy to put on this His-tag. What is actually on this His-tagged protein? So is it just the six histidines are all the same? When you look in the literature, and someone says they put a His-tag on the N terminus of their protein, how do you think about that? So we have some His6-tag, and then that tag, say, is attached to the protein here. What's going on in between? AUDIENCE: Maybe a flexible linker of some kind. ELIZABETH NOLAN: Pardon? AUDIENCE: It would be like a-- I don't know-- like a flexible linker of some kind. ELIZABETH NOLAN: Yeah. So maybe some kind of flexible linker. And what might dictate this linker? AUDIENCE: If you wanted to remove the His-tag after you would want to be able to hydrolyze it or something. ELIZABETH NOLAN: Yeah, so maybe you want to remove your tag down the road, right? So a protease is often employed. So maybe there's a linker, maybe there's a protease cleavage site. OK. OK. And we're not going to talk about cloning really in this class, but just thinking back a step, you need some plasmid DNA to ultimately get here that has your gene, right? And so you'll insert the gene into the plasmid, and many commercial plasmids have something called multiple cloning site. And, for instance, if you want a His-tag or a GST tag, you'll use some different plasmid that has that encoded, right? And then you make a decision about how you put your gene in. And these multiple cloning sites have multiple sites for restriction enzymes where you can put the gene in, which means, even if the same plasmid is used for 10 different proteins, what's happening here can vary a lot even if you have one protein and put it in different sites. So maybe you have a short linker here because you used a site like NDE1 or SPE1, I'm just making those up, right? Or maybe you have a longer region here between the tag and the protein. And so it's very important to go look back at the map of the plasmid that was used and ask where was the gene put in and what does that mean? So is this His-tag a 2 kilodalton perturbation. Is it a 5 kilodalton perterbation to your protein? And some of these plasmids have multiple types of tags, right? So it might have a GST tag and a His-tag, and depending how you put your gene in, you may have two or you may have only one, right? So I'm just pointing out there's a complexity here. So when someone just writes in their paper, oh, I His-tagged the protein, you need to think beyond just sticking six histamine residues on the N or C terminus there. Did you have a question? AUDIENCE: So does that mean that's like another step where they purified the specific plumbing site, one that they wanted? Or does that mean you kind of roll with the heterogeneous-- ELIZABETH NOLAN: No. So you put your gene in particular sites with this type of cloning, right? And so one thing practically, just say you want to express some new protein and you don't know much about this. You might choose to make several different constructs where you put the gene in different sites or maybe you use primers that allow you to add some linker regions because you don't know what will give you better solubility and better yield. And then ultimately, you pick one. So, for instance, like for me, with working with, say, a His-tag for a protein, there's plenty of plasmids available where you can pick N terminus or C terminus. So one plasmid to put the tag on the N terminus. A different plasmid to put the tag on the C terminus. I'll clone the gene into both and test overexpression with both and just see if one's better than the other in terms of yield, in terms of solubility there. And then you make a call. Maybe you purify both and see if there is an effect on behavior, like oligomerization or activity if it's an enzyme here. So if you get into protein purification, it's good to talk to people who have purified many different proteins because the strategy is a little different for each protein. And then you just get more exposure to all of the possibilities and troubleshooting there. So coming to the paper, what was the big motivation for developing this method to have an affinity tag attached to the ribosome? Right? So Youngman and Green went to quite a bit of effort to devise this new system. What was their motivation? And what really was the big issue they were seeking to overcome? AUDIENCE: They wanted to get ribosomes with mutations on it. So they want to synthesize bacteria ribosomes with it and purify it. ELIZABETH NOLAN: Yeah. So they want a mutant ribosome. And they want to make this mutant ribosome in vivo and then purify. So what's the complication with making the mutant ribosome in vivo that they seek to overcome here? AUDIENCE: You have the wild-type ribosomes in there also. ELIZABETH NOLAN: So well, right. That was their decision, right? They want to express this mutant ribosome in the background of the wild-type ribosome. So why do they want to do that? AUDIENCE: Because it could be lethal. The mutant ribosome, it would be a toxic mutant. ELIZABETH NOLAN: Yeah, toxic mutant. What do you mean by "toxic mutant?" AUDIENCE: If you create a mutant ribosome, that was the only way for the cell to express them, they'd be toxic and they wouldn't be able to go through translation. ELIZABETH NOLAN: Yeah. So maybe the mutant ribosome doesn't work very well, and that ends up being lethal to the cell there. So can we imagine why that might be an issue for the types of experiments we've seen in class? So if you're thinking about trying to understand the catalytic mechanism in that function, there's a likelihood the mutations may dramatically affect that activity, right? If you want to put a point mutation into the peptidyl transferase center, into the decoding center, that could be deleterious to your cell, but it could be very important for your mechanistic study. So they want to avoid this lethal phenotype. So what is the complication in terms of doing this in the presence of wild-type for some sort of measurement? AUDIENCE: It gets kind of muddy. ELIZABETH NOLAN: Gets kind of muddy. Yeah. What does that mean? AUDIENCE: You don't have a pure mutant in there, and they're not significantly different from the wild-type. ELIZABETH NOLAN: Yeah. So if you were going to do a standard ribosome purification-- because ribosomes have been purified for many years without an affinity tag-- you're going to have a mixture of your mutant and the wild-type. And so that has a strong likelihood of being a problem for your analysis, right? So they gave an example in this paper where they actually made a mixture and did some analyses, where you could separate the wild-type from mutant activity but that's not necessarily the case. And so, as pointed out, they're both very large. They're very similar, right? There's no good way to separate a ribosome with a single-point mutation, say, in the peptidyl transferase center from wild-type. So let's just imagine you have a mixture that's predominantly your mutant ribosome but you have some background contamination of wild-type. Is that an issue? Say you want to measure rates of peptide bond formation. AUDIENCE: It's going to be an issue. ELIZABETH NOLAN: Yeah. So why is it probably going to be an issue? AUDIENCE: Because your mutant might be a lethal function because of your background and resume the function and [INAUDIBLE] mutant [INAUDIBLE].. ELIZABETH NOLAN: Yeah. So imagine you have a mutant ribosome that has very low activity or none, and you have some small amount of contaminating wild-type that has wild-type activity, right? How do you know-- I mean you might misinterpret your data, and what you're seeing is the wild-type and not the mutant. And this issue is much more broad than the ribosome. So one of my favorites is the contaminating ATPase. And maybe you have an enzyme that hydrolyzes ATP, but maybe you have a small contamination of an enzyme that does a much better job at hydrolyzing ATP that's in your reaction, right? So what are you seeing there? So that's something to keep in mind in terms of potential contaminations, right? And so they go through some justification about why they need to do this method based on available methods, and all of those available methods have strengths and weaknesses. So they talked about using systems without the wild-type ribosome. They mentioned in vitro translation, and I'll just say, in passing, those in vitro systems have improved a lot since the time of this paper. OK. So in terms of their strategy. Let's comment on the various aspects of this strategy. All right. So effectively, they want a way to express the mutant ribosome in vivo in the background of wild-type. They want a way to separate that ribosome. And they want to come up with ribosomes that are active. OK. So this cartoon basically summarizes their solution to this problem, and we should work through the various components. So the first thing is they attached a tag to either the 23S or the 16S. And we'll focus today's discussion on the 23S because that's what they did more characterization on in the paper. So how did they decide on the tag and where to place the tag? AUDIENCE: You don't want the tag somewhere that's going to interfere with function. And you don't want the tag itself to be reactive where it will interfere. So you need to be out of the way and kind of passive so it's not interfering with function. ELIZABETH NOLAN: So that's one point, right? We don't want this tag to interfere with function. So that's one aspect. What's another aspect? AUDIENCE: You still have to be accessible ELIZABETH NOLAN: Yeah, the tag needs to be accessible because something's going to have to bind this tag, right? So on the basis of those two criteria, we can imagine wanting this tag somewhere on the surface, right? We don't want it where the 30S and 50S interact. We don't want it in a position that's critical. So like maybe if it ended up near where EFTU first binds or EFG, that would be bad. So accessible and in a place where it won't interfere. Beyond that, the ribosome's huge. So how does one pick where to put this tag? What did the researchers do? I'll tell you what they didn't do. They didn't reinvent the wheel. So what did they do? AUDIENCE: Lit review. ELIZABETH NOLAN: Yeah. They went to the literature. And what did they find in the literature? AUDIENCE: Someone had installed a tRNA before or something and it didn't interfere with the ribosome function. ELIZABETH NOLAN: Right. So they took observations that were made from an independent group for an independent project but the observations that were useful to them in their design. So for some reason, this lab stuck a tRNA onto the ribosome and saw the ribosome was still active and functioning well. So the decision was, why don't we use that to place the tag here? So what about their choice of tag? What did they use? A big tag or a little tag? Any sense of that compared to the size of, say, the 50S? Take a look at figure one. AUDIENCE: A small. ELIZABETH NOLAN: Yeah, right. So I'd say very small compared to the size of the ribosome, right? So they decided to take advantage of interaction between this MS2 coat protein and the MS2 RNA recognition sequence, right? So that's one interaction involved here, so ligand receptor interaction. So here, this is the depiction from the paper showing where they incorporated this MS2 stem loop into the 23S rRNA, here. And so what does that tag need to bind, going back to the cartoon? And what is different about this strategy from, say, what you've done with nickel and TA chromatography and His6-tags? AUDIENCE: It has three things. ELIZABETH NOLAN: Three things, yeah. Right? So we have three components, two different interactions, say, between a ligand and its binding partner, right? So here, we have the mutant RNA of the ribosome where there's this MS2 stem loop, OK? That MS2 stem loop binds to the MS2 coat protein, right? And then there needs to be some way to pull this out. And what they chose to do here was take advantage of a second interaction and one that's commonly used in chemistry and biology, which is looking at an interaction between a protein called glutathione S-transferase and glutathione, here, right? So effectively, they have a solid support or a resin, so like the nickel NTA column, but in this case, it's modified with GSH. They have to prepare this fusion protein that is a fusion of GST and MS2, and then they have the ribosome with the tag. So why might they have done this with three components rather than two? AUDIENCE: Since GST/GSH, these are pretty standard affinity tags, it was probably easier to acquire GSH then to try to get MS2. ELIZABETH NOLAN: Yeah. That's a practical analysis there. So could they have made a resin with MS2? Maybe, right? So how did they go about doing this affinity purification? What were the steps and why? So imagine they've done the molecular biology required to express this tagged ribosome. They expressed the tagged ribosome in E. coli. Then what? How are they going to get this tagged ribosome out? AUDIENCE: They first purify the crude ribosomes [INAUDIBLE] including the [INAUDIBLE] and normal ones. And then they load these crude samples through the column. ELIZABETH NOLAN: OK. So the crude sample went through the column, right? What happened before that? So can you just put the crude ribosome through the column, if your column is the resin with GSH? AUDIENCE: You have to preload it with a fusion protein. ELIZABETH NOLAN: Yes. So what did they preload with the fusion protein? AUDIENCE: The column. ELIZABETH NOLAN: Yes. That's what they did, right? We can imagine just looking at this without further details. There's two possibilities with three components. So they could have, as stated and as what they finally did, they could add this fusion protein to the column, right? So the fusion protein binds GSH. And then you take your crude lysate, crude material from the E. coli and run that through the column to trap the ribosomes. What's the other possibility? They could have taken this fusion protein and put it into the crude mixture, and they talked about that. So what was one of the complications with this fusion protein that led them to incubate it with the column? Was this fusion protein well-behaved? AUDIENCE: It forms insoluble aggregate. ELIZABETH NOLAN: Yeah, right? It gave them some headaches. It aggregated. Is that something that can commonly happen? So they likely tried both ways, and they observed this problem with the fusion protein having aggregation, right? And they avoided that as being a complication to the purification by incubating the column with that fusion protein first. So after they take their crude sample and have that bind to the resin in the column, how did they get the ribosomes off the column? So what are the possibilities? And what did they end up doing and why? AUDIENCE: So you could just use an excessive of free MS2 ligands to cause the ribosomes to disassociate from the column. Or you could use an excess of the GST to cause that complex to dissociate from the column. ELIZABETH NOLAN: Yeah. So thinking about the latter possibility-- so what Rebecca has done is identify the two different ligand receptor interactions, right? We could disrupt this one between MS2 fusion coat protein and the ribosome or between GSH and GST. So in terms of this one here, does it make more sense to elute with excess protein or excess glutathione, which is effectively a tripeptide? AUDIENCE: Glutathione. ELIZABETH NOLAN: Yeah, right? So just that much easier to come by a lot of glutathione than a lot of GSH-- or sorry, GST. So which one did they choose? How did they elute the ribosome off the column? How long did you each spend on the paper before coming here? AUDIENCE: GSH. ELIZABETH NOLAN: Yeah, they used GSH. So they eluted with excess GSH. So what does that mean in terms of the purified ribosome? Is it just the ribosome with the stem loop? AUDIENCE: They still have the fusion protein. ELIZABETH NOLAN: Right. We still have the fusion protein on. So if you were making this decision at the bench, you have two different interactions to consider, what would you choose and why? Why and I guess, really, why did they choose to disrupt the interaction between GSH and GST? AUDIENCE: Because it's difficult to have enough MS2 than to purify with glutathione. ELIZABETH NOLAN: Here, right? Because if you were going to, say, elute with excess of MS2 stem loop, where would that come from? Or excess MS2 protein, right? AUDIENCE: Also, it's actually MS2. ELIZABETH NOLAN: Yes. It's not very practical here, right? At the end of the day, it would be best to have the ribosome without this fusion protein attached, but disrupting this interaction isn't very practical. So it would be quite expensive either way if you were making a lot of some sort of stem loop. And then how would you even know you have the stem loop or the protein here? OK. So I'm just curious, for those of you who have done like nickel NTA chromatography, do you have a sense of the affinity of the His-tag protein for the resin? So what happens as this tied protein goes down the column? All right. So you have some column with your resin plus His6 protein. So was it a strong or weak interaction? AUDIENCE: Strong. AUDIENCE: Strong. ELIZABETH NOLAN: How would you define strong? Or why do you say it's strong? AUDIENCE: The chelation. ELIZABETH NOLAN: Well, you're forming a complex, right? You're forming-- the His-tag is binding the nickel NTA. AUDIENCE: The Kd is probably [INAUDIBLE].. ELIZABETH NOLAN: Is it? AUDIENCE: I don't know. AUDIENCE: It is a dynamic process where they're releasing and binding and releasing and binding again. ELIZABETH NOLAN: Yes. So there's an equilibrium, right? And we talk about binding to the column but how tightly is this tagged protein binding? And is it just binding there and getting stuck? I mean, it needs to stay in your column, right? In this case, it's not very strong. So if you look at reported Kd's for, say, His-tags to nickel NTA, they're on the order of one to 10 micromolar. So orders of magnitude lower affinity than what you just suggested. So why does the column work? AUDIENCE: Because everything else binds worse than that. ELIZABETH NOLAN: Well, you hope that. You hope. I mean, sure. I mean, if you know that histamine-- a protein that's histamine-rich it's going to stick, but why does it work? So is it surprising that a micromolar affinity can allow this to be trapped on the column? AUDIENCE: Is it because you have six histamines tied down [INAUDIBLE] ELIZABETH NOLAN: Pardon? OK. It's not the amount of histamines. What is in your column? AUDIENCE: You've got a lot of binding sites. ELIZABETH NOLAN: Yeah. Right. There's a lot of binding sites in the column. So you're going to have, as Rebecca said, dynamic. This is coming on and off the column, but there's a lot of binding sites. So if it comes off, it can go back on there. So what about the GST and GSH? AUDIENCE: I know it's one of the strongest attractions. ELIZABETH NOLAN: Yeah. This is much stronger than nickel NTA and the His-tag protein. So orders of magnitude higher affinity here for that. OK. But it's something to think about when you're choosing an affinity purification method there and to think about what's actually happening on this column and dynamic process. So jumping ahead a little bit, they did their experiments first just tagging the wild-type ribosome, OK? And that's very important because they're trying to make a new purification method, and the first thing that needs to be asked is does the wild-type ribosome plus the tag behave the same or differently from the wild-type ribosome without the tag, OK? And you want to know that because if the tag is causing a problem, maybe it's not a good design. And you don't want to go forward making mutants with that kind of modification, OK? So what do they need to do after doing this purification, right? One is just analyzing the purity of the material that they've come up with. And then the other things they did was look at the subunit integrity, right? And so something to keep in mind is that the ribosome has two subunits. It has many ribosomal proteins. Does putting the tag on only one subunit work well? And then, of course, they need to think about the activity. And so they presented a number of different assays in this paper ranging from looking at kinetics of peptide bond formation to looking at kinetic studies of release with one of the release factors here. So let's think about the purity analysis. And what we're going to focus on is there chromatogram, right? So as we know, they took their column of GSH. They first loaded that column with the GST/MS2 fusion protein, and then they added their crude sample and eluted with GSH, right? And so the data they present for the chromatogram for monitoring fractions of that column is shown here, OK? So what does this tell us? What do we see in this chromatogram? So what are they monitoring? So first you want to read your axes, right? So what are they monitoring. AUDIENCE: A260. ELIZABETH NOLAN: Yes, that's the y-axis. And why A260? Going back to 20 minutes ago. AUDIENCE: DNA. ELIZABETH NOLAN: Nucleotides, right? Do we want to monitor DNA here? It will work for DNA. AUDIENCE: Oh, RNA ELIZABETH NOLAN: Yeah. So we have the 23S, 16S rRNA. So we're looking at A260, which makes sense. We're trying to purify the ribosome. Volume, what is this volume? AUDIENCE: The elution volume. ELIZABETH NOLAN: Right. The volume eluted from the column. So looking at this trace, what do we see? AUDIENCE: There's a peak [INAUDIBLE] maybe there's some [INAUDIBLE] introduced later. ELIZABETH NOLAN: OK. So you've mixed what you see with an interpretation. So let's just stick right now to what do we see in the trace? And then we're going to think about where these things come from. And that's just something important, I think, with the problems we give in this course. First, you want to ask just what does the data say? And then, how do we interpret this data based on our knowledge of a system here, OK? So do you see what I mean, how you mixed what you see here and a potential interpretation? So just what do we see? AUDIENCE: Two peaks. ELIZABETH NOLAN: So there's-- Yeah. Rebecca? AUDIENCE: Just the large broad peak that results immediately, and then a smaller separate peak. ELIZABETH NOLAN: So that's a very nice description. We see a broad peak with high A260 absorbance, then it elutes immediately. And then later, there's a peak just after 30 mils that's sharper, right? So what are these peaks? What came off here? AUDIENCE: Everything else. ELIZABETH NOLAN: Yeah. So what's everything else? AUDIENCE: DNA, [INAUDIBLE]. ELIZABETH NOLAN: Yeah. So things that didn't stick to the column, right? They got washed out. So maybe the native ribosomes, right? Maybe there's DNA in there, tRNAs. Could there be EFTU with tRNA bound, right? And there are things we're not seeing, right? So what do we think about the second peak? AUDIENCE: It's much less broad. It's pretty sharp, so that will tell you that it's likely only one thing that has bound to the column a lot. So that would probably be your His-tag. ELIZABETH NOLAN: Is this a His-tag here? I know we're going back and forth because you're all more familiar with His-tags. AUDIENCE: Your affinity tag. ELIZABETH NOLAN: Right. So this is likely what was tagged, right? And with GSH elution, we disrupted the binding interaction with GST, and it came off the column. Do we know that it's only one thing? No. At this stage, we don't, right? So what are possibilities? What could this be? So we have the tag on the 50S. Because the only-- I mean, it's coming off here just based on the amount of GSH required to push it off the column there. AUDIENCE: It could be a mixture of intact ribosome, which is the cell that gets tagged. ELIZABETH NOLAN: Right. Yeah. So we don't know right now in terms of the whole composition of this peak, right? It could be intact 70S because 30S came down. It could be 50S alone. And there's always the possibility of some other contaminant that just for whatever reason came off the column then there. OK. Would you be excited by this chromatogram if you were the person at the bench doing this work? Yeah. You'd be super excited, right? So it looks quite good. So of course, there needs to be some more analyses done. And one analysis we're not going to go into in detail but they needed to ask, is this really all tagged ribosome or is there also some contamination of wild-type? And so they designed some analysis using a technique called primary extension to look at that there. And what they saw is that they primarily had the tagged ribosome, which was good news. So getting at the question in terms of what's actually in this peak, they looked at effectively, say, subunit integrity. And how did they do this? They used centrifugation. And does anyone recall what type of centrifugation they used? AUDIENCE: Sucrose density gradient. ELIZABETH NOLAN: Yes. So they used a sucrose gradient, right? And how does that let you do separation? AUDIENCE: By density. ELIZABETH NOLAN: Right, by density. And we have the different subunits of different sizes, different density there, right? So this is the data they show. And so, again, we want to look at these data and ask what do we see and what does that tell us, right? So these are the results from the sucrose gradient centrifugation. And they looked at untied wild-type ribosomes, so isolated 70S and then the tied ribosome. So what was the key part of the sample preparation here? How did they prepare these samples to be able to look at the subunits individually? Because the question they're getting at is, what is the ratio of the 50S to the 30S in the sample that was purified from the column? AUDIENCE: They dialyzed using magnesium buffer. ELIZABETH NOLAN: Yeah. So what was it about the magnesium in the buffer? AUDIENCE: It causes disassociation between the 30S and the 50S. ELIZABETH NOLAN: So why? Maybe you said it, and I didn't hear you. What was it about the buffer that allowed this? AUDIENCE: Is it the positive charge? ELIZABETH NOLAN: Well, it is the magnesium, but was it low or high magnesium buffer? AUDIENCE: Oh, low. ELIZABETH NOLAN: Low, Right? So we learned that magnesium is important for interactions between these subunits, right? And you've seen some of the experimental details in the paper for recitation two and three, they used different concentrations of magnesium. In this case, they want to separate the two subunits. So they basically used a low magnesium buffer to allow them to dissociate. So in these data, what are we looking at? The axes aren't shown, but what are they? AUDIENCE: I have a question. Why not just a no-magnesium buffer? ELIZABETH NOLAN: Could that be bad for the sample? How low is the magnesium in the buffer? AUDIENCE: One millimolar. ELIZABETH NOLAN: One millimolar. So where else might the magnesium go? Is it only important for this interaction? AUDIENCE: I mean I guess it's useful. It can be used in a lot of different places actually in the cell. ELIZABETH NOLAN: OK. But we don't have the cell here. We have the purified ribosome. AUDIENCE: Is it possible that it's holding together the actual conformation of the 50S and the 30S? ELIZABETH NOLAN: Yeah, right. I mean, that may-- I actually don't know what would happen if this goes into a no-magnesium buffer, right? But I think there's about a dozen contacts where magnesium is used between the two subunits. And you can imagine there's plenty of interactions of magnesium or cations in other places of each subunit. Joanne, do you happen to know what happens if the ribosome-- I think you get a big unfolded mess. JOANNE STUBBE: It would be hard to get rid of the magnesium because it has key binding sites along the place where it comes off and goes back on. ELIZABETH NOLAN: Yeah. So what are the axes? If we were to have them, what are we looking at? AUDIENCE: I think the gradient. ELIZABETH NOLAN: Yeah. Right. So we have basically, yeah, the percent sucrose, say, the gradient-- or the density, right? And how are we seeing these peaks? AUDIENCE: We see two peaks and the upper graph we see the peaks are more similar in size. And then in the second part we see-- well, we see two peaks but one peak has a shift over. ELIZABETH NOLAN: Right. So just backing up. That's all certainly the case. How are we detecting these peaks? What's the readout? So we have some sucrose gradient here. How do we see the peaks? AUDIENCE: PUB. ELIZABETH NOLAN: Well, what, more specifically? AUDIENCE: Oh, A280. ELIZABETH NOLAN: Yeah. So here, they're using the absorbance at 280, right? So a little different than the chromatogram, but that's OK, right? There's 280 absorbance here and maybe their instrument for this didn't allow 260. So as we just heard, if we look at the untagged ribosome, we see that there's two peaks, and they're nicely labeled. It's nicely labeled with the 30S here in terms of percent sucrose in the gradient and 50S here, right? And so we see kind of in our standard control, the peak for the 30S is smaller than the 50S, and this is where they're placed. And so what we want to do is compare this data to the data here for the tagged ribosome. OK. So what are the two major observations, two major differences? Rebecca. AUDIENCE: The second one's enriched for the proteasome, there's relatively higher concentration of 50S. ELIZABETH NOLAN: OK and-- AUDIENCE: And it's [INAUDIBLE]. ELIZABETH NOLAN: Yes. So or another way to say that, maybe, it's depleted in the 30S, right? But I think you're coming across the same observation, right? So first we see there is a peak corresponding to the 30S and then there's this peak here which is shifted relative to what we saw for the 50S and the untagged ribosome. And the peak heights or peak areas, however you want to analyze it, that ratio is very different than what we see up here. So why is this shifted and why is there more 50S? AUDIENCE: Can I ask a question? So can you definitively say that it's enriched or depleted in 30S? I don't know a lot about biochemistry in general, but can you say that possibly the multiple histamine-- oh, it's not histamine. AUDIENCE: Is it just tagged because of the shift. I mean, it seems like such a small derivation. ELIZABETH NOLAN: OK. So what else is there in the sample? So we have that stem loop tag, but based on how they eluted this, what else is there? AUDIENCE: The fusion. ELIZABETH NOLAN: The fusion protein, right? So then the question is-- and this is just getting to the point of needing to look at all the details in terms of what's done in someone's experiment to try to figure out what's happening. Is the MS2 coat protein alone enough to cause a shift? So what else might that be? AUDIENCE: Could it be the GST? ELIZABETH NOLAN: Well, you know that the GST/MS2 is there because it was GSH that was used to elute that whole thing. So the ribosome and the GST/MS2 fusion. So I guess the question is, just thinking what is the size of the fusion protein? And then how does that compare to the ribosome? And what does that mean in terms of a shift to a different percent sucrose? So just what are possibilities? We learned that the fusion protein had some problems, that it liked to aggregate. Is it possible that contributes? Is it possible that not all of the 70S was dissociated, right? So there is not a label for where the 70S would come here. These are just things to think about when looking at the data here and to look at their explanation. So is this good news or bad news? And is there a solution? Yeah? AUDIENCE: Sorry. I had a question. Since they're monitoring the A280, so the protein absorption, since the 50S would have the fusion protein associated with it still, wouldn't that affect the A280? Can we definitively say that it's actually depleted in the 30S or is maybe just the absorbance of the 50S higher for the tagged protein because of the fusion? ELIZABETH NOLAN: Yeah. So that's a good point, right? The fusion protein is going to have some absorbance defined by its extinction coefficient. And then the question is, what is that in magnitude? And how does that compare to, say, the total extinction coefficient of the 50S, right? So there are a number of ribosomal proteins, 20 to 30, and we don't know how many of those got through this purification, but they're there. So that's how I would go about analyzing that. I actually don't know what the relative absorbance of the fusion protein to the intact 50S ribosome is but that's a caveat to consider and could contribute. So what do we think? What are you going to do for an assay? If we were to set these ribosomes up, the wild type and the mutant one, and, let's say, do a peptide bond formation reaction, like what they presented in the paper as is, what would you expect? Are you going to see the same activity or less activity? Working under the model that we don't have enough 30S here to fully form 70S ribosomes. AUDIENCE: You'd expect the 30S to act like the limiting agent in peptide bond formation where they come together. And you can only make as much ribosome as you have 30S. So you would get less peptide bond formation. ELIZABETH NOLAN: Right. So you'd have fewer 70S assembled ribosomes that can do translation. Right. So what was their solution to this problem? Or if you don't remember, what would you do? What happens if we don't have enough of a given component? We'll finish on this note. AUDIENCE: You just add more of the 30S. ELIZABETH NOLAN: Yeah, right? So you can imagine adding in more 30s there. We know how to purify wild-type ribosomes, can dissociate the subunits and imagine purifying 30S and adding that into the reaction to have the correct stochiometry there. So that's exactly what they did in their assays. And so I encourage you to take a look at the assays they did for peptide bond formation and for peptide release. And in the posted notes, there's a schematic overview of everything that was done to get the components of their syringes. So these were quench flow experiments, like what you talked about in recitation in prior weeks and we talked about with EFP. And there's also an example where they used puromycin, like what we saw with EFP, in the experimental setup, so that antibiotic that causes change termination there. And so what they found is that these tagged ribosomes really work quite well in terms of the kinetic comparison there. And they moved on with this methodology to use it to purify new ribosomes for other studies. So it turned out to be a useful method for their lab, and I would argue, what would be a useful method for other labs, based on the information they presented in their paper there. OK? So you're off the hook, and I'll see you in recitation next week.
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MIT_508J_Biological_Chemistry_II_Spring_2016
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R13_Fluorescence_Methods.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JOANNE STUBBE: So the key question is, do these-- and I think this is a general question you can ask, metabolically, inside any cell is do these enzymes that are on different polypeptides cluster. And is there an advantage, kinetically or whatever, is there some kind of an advantage to have clustering inside the cell. And where have you seen something like this before? Do you remember? Do you remember the section. Where have you see multi enzyme complexes and clustering before? Yeah? So or the classic one, PKS has been around. But it's completely analogous to fatty acid synthesis, right? And so in bacteria, they're all single polypeptides. In humans, they're all activities that are on single chains. OK, so that's sort of what's going on here with the purine pathway. We'll see there are ten. This just sort of helps us focus if we get to the data at the end, which I think we will from what we did the last time. Is that you have six enzymes for 10 activities. So that just means you have more than one enzyme per polypeptide, OK? And so I guess the key thing that I wanted to focus on is do you think it's important to cluster? Here's a pathway. These are the names. We're not going to go through the names. The names really aren't important for what we're doing. There'll be two names that we'll be looking at over and over again. These are the papers that you guys did, in fact, read. One is the original paper, which got a lot of press. And I just want to show you that there have been there's actually been four papers published in the last six months on this topic. And one of which was published. Yeah. One of which is published with Science, where they are now claiming that this complex is localized to the mitochondria. OK, so you take pictures. And it's that this is looking at super resolution fluorescence methods. And you can clearly see clumps of blobs focused on the mitochondria. Why would you want it to be at the mitochondria? So then you have to ask your question. You might need purines, because that's where you make, through a proto mode of force in respiration. Remember, when you convert oxygen to water, you get a huge amount of energy released. You make ATP. But it's going to be made from something. So maybe you would want. That's the way they rationalize it. And they do. And then they connect it to the other latest hot topic, which is EM torque, which is the major signaling switch for fatty acids and for amino acids. And now in the last two years, purines and pyrimidines, I decided-- I've done a lot of reading about it, decided and believe. I mean, I believe it. But I don't believe the connections yet. So again, this is what you're going to see in the next decade is connecting signaling to primary metabolic pathways, like the purine pathway. That's going to be a big thing and how do you connect them is going to be the key question. So anybody that wants to do some more reading, this is an updated version. I kept updating this three or four times. And so I think these are the key questions we want to focus on. And so what I'm going to do. Well, define the questions a little bit and whether the things we need to think about just to determine whether this is really important, biologically. Then we'll define fluorescence and what you can do with fluorescence. And then we'll come back and look at the data. In the paper, we're also going to look. We probably won't get all the way through all of the data. But we will look at some of that data again in either the next lecture or Wednesday's lecture. So you will see it again if we don't get through the data. So they claim they have a multi enzyme complex. Did you believe that from the data? I mean, they didn't look at all 10 enzymes simultaneously, right? Or six enzymes. AUDIENCE: Whenever they show an image of the cells, and then they fluorescent, trying to show local sections, it's always so hard for me to figure out-- JOANNE STUBBE: What you see. AUDIENCE: Yeah. JOANNE STUBBE: OK, so that was said. We'll look at some of those pictures. But I completely agree with that, that you can't see anything from fluorescence pictures. So everybody, all chemists or chemical biologists, now have huge numbers of these pictures in their papers. And with Alice's group, I'm always on their case that I can't tell a damn thing. This is on thesis. I can't see anything. And Alice says she can't see anything either. So it's very hard to see things in these pictures. The contrast isn't very good. And what her lab now does is it goes to EM, where you can see things much more clearly. The fluorescence things are tough. So you're not the only one. And if somebody says it's obvious that this. And you don't see it. Raise your hand and say, I don't see it. Show me what I should be looking at, OK? So that's a good take home message, because everybody and his brother is doing this. And this goes back to knowing how to do it correctly. We're not going to talk about any of that stuff. I mean every one of the methods I'll sort of show you that's out there. You have to really study it to make sure you're handling it correctly. So I mean I think, to me, this has been a problem that I've been interested in. And I started working on this a long time ago in the purine pathway is not are things sticking together important. Actually, I don't think those are important. You immunoprecipitate all these things. OK, so you say obviously these are talking to each other. But the key thing is the kinetic competence. And lots of times when you mess around. You get it in a state. And you post-translationally modify it. So it's sitting in this state there probably isn't on the pathway. You need to then show it's on the pathway. So I think a lot of protein, protein interactions, especially now that we know that proteins move around. And they're in this complex. And they're in that complex. And they're in that complex. The key, I think, is to transient interactions. So why? So this is just my personal take on this. I'm letting you think about this. But is it easy to look at transient interactions? No. OK, so anyhow, I think people need to start doing a lot more thinking about how to look at that. And one way you could look at transient interactions is if you can fluorescent label something. And they come together on a certain timescale and then move apart. And can you do that inside the cell with the right spatial and time resolution. You might be able to start looking at that. So that the methods that are being developed and continue to be developed are incredibly powerful. And I think will allow us to ask this question happens inside the cell, which you've all seen pictures in your introductory courses of, man, how complicated the inside of the cell is. That's part of the issue. So the issue is that you might have a purinosome somewhere in the cell, depending on the growth conditions. But those enzymes might be involved in other things. And so you have only a tiny amount of it, as opposed to trying to make the cell by growth conditions, putting it into all one state. So you can see it. So the question is, how do you see it? And so that's the key issue. And if you perturb it enough. And you do see it. Then you have to ask the question. And this is a question that you might want to think about in terms of these two papers you were reading. That's what. If you do this, that's what the Marcotte paper said, that the cells were incredibly sick when you take out all the purines. And in fact, Alice is-- because of this mitochondria connection between the purinosome and Alice's interest in the mitochondria, she's had people trying to repeat this. And Vicki Hung worked on this and couldn't repeat it. So she didn't spend that much time on it. But all I'm saying is it's not a slam dunk to be able to do this. But that being said, I think this has been an issue that people have been thinking about for decades. And it's just really hard to test experimentally inside the cell. This is where we need chemical biologists to figure out new ways of being able to look at this, so that you can actually make a measurement that's interesting. So I guess the question I want to start with, before we researched looking at fluorescence, is why do you think it would be important to do this. Or do you think it would be important to have a complex. What's the advantage of doing that? Yeah. AUDIENCE: You were saying in lecture that you want to increase the effective molarity. And so by having all these things right next to each other, there's-- obviously you're going to have more interactions per second. JOANNE STUBBE: Well, you may or may-- you may not. It depends. So I think this is the key question. Is diffusion fast inside the cell? AUDIENCE: Yeah. JOANNE STUBBE: Yeah. It's still very fast. For small molecules, it's incredibly fast. Even for proteins, it's incredibly fast. So even if this guy is over here. If you're turning over here at a much slower rate, and you have enough of them so you can interact at diffusion control, do you need this organization? There are a lot of smart people who think you don't need that. There are a lot of smart people who think you do need that. But this is the question I want to raise. However, so catalytic efficiency is absolutely it. But where might you really need catalytic efficiency. And so that goes back. There are places where you really need this. So if you look at the first intermediate in the pathway, this guy, what do you think about that guy? Do you think he's stable? So if you look at the first intermediate in the pathway, which we'll talk about next time. So this is amino phosphoribosine-- phospho-- I'm drawing a complete-- I think I'm tired. Anyhow it's the amino analogue of ribose 5-phosphate. Phosphoribosylamine, that's what it's called, PRA. OK, do you think that's stable, as chemists? So what do you think that could do? AUDIENCE: Could you release the amine? JOANNE STUBBE: Yeah, so how would you do that? AUDIENCE: So if it's proteinated, and then the ring opens till-- JOANNE STUBBE: OK, so that would be one way. You want to release it that way. OK, so it would have to be under conditions. We could do that under neutral conditions. What else can happen to this ring. That doesn't happen. There are lots of ways this molecule can break down. OK, it depends on the details of the environment. How else could this molecule ring open? You wouldn't need to ring open here. You just go through an oxocarbenium ion and have water attack. So what if it opens that way. So that's the way you form aldehydes. All sugars are in equilibrium with aldehydes. These things are in equilibrium. So you have a ring open species. But then what happens if a ring closes? It can ring close from the top face or the bottom face. You have an imine. What can happen to the imine? It can hydrolyze. This molecule, and this is a molecule my lab worked on decades ago, has a half life in solution of 10 seconds. So is 10 seconds short or long, biologically? What do you think? AUDIENCE: I'm going to say short. But I don't know JOANNE STUBBE: Yeah. I think it's amazingly long inside the cell. So I think as a chemist, nobody could ever. Nobody ever saw this intermediate. My lab was the first one that figured out how to look at it. And I won't go through that. But the fact is that 10 seconds is a long time inside the cell, if you think about how small the cell is and how fast diffusion is. OK, so one place, though, where you might want to have organization is if you have something chemically really unstable. OK, because then when you generate it, it could potentially be passed off, or as you say in the immediate vicinity, it's a competition. But if it's right there, you're effective molarity, that would get into that first question, the effective molarity. It would be high enough to get passed on. It would get high enough to get passed on to the next guy in the pathway. So that would be one thing is instability. And in the purine pathway. We will go through this a little bit. But really, that's one of the things that's most amazing about Buchanan's elucidations of the pathway is only intermediates are unstable. Nobody, still, if you're looking at omics, looking at nucleotides, nobody knows how to deal with these molecules. They're all chemically unstable. And they don't get that they're chemically unstable. They don't ever see them. The reason they don't see them is because they don't know how to handle them to keep them alive during the analysis part of the project OK, so you have this instability problem. And in the purine pathway, the instability problem is a real problem for not just this guy. This guy is obvious. But for other guys. OK, so then the next question is where else. And if you're thinking about metabolism in general, where else might you want to have organization of your enzymes? You might want to have it if you generate an intermediate in the pathway. And then there. It's a branch point for other metabolic pathways. OK, so there's an intermediate in this pathway that can go to thiamine biosynthesis to histidine to tryptophan in biosynthesis. I'm not going to go through that. But that would be another place that I think it's obvious that you could sequester, under a different set of conditions, and prevent the other pathways from happening. So if you have an intermediate branch point, you can prevent other pathways. So those two things I think are important. One of the questions is, do you increase the flux through the pathway? OK, so there's been a lot of engineering people. People really care about this in terms of engineering. If you want to engineer a metabolic pathway, should you be linking all your proteins together? And there have been a lot of papers published. If you look at bioengineering papers, where they link all the pathways, all of the enzymes together in a way, because they want them to cluster, because they think they're increasing the flux through the pathway. And so there are some people that do calculations that show you increase the flux. Other people do calculations so you don't increase the flux. So I think, again, this is an area that I think is very active. And it's pertinent, because everybody and his brother is trying to make biofuels. You'd need to do a lot of engineering from a lot of enzymes from different places, putting them together. How do you make them efficient? OK, so we asked the question about flux. And I think, mathematically, people are looking at that. You need to know a lot about the kinetics of your system. These systems, there's a lot known about the kinetics. So and then this goes to the question of how, what is unstable. And you need to think about diffusion. I think this is not so easy to think about this. But we do need to think about flux through the pathway. And then the other thing that's interesting in terms of regulation is it turns out in eukaryotes, where things are much more regulated than in prokaryotes, because of the increased complexity of everything. Almost all of these pathways are organized on multiple activities on one polypeptide. That's telling us something, I think, since we see this over and over and over again. So there must be some reason to do that. So for all of these reasons in terms of the purine pathway this has been sort of a target for people for a long time. That's one of the reasons I decided to talk about it, because this was one of the first papers where people were excited that they thought they had evidence for this kind of organization in the cell. Not in the animal. But in the cell. OK. Let's see what I want to say next. I'm trying to keep this on some kind of a schedule. OK, so this is the hypothesis. The hypothesis is that these things are organized in some way. And this was taken out of-- probably it was a review paper. It wasn't taken out of a paper you had to read. Here's the cell. That's the nucleus of the cell. And what do you see. I think you can see this right. You see these little dots which they call punctate staining. So what else do you need to know that they don't have in this picture that's really sort of key to thinking about this model. So here they've just have a bunch of enzymes stuck together and all in a little ball. OK, so if you read the paper there was a couple of things. AUDIENCE: How you're getting the fluorescence? JOANNE STUBBE: How you're getting them? AUDIENCE: If it's by effusion or fluorescence. JOANNE STUBBE: Yeah, so how you're getting the fluorescence becomes key. OK, so we're going to talk about that. How did what was a major way they got the data. We'll talk about this in a minute in more detail but. But whenever you're going to use fluorescence, you have to figure out how to get a probe onto your protein. So that's like a major focus. And this again is where chemical biology needs to play a role. We still need better ways to be able to do this. You've seen over the course of the semester. I think in a lot of ways you could potentially do this. We'll come back to that in a minute. But if you look at this, what's missing? And this is something that drove me crazy when I reviewed the original paper. AUDIENCE: I just noticed, so you're getting that. But they didn't stain the membranes really. There's not a good-- I mean, you can kind of see the shape of the cell. But it would be nice to have a clear sort of-- JOANNE STUBBE: OK, so they might have done that. Did you look at the supplementary material? They might have stained the membrane. OK, so I think everybody would believe you see little blobs. OK, so what do you need to think about in terms of the little blob. AUDIENCE: The size. JOANNE STUBBE: The size. Right. Yeah, so that's one thing. They don't ever they don't ever talk about this. They might in some of the very later papers. But if we know this, we have structures of all the enzymes in the pathway. So you could make a guesstimate about how big these blobs should be, if you had one of each of these. And these things are huge. So this would tell you that you would have many, many of these. This is one thing that I think they need to do some more thinking about that they could have many, many of these things. And then the question is, why would you want many, many of these things. And how were they organize? Are they just sort of randomly organized or are they really organized in something like that with this big huge protein in the middle. That's one of the ones they look at. FGAM synthase, that has a molecular weight of 150,000, which is huge for an enzyme. And so for a long time-- and the catalytic activity-- my lab has studied that-- is way over here. And so you have a lot. Could it be a scaffold. OK, so that's where that idea actually came from. So but the hypothesis is that these guys are organized. And they're under certain growth conditions. That's the key. And we'll look at those pictures that come together if they do this when you need to make purines. And then they can go apart. OK, so the key thing, I think, is. And I wanted to just remind you why we're spending this time looking at fluorescence. And we probably should have spent two or three recitations on fluorescence methods. But we didn't. Is that we've seen this many times before. We've seen stopped-flow fluorescence in the Rodnina paper, where we were looking at the kinetics of fidelity of EF-Tu. And somehow they put a fluorescent probe onto the piece of tRNA. That was not trivial. How you got the probe there. And that probe could-- and we'll talk about this in a minute. But it could change. It changes when it's in different environments. And so you can use it as a way to monitor changes. So reactive oxygen species, we just looked at this. And I decided to put this up, since we didn't have the structures up last time. Fluorescein is one of the dyes that. This is fluorescein that people use. This is a version of fluorescein. But we talked about how do you know that epidermal growth factor is generating hydrogen peroxide? OK, so what we need is a sensor of hydrogen peroxide. So we talked about that last time. And this is the sensor that people use. Why did they use it. We talked about it. But we didn't have the structure. So they use the dye acetate of this molecule. This one they use, the triacetate. The one that they use in paper was the diacetate. Anyhow, you need to get the fluorescent probe into the cell. So that's something you're going to have to deal with. And so if you acetylate it, you don't have phenols or phenolates which might not get through the membrane, which apparently they don't. So then when they get in the cell, what do they do? They hydrolyze, OK. So what happens is when they hydrolyze, they are now-- you have these hydroxylated compounds that are able to be oxidized by an oxidant. And one of the oxidants that can do this. And there are others that can do it as well, is hydrogen peroxide. So people use this as an indicator of hydrogen peroxide. But it's not specific. Yeah? AUDIENCE: So are they also trapped after that esterase, like from diffusing that out to the-- JOANNE STUBBE: No. I mean, I don't think they diffuse back out, because I think they're the phenolates. So I think the diffusion out, like with many of these things, like if you use-- lots of times you esterify phosphates to get them into the cells. Once they hydrolyze, they charge. They don't get back out. So I don't really know. But that's what I would guess. So I guess the key thing and the basis for some comments that I made in class was that we don't really have. We don't know that this is specific for one reactive oxygen species. And so there are lots of people in the chemistry, biology interface trying to make specific sensors. OK, that's not easy to do. The hydrogen peroxide, they're getting better. In fact, Ting's APEX, which is a peroxidase, sort of similar to what we talked about with peroxireductions in the myeloperoxidase can actually function as a hydrogen peroxide sensor. So anyhow, what happens is that when it gets oxidized, it becomes fluorescent. So it's not fluorescent. It becomes fluorescent. So it just gets turned on. And you can see something, OK? So that's something we talked about. In Liz's part of the course, we talked about the fact that we can watch protein unfolding in the e. Coli proteasome. OK. And what did you look at in the proteasome, clip X clip P? You looked at titin. That had a little tryptophan on it. And tryptophan can absorb. It's not a very good thing, because it absorbs in the UV. But tryptophan fluorescence is used a lot. There are lots of tryptophans, so it's also really hard to use. But titin was this little tiny protein. And it was the only tryptophan. And they also did experiments with green fluorescent protein, which is what we're using in this paper. We remember. They pull on it. And you pull and you pull when you pull and then all of a sudden it unfolds. And you lose your chromophore. So you go from the on state to the off state. So all of these things. Binding measurements. You talked about. You had one problem set. I don't know whether you guys did that problem set or not. But there was a-- what was the calcium sensor? Does anybody remember? Anyhow, there was the calcium sensor, where you were asked in the problem set for a something or other that you asked to measure the KD for. And you can do binding assays. So fluorescence is an incredibly powerful tool as is the take home message. And we've seen it throughout the course. We just haven't talked about it. So now the key thing. And we're going to talk a little bit about fluorescence at probably a freshman level. Many of you guys, who were the undergraduates. You guys, have you done fluorescence experiments? You haven't done in the lab? I thought we had two Eureka labs that were fluorescence oriented. AUDIENCE: [INAUDIBLE]? JOANNE STUBBE: No? AUDIENCE: Yeah, yeah. so we-- [INTERPOSING VOICES] JOANNE STUBBE: So doesn't Tim's? He does sensors to sniff. I don't know what to sniff, but to sniff something, TNT or-- AUDIENCE: Right. but we didn't use fluorescence with that. JOANNE STUBBE: You didn't use fluorescence for that. OK, or the Tokmakoff lab? AUDIENCE: The one experiment we did in lab is we labeled a protein, the green absorbing dye. And it used laser anisotropy to measure KD rotations. And so the-- JOANNE STUBBE: OK, so you guys are experts, then, on fluorescence. Well, hopefully you-- anyhow, so one of the questions is we need to ultimately the key thing for any of this is we're going to have to have a fluorophore. So that's it. So we need the key starting point is a fluorophore And what are fluorophores. So you want something that's usually aromatic and large. It could be-- it could have a lot of nitrogens in it. Oh, I knew I forgot something. So there's a book called, Molecular Probes. OK so I gave you a handout on fluorescence. I forgot to bring the book, if anybody wants to see it. This book is worth its weight in gold if you're a chemical biologist. Because this has everything in the world you need to know about fluorescence. It's described in a thoughtful way. They sell all the probes. If you want to do something to tweak something, they'll help you do all of that. So this book, this is molecular probes book. I think it's online now. I have a copy that's five years old. I use it a lot. It's a really important book. And I got this out of the book. And it just shows you in the book, they have all these pictures of these fluorophores. So they're just big, huge, greasy molecules. You have to worry about solubility a lot of the time. So you have to stick sulfates, or something that ends up making it soluble. So that's going to be a key thing. So we need to have a fluorophore. And we have many options that we can buy these things. OK, so what's this? OK, so what we want to think about is this. So in your, the latest version of your handouts, I've written down what I'm going to say. Butt it's pretty simple for-- I'm talking about this in a pretty simplified viewpoint. But what we're going to see is these fluorophores are going to allow us to. They allow us to do assays. I'm going to show you a quick example of that. That is you can have something that is. You can have a molecule that is quenched, so you have a quencher on one side. I'll show you. And I'll show you the way the quenching comes from, something fluorescent on the other side. You can't see anything. You cut it in half. It could be a protease. It could be a nuclease. The quencher goes away. And you see fluorescence. You could have a sensor for metal binding, which Liz talked about. So you have two fluorophores. OK, you've got to figure out what the right fluorophores are. Something binds. They change confirmation. And they change confirmation in some way that you can actually detect a shift in the wavelength. And then you're looking. In our case, we're just sticking something on the end to see something. You were making a protein fluorescent. That's all we're doing. So you can use it for assays. You can use it for FRET. And in the current-- so you can measure distances. We're not going to go into that. But any of you that are interested in the current version of the handout, I have sort of short tutorial on what FRET is and where you should go to look this up. And then we just basically have a fluorescent tag. OK, and we'll come back and talk about the tag. We already talked about the fact that we have green fluorescent protein, red fluorescent protein tags. But we'll come back and talk about the other tags. So we have a fluorophore And so what does that mean in terms of what's going on. So you have your molecule. And your molecule has a ground state, which we'll call this S0. This is the ground state. And you have many vibrational modes. And you have this big huge fluorophore that can absorb your electron. And your fluorophore can't absorb a photon. And so what happens is. So we're going to have excitation with a photon in a certain way, in a wavelength that can be absorbed by the electron in your molecule to the excited state, which they call S1. And so you can have your electron going to an excited state. And we have a wavelength of light when that happens. And that depends on the structure of your molecule. So you don't want to be in the UV region. You want to be out in the region where you have less interference. And so that's the key game you have to play to get into that region in the visible. You really have to put a lot of stuff on here. You just can't make a small little molecule that absorbs at 600 nanometers. So that's part of the problem. So you're making big things of necessity, so you can actually see something happen. And so then what happens under those conditions. So we're going to have the excitation wavelength of light at a certain lambda max. You absorb. It's just like absorption. You have a certain wavelength that it absorbs more frequently. Then what happens in the excited state on a very fast timescale, you lose energy. OK, so under these conditions, you're doing a relaxation. And then we'll see in a minute. I'll talk about what are the mechanisms of relaxation. But that can tell you. You can use those relaxation mechanisms in a different way to design your fluorescent experiments. So what you see in this cartoon is that you're relaxing on a very fast timescale. And physical chemistry has told us that to see fluorescence, it needs to go down. So these are the vibrational modes. So you're exciting your electron electronically and vibrationally. And then you need to go down in vibrations. You're losing energy somehow. What happens to that energy? OK, we can talk about what can happen to that energy. And when it gets to the lowest level of the excited state, you have fluorescence. OK, and so that also happens on a pretty fast timescale. So the key thing here. So when you get to the lowest. So this is the lowest level, it fluoresces. And so this is where the photon emits. OK, so the photon wavelength for emission or h nu emission. And the key thing that you've probably heard about, again, when you were introduced to fluorescence is because you're losing energy here, what happens to the energy? You're going to a longer wavelength. OK, so the excitation and the emission wavelengths are distinct. And that's called the stoke shift. So it's the wavelength of excitation vs. the wavelength of emissions. So you have a stokes shift, which is the wavelength of excitation minus the wavelength of emission. And so you need to look at molecules. People have spent a lot of time. You saw those 25 lists of things where people have designed things that actually work quite effectively. OK, and so then the question is you losing energy. You are always going to be at longer wavelengths. OK, so that's good, that makes it easier to see, because there aren't that many things inside the cell that give you a background, which is what you need to worry about in all of the experiments you're doing inside the cell. The brightness, we'll come back to that in a minute. So what kinds of models can give you. What kinds of mechanisms are there for relaxation of the excited state. And so there are a number of mechanisms that can be involved. And one is, again, non-radiative relaxation. And how does that happen? So you're changing vibrational modes. And when you're in the excited state, if you're in solution, you have interactions with solvent or other molecules, all of which can affect this kind of transition. If you're in the active site, there can be other things. So the key here is the environment. And again, it could be solvent. It could be protein. And the only way you can tell is by actually looking at the fluorophore on your molecule to end up seeing what you end up seeing. OK, so a second way that you can see. And you probably saw this in your introductory. Yeah? AUDIENCE: So what would be an example? Like, if a unit of the energy being released is a photon in one case for non-radiative, what's the unit of energy? JOANNE STUBBE: What is the unit of energy? So energy, heat is one way that you lose all of this. So it's vibrational energy. I would say, it's mostly heat. So you're changing excitation levels somehow. And the beauty of fluorescence. And this is the key to the sensitivity is you're not doing anything to your molecule. So your electrons got excited. They give off a little heat or whatever. They somehow change a little bit. And then they go back down to the ground state again. So what can you do? You can excite them again. So this can happen over and over and over again, unless the molecule in the excited state becomes destroyed. So that's called photo bleaching. So the key thing here. And this is, I think, this ability to recycle is the key to sensitivity. But again, I haven't used fluorescence inside the cell. I've never done this myself, experimentally. So I don't really know. But you hear about photo bleaching all the time. So I think this is not a trivial thing that you can just blow off. It would be nice. But what you're doing is you're using the same excitation. And then loss and excitation and loss over and over and over again. And so it provides a much more sensitive assay than what you normally see for something like absorption. OK, so let's see. There was one other thing. Oh, so we talked about this mechanism, non radiative relaxation. How else could you relax? You can go from a singlet state to a triplet state. OK, I'm not going to talk. But intersystem crossing, yeah. So you can go from the singlet excited state to the triplet state. I'm not going to talk about this. But the triplet state then can phosphoresce. We're not going to be discussing that at all. But that's one possibility. We just talked about the fact that you can have something in there that quenches the fluorescence. It interacts with something in a distance dependent fashion. And that, again, affects the intensity of your fluorescence. So you also have reaction with the second molecule. And that can become. And it could be good or bad. If it reacts with oxygen, what happens is oxygen, the energy is immediately transferred to the oxygen. So that's why in many fluorescence experiments, you remove oxygen from all of your samples. It acts as a quencher. So you have. And it could be oxygen, which acts as a quencher. Or it could be another fluorophore. In which case. and if everything is set up correctly, you can get the energy to shift the energy of emission can get shifted to longer wavelengths. So that's what FRET is all about. OK, so it could not. A second molecule could be another fluorophore. OK, so those are sort of ways that you can relax. And then you can set up different kinds of experiments, depending upon what the objective is of using fluorescence. So I've written this out in more detail. And for those of you who want to look at FRET, I've defined FRET. I've given you the equations. And people use this quite a bit inside the cell. You need to study this. There are a lot of issues associated with it that you need to think about. And I'll come back. You need to think about. It's not. There are a lot of constants that determine the rate constant for your FRET, OK. And so you just you need to think about all these constants to be able to interpret the data in a thoughtful way. And I've given you a tutorial that I felt was pretty good that I get off the web that just shows what FRET is. And that we have many, many dyes that we can measure distances from 10 to 100 Angstroms using FRET. That's not in this paper. So I didn't. And this just sort of is a cartoon of what I was just telling you. So here, you might have an interaction. But if you cut it, the interaction could be gone. Here, you might have no interaction. But when some small molecule binds, you see an interaction. And you can pick this up using fluorescence changes. OK, so people do these kinds of experiments all the time. And this kind of an assay is extremely-- there are two kinds of assays that one does. So if you work in a pharmaceutical company, people do this all the time. They want a very sensitive assay. Everybody uses fluorescence. They might use an assay like this, where you go from nothing to something. OK, so you have high sensitivity. And the other thing they use is, which I gave you in your handout, is fluorescence polarization, which I'm not going to be talking about. But those are the two major methods that people develop assays around in the pharmaceutical industry. So fluorescence is here to stay. We still need better tools. It can be quantitative. You can measure a quantum efficiency of the electron, light, that's involved in the excitation and the photon that's involved in the emission. If it's 100 percent efficient, then you're quantum efficiency is 1 anyhow. So you have a whole range of quantum efficiencies. OK, so now what I want to do is we're late. But we'll at least get to the other sources telling you what I just told you. OK, so I want to just introduce to you some of the issues that we're going to be facing. And we are going to talk about this in class, probably Monday or on Wednesday morning, OK. So I'll extend this in class. But they've attached green fluorescent protein to all of these things. So this is issue number one. What should they have done in these papers that they didn't do? If you read the paper carefully. I mean, it's hard to read a science paper, because all the key pieces of data are in supplementary information. So they made a few. In all of these, I can't remember what they made. But they made fusion proteins, right? So here, you have a purine enzyme. And here we have some kind of fluorescent protein. So that's the probe they're using. OK, so what's wrong with that, with the way they did their experiments? Can anybody look at the details of what's going on? So what, if you made this fusion, what would be the first thing you would do with a fusion protein? AUDIENCE: My first thought would be, if it changes the activity of the original protein. GFP's a very large [INAUDIBLE]. JOANNE STUBBE: Right. Exactly. GFP I'm going to show you in a second. I think I can show you this in a second. These are just the ways they were looking. But you have all these probes. GFP, it's over here. These are the organic dyes. Here's an antibody. We'll come back to that. So GFP is big. So does it change activity? They didn't assay that. To me, that's mind boggling. OK, because I've dealt with these. I know these proteins, that one protein there. So two of them they're dealing with. One of them is a trifuctional protein. The other one's 150 kilodaltons. These proteins are not easy to deal with, OK. So to me, this is a key thing. So this goes back to the Marcotte paper, where he's saying, well, I mean, maybe these things don't express very well. And they aggregate. They don't fold. We saw how complicated the folding process is. What is the second thing? How did they get the proteins into the cell? How did they get? They don't get proteins into the cell. How did they get? Yeah, how did they get GFP constructs into the cell? AUDIENCE: Transient transfection? JOANNE STUBBE: Yeah, transient transfection, what is the issue there? Without going into details, but what's the issue? AUDIENCE: Like, when the cell's normal mechanism, like the cell's own enzymes maybe-- JOANNE STUBBE: So you do have a normal-- you do have the normal enzyme. They didn't make any effort to knock out the purine enzymes. OK, but I think the key thing with transient transfection is the levels. First of all, a lot of cells don't have anything. But then you don't care about that, because you don't look at them, because they're not fluorescent. OK, but do you think the levels are important. I think the levels are incredibly important. So the question is 100-fold, 1,000 fold over the endogenous levels. And so to me, the first experiments I would have done before I did any of these other experiments is I would have looked at. You might have chosen the trifunctional protein, which they did, because it has activities 2, 3, and 5. And this other big huge protein, which-- so these are the proteins they focus on is activity 4. So 4 is huge. You might think it could function as a scaffolding protein to interact with activities 2, 3, 5. All of that's totally reasonable. OK, but they didn't deal with those issues. So you need to figure out how to attach something that's fluorescent. So one way is genetically. OK, and we've seen this. So we're just fusing GFP onto the protein of interest. Another way in this paper, also, and you mentioned that, is they were using endogenous antibodies. OK, so antibodies can't get into cells. So how do you assay this? So these are also tough experiments. So somehow you fix the cells. So they aren't falling apart when you're trying to perturb the cell to allow the antibodies to get in. And then you permeabilize the cells. Have any of you ever done that? I've done it in yeast. In yeast, it's brutal. I mean, it works. But it's the conditions are like it's a witch's brew. Anyhow, so then you get the antibody in. And that's what you're looking at. And if you look in the-- I have. We're not going to get that far. But I have pictures of-- So when they compared the transient transfection with the endogenous levels, that might give them some feeling for what levels, the levels of expression actually are. And of course, the way that people really want to attach things is using small things, whatever these lists of dyes are that we have. And what are the methods that you guys have learned about to attach these fluorophores. So instead of using a genetic fusion, which is probably. That's a really good way, except the protein, the green fluorescent protein is big. Green fluorescent protein is also a dimer. So people have spent a lot of time engineering green fluorescent protein to be a monomer. So the ones you buy commercially now are all monomers. That would add complexity to everything on top of this. How would you attach some of these things? So we know what the structures of these things are. AUDIENCE: You can do like a halo tag. JOANNE STUBBE: So you could do a halo tag. Have you talked-- we haven't talked about that. So give me another method. Give me a method we've talked about. AUDIENCE: A handle, [INAUDIBLE] handle to attach? JOANNE STUBBE: Yeah, but how would you do that? How do you attach these handles? You want to attach a fluorophore. OK, so it turns out that all of these things here, which you can't see. But these little aromatic things have been synthesized. So click it on. If they could have a settling there. But then it needs to be clicked to something. AUDIENCE: [INAUDIBLE]. JOANNE STUBBE: So you can't just. So how do you click it? AUDIENCE: [INAUDIBLE]. JOANNE STUBBE: So but is that easy to do inside the cell? No. And in mammalian cells, it's impossible. OK, so you can't use unnatural amino acids inside the cell. The technology is not there at this stage. So the question of how you attach this. You could make your. If you could make your protein outside the cell. You might be able to do that. But then you have the problem of getting your protein inside the cell. So getting a probe that's fluorescently, you're labeling the protein of interest is not easy. And Alice Ting's lab, again, has spent a lot of time, not that successfully. But using ligases that you can then incorporate into the cells that can then react with things you put onto your protein to attach fluorophores. But this is an area that's really important, because in my opinion, looking at regulation inside the cell, we don't really want to perturb. We don't want to be at very high levels. And we want to be able to see something to understand regulation. So I think. So anyhow, the issue is that we want to be as small as possible. We don't want to be Brad's lab. What is Brad's lab? Does he use these nanobodies that are antibodies? AUDIENCE: Like an [INAUDIBLE]? JOANNE STUBBE: No. They have all these. They have things called the nanobodies now. So I think they are like the little guys you make on your solid phase peptide synthesizer. But they are specific. They specifically bind to proteins. So there are only five examples that I've seen in the literature. So they act like antibodies. But they're-- huh? AUDIENCE: Like [INAUDIBLE],, like little-- JOANNE STUBBE: They're little tiny proteins that are maybe. I don't know. 50 amino acids that somehow, some guy at the University of Chicago-- not Kent-- developed these things. And they specifically. They act like an antibody. They can specifically interact with a protein of interest. And then you attach a green fluorescent protein onto it. So again, what you have something smaller. So because with these antibodies. What you see is the non-specific, right? I mean, we've seen that. And with fluorescence, that means you have fluorescence background in everything you do. So anyhow, I think we're not that. So that's just you're using fluorescence microscopy. This tells you why you're interested in fluorescence microscopy. And we'll just close here. And we're going to come back and talk about this in class. But this is sort of the example of the data that you need to think about. So what we hear is in the presence of purines, you don't see any of these little dots. You remove the purines. OK, so this is not so easy either, because the way we grow cells, we don't have defined media, right? I mean we're using. I don't know what you guys use now, but fetal calf serum or something. It's got all this stuff in it that we don't really know what it is. We don't use defined media. And apparently, when they-- the Marcotte paper-- when they were describing this, said it was not so easy to remove the purines. And the method they used to remove the purines also removed other stuff, OK. So you're stressing the cell. That was the take home message. So under those conditions, you see something different. OK, and so then they did another experiment, because they were worried about levels. Here, they are they have an antibody to the trifuctional protein. And so this is what they see under low purine conditions. Does this look like this? I don't know. So you can't tell by looking at one picture. OK, so you've got to do statistical analysis of all these things. So I think this sort of-- we'll come back and talk about this in class. But I think this is the first example where people are trying to look at this. The data is interesting. But we've already raised issues of what some of the problems are. And hopefully, you can think about more of the problems.
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MIT_508J_Biological_Chemistry_II_Spring_2016
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27_Metal_Ion_Homeostasis_3.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JOANNE STUBBE: At the end of the last lecture, we were talking about some generic properties of metals. And we were talking about the Irving-Williams series that was talked about, and will talk about in recitation this week, and this issue of how do you control metalation inside the cells when inherently copper is going to bind more tightly than zinc. And so if you have same amounts in solution, copper is always going to win, even though it clearly is dependent on the environment, the ligands, et cetera, which we didn't talk about. So the Irving-Williams series is an overview based on binding. Without that much details, it assumes octahedral environment plus 2 oxidation state, and it gives you an intuitive feeling for what the binding might be. And we focused on one creative solution that is controlling the location of the folding, so that you could pick up the appropriate metal and get correct metalation. In this case, on the cyanobacteria that they looked at in the periplasm, they have two proteins that have the same structures, virtually the same ligand spheres. Yet, they're able to selectively metalate. So and there are a few more things I want to mention about metals, in general. And the third thing is tuning metals. And the things that are going to tune the metals, we'll see when we look at iron specifically, but this is true of all metals, is the ligands, so the first coordination sphere, the geometry around the metals. So in the cases we were looking at the last time, with the Irving-Williams series, they were looking at an octahedral environment, but most of you know you can have tetrahedral environments, trigonal bipyramidal. You can have square planar. There are all kinds of environments, so you need to think about the geometry. We'll see later on, with transition metals, we need to look at the spin states. It can control the oxidation and reduction. So we'll look at spin state oxidation and reduction. And we need to also think about-- and this is something that a lot of chemists are now finally trying to build into their molecules-- it's not just the immediate environment around the metals, but the second coordination sphere. And this is hard to build in, from a chemist's point of view. It's not so hard to tune in from a protein environmental approach because you could have hydrogen bonding to an oxygen, which could then tweak the PKA of the ligand bound to the metal. So why do you have such a big protein? And why can't chemists recapitulate rate accelerations that are actually observed in enzymes? And it's simply because you need the whole protein. So it really is not just the first coordination sphere and the second. You can make mutations far removed from the active site, and actually affect, in this case, the properties of the metal, or the properties of whatever groups are actually involved in catalysis. And this is an example I took out of their very recent literature which I think is quite amazing. And I also think it's indicative of where chemistry is going in the next five years in the organometallic area. So this is a paper that was published by Yi Lu, who's at the University of Illinois. And what he was trying to do is he took a small little protein. The protein happens to be azurin that binds copper. It really doesn't matter what the protein is. But it does oxidation and reduction of copper. And what he was able to do is by changing the metal-- so he could either use copper or he could use nickel. And by changing the first and second coordination sphere around the metal by site-directed mutagenesis-- and at most, he made five mutants-- what he was able to do is tune the redox potential over 2 volts. So that is pretty astonishing, I think. And so he has structures. Here's a thing with two histidines and a methionine. He has structures of all of these species. And we would love to be able to put something-- chemists would love to be able to put things other than copper, or iron, or transition metals into proteins and control the redox potentials. And I think this is just the tip of the iceberg. I think this is an incredibly exciting result. And it also just allows me to show you that we've talked a little bit about iron sulfur cluster proteins, and we talked about that last time. We can go over 1.2 volts by changing the environment of the iron sulfur cluster. So you saw there were a number of flavors of iron sulfur clusters, and they all look pretty similar. But it's the protein in the environment that's tuning that. And the question that chemists are asking is, what are the basic principles that govern redox chemistry? Instead of having to select for something to change the redox potential, can we eventually go in and just rationally make a change? Especially now since you all know we can put in a natural amino acid. So we can put in ligands that aren't the normal repertoire for protein. So I think this is an incredibly exciting time because metal-based reactions offer huge numbers of opportunities. And I think we aren't going to be limited to the repertoire observed in biology. So and the last thing I wanted to talk about, in terms of generic systems, is that we had a cartoon and you looked a little bit at the diversity of metallocofactors. So we have a huge diversity. Where does the diversity come from? And in general in biology, there are biosynthetic pathways to make the metallocofactors. Also the organic cofactors, as well. So even something much simpler than the cofactor we looked at that converts nitrogen gas into ammonia-- remember, that had a lot of iron sulfurs and a molybdenum, and a carbide in the middle, and a citrate on one end. Very complex. Even in some of the simpler systems, it's likely there are going to be biosynthetic pathways to control all of this. And we're going to be focusing on, starting in today's lecture-- and the reason that I focused mostly thus far on iron sulfur clusters is we're going to see a major regulator of iron at the translational level are iron sulfur clusters. So I think looking at an iron sulfur cluster allows you to then maybe think about-- I think people don't think very much about this. What do you need to actually make one of these clusters? So this is a four iron, four sulfur cluster. Let's see. I got to put this up here a little bit more. So we have a cubane structure. And these are attached to proteins through-- irons are attached to the proteins through systanes. And this one is attached-- I'll draw it out here-- through a systane. And in many of the clusters now where chemistry happens, you're going to have binding through systanes. So each one of these guys is a systane. But you also have an iron that's not coordinated to a systane. So here you have any unique iron, and that is going to be key. This is iron. One of the first systems that people looked at involved-- maybe you remember back to the TCA cycle-- is a cytosolic aconitase. And people had really thought about iron sulfur clusters as only being involved in oxidation and reduction. And this one was the first example where citrate could bind here. So you had a unique iron allowing you to do chemistry. And now we know there are 100,000 types of enzymatic reactions that use iron sulfur clusters. So there's a couple of points I want to make because I think it's confusing, and we're going to use this when we think about the regulation. People write the oxidation species as plus 1 or plus 2. That's a typical oxidation of the iron sulfur clusters that we'll be dealing with. And so the question is, where does that number come from? And so what they do is ignore the cysteines. And so what you have is you have four sulfides, if I've drawn this right. One, two, three, four. So you have four sulfides, so that gives you a minus 8. And so then what does that tell you? You ignore the cysteines altogether. Then what does that tell you? To get a plus 2 state, what does that tell you about the oxidation state of the irons? So if we need a plus 2, what are the two common oxidation states of iron? AUDIENCE: Plus 2, plus 3. JOANNE STUBBE: Plus 2 and plus 3. So if we have plus 3 plus 3 plus 2 plus 2, that gives you plus 10, and that gives you the plus 2 oxidation. People get confused by that is the only reason I'm going through that. So if we want an overall plus 2, then you can have two iron 3s and two iron 2s. Now, it's not as simple as that because in some cases, the electrons can be delocalized, so you have 2 and 1/2 states. They're moving around a lot. And we're not going to talk about anything like that. You need to take bio and organic chemistry if you want to think about that. So if you look at this complex, you have a cube. It's got irons and it's got sulfides. That's it. What do you need to make something like this? What do you think you need? And this is true of all these cofactors. It's not just this one. This is the one we're going to actually worry about by the end of this lecture or the beginning of next lecture. Where does the iron come from? So you need something that can deliver the iron. Do you want iron 2 sitting around? We're going to talk about that more in module seven, but no, because iron 2 can undergo redox chemistry. So you need to deliver iron. What about the redox state of the irons? So I just told you they can be plus 2 or plus 3. How do you get that? How would you deliver the iron in the first place, given one of the rules we've already talked about? So you have iron inside the cell. Would you want it to be delivered in the plus 2 or the plus 3 state to a protein with no metal in it? AUDIENCE: Plus 3. JOANNE STUBBE: OK. Everybody agree with that? See, why do you say plus 3? AUDIENCE: Well, just because it won't undergo redox chemistry with the proteins. But I guess it is also insoluble, so then you have a problem of how do you go about delivering Fe 3 plus? JOANNE STUBBE: So you have, how do you deliver it. But you also have an additional factor, which you have to pay attention to, is the exchange rates of the ligands. So wherever it starts, it's got ligands on it, whether it's a protein or whether it's something else. And you have to go some kind of mechanism, associative or dissociative, to do exchange into the metals. So in general, we'll see one of the rules is that it's almost always iron 2 that's delivered. And so we need to control the redox state. And we'll see this big time when we look at humans, and how does iron get into cells. You've got toggle between plus 2 and plus 3 all of the time. And part of the rationale is related to ligand exchange. What else do we need to deliver? We need to deliver sulfide. Where does that come from? So let me just put down there's a paradoxal phosphate enzyme that can deliver sulfide from systane, for example. So where does it come from? What about these proteins? Can you think of another kind of protein that you've seen before, that if you have an apoprotein-- so the metal's not in there, and the metal doesn't go in during folding. And there is some where the metal goes in during folding, some where it doesn't go in during folding. What else might you have to do to prepare the active site to be able to bind the metal? What kind of a protein might you have to use? AUDIENCE: Some sort of chaperone [INAUDIBLE].. JOANNE STUBBE: Yeah, so some sort of chaperone. And you've all looked at the heat shock proteins. HSP 70, HSP 40. Almost all of these things have chaperone proteins with HSP 40, HSP 70-like activities. So you require some kind of chaperone. And this could be HSP 70, HSP 40. So I'm showing you this for an iron sulfur cluster because that's who we're going to focus on in the case of iron homeostasis. But this is true for many, many metal clusters that are generated. And this just shows you the complexity of it. So here are the pathways in bacteria for generic iron sulfur clusters. There are two pathways, one housekeeping, one under stress conditions. And you can see how many gene products you need. And they're involved in different kinds of proteins that do all of this stuff. We have things like scaffold proteins, so if you have to make something really complicated, you make it on a scaffold first and then you transfer and you transfer it. So ligand exchange becomes extremely important. What about this guy? This is the cluster for that huge formation of that beautiful cofactor you saw in the case of nitrogenase and how to get a carbide in there. So anyhow, I don't want to see say anything more about that. But this is all controlled, and I think this is something that when people want to study metallocofactors in the chemistry, they always encounter problems of how to assemble the cluster. Because if you heterologously express a protein, it's not a given that the cofactor is assembled correctly. So that's what I wanted to say about the generic properties of metals. And what I want to say now is just give you a very brief overview of metal homeostasis. So this is in general, and then we'll come back and talk about it specifically with iron. And so I'm going to say-- I'm not going to draw this out. I'm just going to say, see the PowerPoint. But I want to make a couple of points. And one of the first things that we need to think about that's not shown in this picture is any kind of regulation. How do we control whether you want metal or you don't want a metal? It's the same thing with cholesterol. How do you control whether we want it or we don't want it? So control of metal levels can happen transcriptionally, just like we saw with the SREBP. So in that picture that I have over there, I should have redrawn it. So this is all in the nucleus, and there is no nucleus in that cartoon, even though it's a eukaryotic cell. Transcription factor is they bind metals in some oxidation state. And they can be either activators or they can be repressors. So there are a lot of people still studying this. So these can be activation or repression. And I think almost all organisms use this in a productive way to control the metals. We're not going to talk about transcriptional regulation at all, but it's out there. It's a real challenge if you have weakly-bound metals, like you are learning about in recitation this week. The second thing we have in the cytosol which is also involved in regulation-- which is again not shown, so this happens in the cytosol-- is you have a piece of messenger RNA. And it turns out-- many of you probably know, but messenger RNA has a lot of secondary structures. So this is a secondary structure, which is a stem loop. So this is an RNA. And this is the five-prime end, and this is the three-prime end. And it turns out that if you're going to convert your messenger RNA into a protein, you want to use the ribosome and you want to have translation. And we're going to see that there are structures-- stem loop structures, in the case of iron-- that can bind proteins. And when they bind proteins, what happens is you can alter. So this could be a protein. Or you can also, at the three-prime end, bind proteins. And we're going to talk about this. So I'm not going to write all of this down. You're going to see this again. But you can control-- you can stop, for example, the translational process by putting a block over here. And we'll come back to this. Or you can stop messenger RNA degradation by putting a block there. So again, this is just another level of control that you guys haven't thought about. Has anybody seen control of RNA by little structures and small molecules binding? Have you seen that before in some class? AUDIENCE: Ribozymes? JOANNE STUBBE: Yes. Riboswitches. So ribozymes is the catalyst. All we're talking about here now is preventing the translational process. And so what you see, which was discovered by Breaker's lab, is that you have riboswitches. They're much more complicated than this. They have a much bigger structure. But they can bind things like adenosylcobalamin. They can bind flavins, all of which you might want to control just like you might want to control metal homeostasis. So metal homeostasis-- this is the regulation part. So this is regulation. And then if you look at the cartoon over there-- again, I'm not going to write down all the details, but we'll just walk through it. What do we have to be able to do? So we have to be able to take metals up into the cell. Do you think there's one uptake system? What do you think? Yeah. So what you would be surprised is if you want to take iron into an E. coli, they're at least iron 5 transporters. And the issue is, again, not many people have measured the specificity of all these things and the binding, and it all depends on the environment. So this is, I think, an incredibly important area that needs to have a lot of attention. We'll see in humans. We'll also see in bacteria. Bacteria are desperate for iron. They have an amazing number of ways to take iron into the cell. So once it gets inside, what happens to it? So the metal can come in. Here's a metal that's sort of free. It can form what people call a labile metal pool. It comes in maybe as aqueous or one of the ligands. They always have ligands. it can then form an interaction with all of your metabolites. So and again, you're thinking about KDs for all of these things. Where does it stay? it depends on the concentrations, and it depends on the proteins you have, and what the binding constants are. Once it gets in, there-- little things called metallochaperones. So it picks it up. A protein can pick up the metal, and then deliver it in some way to an apoprotein. So you have a protein with no metal on it at all. Can there be another protein that delivers the metal? And the answer is yes. And where this has been studied in detail in the transition metals is with copper. So they have very well-defined copper. And that's, again, a study in ligand exchange reactions because it binds here, but you don't want it to stay over here. It needs to move over here, and how do you control the transfer so things end up in the right place? So say you have an excess of metal. What happens? There are different ways of storing the metal. So storing the metal-- we have ways of storing zinc, copper, cadmium, and you'll see we have ways of storing iron. And so we want to be able to control that. And we'd like to be able to get it out of storage when we need it. And in many cases, depending on the metal, it's really important we store it because the metals can be toxic in the reduced state. And so you want to prevent toxicity to the cell. So the storage also plays a key role in eukaryotic systems. We'll see you also have iron transporters into organelles. It could be into the vacuole. It could be into the mitochondria. It could be into the lysosome. And so that's all controlled. Just like you have the importers, we're going to see you have exporters out of the organelles. And not shown here-- again, this is not a very good cartoon-- you could have exporters. You could control elevated levels-- in some cases, not in humans-- but by exporting the metals out of the cell. So there are many levels. Doesn't matter what the metal is. You see these kinds of mechanisms in all cases. And we'll focus on the case of iron in both humans and bacteria. So that's the end of the introductory lecture on metal homeostasis. And now what I want to do, we're going to focus on iron. And as I told you at the very beginning, we're initially going to look at a few pieces of information about iron specifically, rather than metals in general. But all the principles we talked about in general are applicable to iron. And then we're going to look at what happens in humans. And specifically, after you get the big picture-- how much iron do we have? Where do we get it? How much comes from the iron? Is it recycled? Et cetera. Just like we did with cholesterol. And we're going to be focusing on uptake into the cell, and we will see there are two ways of taking up iron into the cell in the plus 2 state. And there are iron 2 transporters. And we'll also see that there's a protein called transferrin that binds iron in the plus 3 state. So the oxidation state is distinct. You then have an iron 3 protein complex, and that gets taken up into the cell by receptor-mediated endocytosis by mechanisms quite similar to what you've seen with the LDL receptor we have, a receptor that recognizes this a little protein called transferrin. And we'll look at that. And then we're going to do in the end is talk about, how is iron regulated? We'll see there are a number of mechanisms that regulate everything iron homeostasis, and we're going to focus on one regulation at the translational level, like we were just talking about up here. So that's where we're going in the next couple of lectures. And so I just want to make a few points about iron. And so the first thing we're looking at is some general issues about iron, the properties of iron that we need to think about. So we're going to look at the properties. And what did we learn in the last lecture? We learned that iron is abundant. We know that 80% of the core of the Earth is iron. But we also learned that the crust of the earth-- the fourth predominant metal is iron. And so it's all over the place. We also learned it's unavailable because we move from an anaerobic into an oxygen world, and it becomes oxidized, and the solubility goes way down. So iron is abundant. And we know that we have many, many cofactors, but it's unavailable. And if you look, an example of this-- if you take iron 3 aquated at pH 7, what you see is the solubility is 10 to the minus 18th molar. So it's not very soluble. And again, this poses the problem, not for us, but for bacteria who are desperately trying to get iron, how do you get iron from the environment where it's insoluble? So what is one of nature's solutions that you've already discussed to obtaining iron? You talked about in the first half of the course. AUDIENCE: Siderophores? JOANNE STUBBE: Siderophores, yeah. So what do we know? So a solution for the bacteria and fungi is siderophores. And which siderophore-- do you remember which one you talked about in detail? AUDIENCE: Enterobactin. JOANNE STUBBE: Enterobactin, yeah. So they estimate that there are greater-- they have all kinds of structures. You saw a structure where you had a cyclical structure with some serines making ester linkages. Does anybody remember what the KD was for iron? This goes into today's recitation section-- today and yesterday's recitation section. How does it bind? What oxidation state does it bind in? So I'm not going to talk very much about siderophores, but we will see that in the next lecture after this one. So the iron, in general, binds in the plus 3 oxidation state. And the KD-- I don't remember what it is for enterobactin specifically. For some reason, the number of 10 to the minus 52 sticks in my mind. Is that correct? Or is it 10 to the minus 38? AUDIENCE: Well, it depends on pH [INAUDIBLE].. It's minus 52 or minus 49 recorded, but then it's in the minus 20s at pH 7. JOANNE STUBBE: So what's the one that's 10 to the minus 52? All right. How about this? So this is an interesting problem. You're going to be looking at this in class today. How do you measure that? Do you think that's easy? Do you think you're going to have any way of detecting things? So this is where you've got to be creative and think about what you're learning about in recitation. So these things-- the bottom line is, everything is dependent on the ligands and obviously on the pH. But they bind like a son of a gun. And that's because these bacteria need to get iron to survive. So I think that's pretty important. The second thing, again, I wanted to point out is there exists a diversity of iron cofactors. And you've seen these in the first few lectures. I've blown through a number of structures. But they're found in general ways. What is the one that you're all familiar with? Where do you see iron that you all think about? AUDIENCE: Heme. JOANNE STUBBE: Heme, right. Why do you think about it? Why do you know heme and not some of the other? So this is heme. This is my protoporphyrin IX. Actually I can draw the structure of [INAUDIBLE] but I'm not going to draw it. I'm good at drawing structures of organic molecules, but I'm not going to draw it. It's not relevant. But why do you know heme? AUDIENCE: It's [INAUDIBLE] It's easy to see. JOANNE STUBBE: It's easy to see. That's it. Why do we know so much about heme? Because it's easy to see. Its extinction coefficient is like over 100,000. So those are the ones that people saw immediately. You prick yourself. It's blue or it's red. Your blood is blue or it's red. So this has a high extinction coefficient. So everybody knows we reversibly bind oxygen, but hemes have an amazing diversity. Where have we seen heme before? We've seen it, if you remember, in cholesterol biosynthesis in the last 19 steps. We got to get rid of three methyl groups. All of those are heme enzymes which catalyze hydroxylation of unactivated carbon hydrogen bonds. So hemes can reversibly bind oxygen, but they can also do this really tough chemistry. And how did they do that? They're controlled by the environment around the heme. So why haven't we seen the other places where-- why don't we think about the other cofactors that involve? So we have non-heme iron, and this can be mono or dinuclear. And why don't we think about those? So no heme. So you have oxygen, nitrogen, histidine ligands, hydrazine ligands, perhaps, sulfur ligands. And you don't see this because they're not colored. In the plus 2 oxidation state, they're really hard to see. But for every heme-dependent system, there are mono and dinuclear non-heme iron systems that are probably more prevalent that can do the same chemistry. So we don't see them, that doesn't mean they're not there, and it doesn't mean they're not important. It's just they're much harder to study. So these things are very prevalent, and they do the same chemistry as hemes. So I showed you one where you could reversibly bind oxygen. Remember those little worms we saw in the slide that can reversibly bind oxygens, just like hemoglobin? You can hydroxylate unactivated carbon-hydrogen bonds. And where do you see that? Nowadays, one sees at all over the place because DNA and RNA modification is all mediated by, in many cases, alpha-Ketoglutarate, non-heme iron, dioxygenase. So I don't want to say any more about that, except they're extremely important, and they're hard to study. But we have really good tools to study all of these things. And then the other one, which we've just been talking about, which is the focus of the section on iron homeostasis, is iron sulfur. And so iron sulfur, for decades, was thought to be oxidation reduction electron transfer, which we talked about. But we now know, again, through these radical SAM proteins, there are just basically hundreds of complex radical reactions that we'd just be scratching the surface in learning. So this is also on there. Again, greater than 100,000 reactions, and these reactions are chemically interesting. So from a chemical point of view, the frontier, in my opinion, is not in the organic side at all. It's in the metal side. I think we don't have that much. You know, we have a little bit of intuition about what happens, but what we're seeing is things that we didn't expect to happen at all. We're seeing it in proteins, and then people are trying to figure out whether they can make the same things happen in solution and take advantage of all of this. So we have a diversity of metallocofactors. What about ligands? And almost anything can be a ligand. So it can be a protein, or it can be a small molecule metabolite. So you can have proteins. What are the ligands you might think you would find on iron? Tell me what the amino-- give me the one letter codes. AUDIENCE: D, E. JOANNE STUBBE: D, E. OK. Give me a D. Give me an E. What else? What else? Come on. Let's go. AUDIENCE: H. AUDIENCE: C. JOANNE STUBBE: H. They don't have to be in alphabetical order. AUDIENCE: C. AUDIENCE: C. AUDIENCE: C. JOANNE STUBBE: C. Try one more. You'll see it in a minute. AUDIENCE: [INAUDIBLE] for water. JOANNE STUBBE: Water? Yeah, water is wonderful. I'm not going to write down water. That's not an amino acid. So how about tyrosine? So what's amazing now is we even see things like arginine. That has a PKA of between 10 and 11. And the Drennan Lab has found several proteins where arginine appears to be-- and other people-- a ligand. So we have a diversity of ligands from the amino acid side chains. If you look at metabolites, we've already talked about citrate. Is that how you spell citrate? Citrate is in the TCE cycle. Alpha-Ketoglutarate-- that's also in the TCA cycle. I'm not going to draw this, but these are major players that mediate chemistry on iron-independent systems. So we have a diversity of these things. What about the geometry of all of these things? The geometry can be octahedral. It can be tetrahedral. It can be trigonal, bipyramidal, et cetera. Almost anything you can imagine, you can find it. Nature has figured out how to use this. In that paper by Yi Lu, where I told you they were changing the redox potential over 2 volts? One of the things they invoked was figuring out how to strain the metal to enhance the ability to reduce it to change its confirmation, which might be more favorable. So you have just really a huge number of things that you can deal with in these metals that I think allow the huge diversity of reactions that we're still unraveling, actually. So the other thing about iron is that, what are the oxidation states of iron? So we have the redox states. I can't remember what number I'm on. So we've just been going over and over again, these are the two common states. Iron 2, iron 3. And in the last lecture, we talked about other oxidation states. And it turns out if you look at the chemistry of what's going on, and you want to hydroxylate an unactivated carbon-hydrogen bond, you frequently see iron 4. And usually iron 4 is not sitting around in the test tube. It's activated, so it wants to get reduced. And that's what allows it to be able to do the chemistry. So unlike these guys, these are the workhorses you see over and over again. That's what we're going to see in iron homeostasis in general. But one also sees iron 1 or iron 0. And where does one see iron 1 or iron 0? Again, remember those ligands on the hydrogenase I showed you? Iron hydrogenase is what I showed you. There's an iron nickel hydrogenase. There's an iron-only hydrogenase. People are really interested in this for the energy problem. Hydrogenases are really, really fast. And what kind of ligands? Remember, we discussed this. And the ligands are going to control the chemistry. What kinds of ligands did you see? You saw a CO in cyanide. So that allows very different properties of the metals, in terms of the spin states you'll see, that allows different chemistry to happen. So let's just recall we have CO in cyanide ligands. So again, this is not the norm. But there are many systems where these have now been formed in unusual bacteria. We don't see these ligands, at least I don't think, in any eukaryotic systems. So the other thing that you need to think about with metals, if you get into it and start thinking about it-- and this is key to really, how do you know you have an iron 4? How do you know you have an iron 0? How do you study whether it's iron 2, iron 3? And that's different dependent on the spins states, because you have dielectrons associated with both the iron 2 state and the iron 3 state. So if you go back into freshman chemistry, or if you've had 5.03, you need to think about the spin states. And what we have is high spin and we have low spin states. And this is dependent on the ligands. So this is going to be ligand-dependent. And if you look at iron 2, you have six electrons in the d orbitals. If you look at iron 3, you have five electrons. And so if you look at the d orbitals in an octahedral field, depending on the ligands, the energetics of the d orbitals are quite distinct. Again, we're not going to talk about this in any detail. But what you can do, then, is if you want to put in five, depending on what the energy differences are, they might be all unpaired, or they could be paired. So this unpaired is high spin, and the paired is low spin. Do you think they're different spectroscopically? The answer is yes. And we have lots of physical biochemical tools that allow us to look at the differences between all of these things. And so this is, again, an active area of research. So the last thing I want to talk about in terms of iron properties are going to be key for us thinking about module seven. So there are two kinds of iron properties that you will be introduced to this semester. So we're looking at, now, the chemical diversity. And so one of the things is that, remember, when we're in an anaerobic world, we could use iron 2 because we didn't have to worry about any redox chemistry. Now in humans, we're in an aeorbic world, and we're faced with this issue of reduced metals and oxygen. So what you're going to see is, in the presence of oxygen-- and we're going to go into this in some detail in module seven. We're not going to spend a lot of time on it. But you learn a little bit about what we call reactive oxygen species. In the presence of oxygen, you can form iron 3, and you can form a molecule that looks like that. That's super oxide. And many people call this a reactive oxygen species. It depends on its environment whether they're reactive. So again, from a chemical perspective, I think thinking about what's possible is really the key in the kinetics, and what's around-- the concentrations, the kinetics, what's around. That's what has been missing in the reactive oxygen field. And for example, in the presence of protons-- we'll talk about this in detail, I'm not balancing the equations-- we can form another reactive thing that's considered a reactive oxygen species, which is hydrogen peroxide. And I'll show you that that really, in one or two cases in proteins, that can be very reactive. But in most cases, it's not all that reactive at all. And what we will see is iron 2 can react with hydrogen peroxide. Again, I'm not balancing my equations. We'll come back to this later on. But here's where we do form a reactive oxygen species. And this is hydroxide radical. And hydroxide radical, we'll see, can react with anything it hits. So this is really reactive. So all I'm pointing out here is you're forming species. They're all reactive. All molecules are reactive to a certain extent, and you need to put yourself into the context. So this is really a reactive oxygen species. And these guys are the focus of module seven where you'll be introduced to the fact that hydrogen peroxide can, in some way, be used to kill bacteria. We're going to see how that's done. But hydrogen peroxide is also now thought to be a second messenger in a signaling agent. So again, it's all about homeostasis. So with iron diversity, we've talked about hydroxylation in the cholesterol biosynthetic pathway. We're going to be focused now on this kind of redox chemistry. And so that's all I want to tell you about in terms of introduction to iron. All the properties we talked about-- wrap, exchange, the exchange reactions, et cetera-- you need to think about when you're thinking about iron, as well. So what I want to do now- so going away from these general ideas about how iron works, and we want to go into an overview in humans. And the first thing, in many of these cases, the pictures are really complicated. So I urge you to pull out your PowerPoint slides and look at them, and then just annotate them a little more. Because I mean otherwise, I won't get through. I'll spend all my time drawing the same pictures on the board. So one of the things we care about in this section is iron distribution. We cared about that with cholesterol, as well. So this was taken out of some textbook, and I assume it's correct. I don't really know that much about iron distribution in humans. But they say the average adult has 3 or 4 grams of iron. You know, I sympathize with you guys for not being able to read my writing. When I write something now, half the time, I can't read it either. So when I was young, my writing was beautiful, and my board work was beautiful. And it's gone because we don't write that much anymore. So anyhow, iron distribution. We have 3 to 4 grams. And we'll see that, in contrast with cholesterol, where we take a lot in from the diet and then we have to regulate everything-- the biosynthesis of this, the uptake of all of this-- we don't take that much in from the diet, and almost nothing goes out of us. So it's really the iron is recycled in general. So this is really different from what we saw with cholesterol. And from this one book, the numbers are all about the same. So I think they're OK. Where would you expect to see the most iron? AUDIENCE: Hemoglobin. JOANNE STUBBE: Hemoglobin, yeah. And that's-- so hemoglobin, 2.6 grams. Where else would you expect to see iron? How about myoglobin? Myoglobin takes the oxygen from the hemoglobin and delivers it to the tissue. Remember, we talked a little about metal storage. So these are metal storage proteins. There's a gram there. That's going to be found in the liver. It turns out that only 4% of the iron is found in proteins that catalyze these many reactions. So next time, we'll come back. We'll have a big overview of iron in humans. And we will also talk about regulation at the translational level.
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MIT_508J_Biological_Chemistry_II_Spring_2016
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20_Cholesterol_Biosynthesis_2.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JOANNE STUBBE: So we're talking about cholesterol homeostasis. And I said at the very beginning, the first two lectures are going to be focused on the terpenome and how you make cholesterol. And in the beginning in Monday's lecture, we had gotten through the first few steps in cholesterol biosynthesis, starting with acetyl CoA. And we'd gotten up to the position where we had condensed three molecules of acetyl CoA using Claisen and aldol reactions to form this molecule, hydroxymethyl glutaryl CoA. And at the end of the last lecture, we were talking about HMG CoA reductase, which is abbreviated in the Brown and Goldstein papers like that, which requires two molecules. This should be "NADPH." I announced that before all of the notes. "NADH" should be replaced with "NADPH." We're doing biosynthesis. We talked about the mechanism of how you go from this CoA analog to a double reduction to form an alcohol. And the question is, why is that interesting and important? And it's interesting and important because it's the major target of $30 billion drugs, the statins, which specifically target HMG CoA reductase. And so what I want to do-- I'm not going to spend a lot of time on this, but I want to say a little bit about this, given its central role in many people's health nowadays. So how do these analogs end up working? And so we have an intermediate in this process, which I drew out in detail last time. I'm not going to draw it out again. So we're going to go through two reductions. The first reduction forms a thiohemiacetal, which then kicks out CoA to form the aldehyde, which then gets reduced again. I'm not going to write that out. This is the first intermediate you see. And what we'll see is the inhibitors all look like this intermediate. So when they look like the normal substrate or an intermediate along the pathway, those are called "competitive inhibitors." So these are competitive inhibitors. If you don't remember what a competitive inhibitor is or how it's described, you might want to go back and look it up in Voet or whatever your basic biochemistry textbook is. And so what we'll see is all of the compounds that are actually used clinically look like this. And so if you look at this, they're not exactly the same. But it's proposed to be a model of this particular intermediate in the reaction. So this is the thiohemiacetal. But here, I've drawn a lactone rather than an acid. And in general-- whoops. In general, lactones are actually given as the drug, rather than the acid. And does anybody have any clue as to why that might be true? Why would you want to use this molecule, as opposed to the ring-opened species, which would look like this? AUDIENCE: Uptake. JOANNE STUBBE: Pardon me? AUDIENCE: I said, uptake issues. JOANNE STUBBE: Yeah. So it's uptake issues. So what happens is you give this-- and you'll see this is true also in the last section in purines and pyrimidines. How you deliver the drug, it needs to be able to get across the membrane in an efficient fashion. And so they give them lactone, but the lactone rapidly ring-opens inside the cell. So that's the analog. I don't know if you can see it, but it's the same analog I've drawn up there and I drew last time on the board. And then these are the drugs. And so the drug-- I've written this part looks like this part. So this is where the competitive part comes from. And what's down here? Down here is, remember, the CoA. And down here in the drugs is stuff. And the key to this stuff is hydrophobicity. And so the key to almost all drugs is hydrophobicity. So it can't be so hydrophobic that it's insoluble. But lots of times, you have pockets that you can't see in protein structures. It inserts itself in, and you get a lot of binding energy. So if you look at these, these, again, are on the PowerPoint presentation. What you see here, these are all the ring-opened here and ring-closed. And then if you look down here, what you see is all the stuff. And you see the stuff is dramatically different. You can look at it, but each company has tried to get its cut of the $30 billion market. And what I'm going to show you on the next slide is this molecule. So again, the key thing is what's common to all of these inhibitors is this moiety. And when you look at the structures, you can see this guy. And then this guy will have slightly different orientations within the structures. So if you look at the structures of HMG CoA reductase, what you can see-- and if you pull this up and stare at it, it looks like a mess. But it really isn't a mess. If you look at this, you're going to see in all the structures, here's the carboxylate. Here's a carboxylate. And this is the hydroxyl group. So that's the key part over here, the carboxylate in the hydroxyl group. There in the hemiacetal, the carboxylate's not there yet. So you can't put in the active species, or they turn over. So what you're doing is throwing in components to prevent turnover but to give you a feeling for where the different substrates bind. CoA is not attached to this moiety, but CoA is attached here. And I think what's unusual about this-- I don't know if you looked at any structures in the polyketide synthases. But you would think at the end of CoA, if you look at the structure, it has an adenine moiety on it. And you would think you would get a lot of binding energy onto this hydrophobic adenine moiety, which can also hydrogen-bond. And almost all structures to that part of the molecule never bind to the protein. It's stuck out in solution. So this is sort of typical. Why nature designed things this way, I don't know. She's got this huge-- she could have used a thiomethyl group, and the chemistry would have been exactly the same. But she uses this huge CoA moiety. And in most cases, you see the chain extended and the adenine on the outside. So if you look at that, the chemistry is going to come over here. And if you look at this yellow, yellow is sulfur. So that's where the sulfur would be connected to the hydroxymethyl glutarate. And then what do you see over here? Hopefully, you can see this. But this is another adenine ring. And then over here is the pyridine ring. And we went through the mechanism last time that transfers the hydride. So this green part is the redundant, and this part is a mimic of the hydroxymethyl glutaryl CoA. And in all cases-- we have hundreds of structures now-- what you will see if you look at the inhibitor binding is you have changes in conformation in this region. And the Km for binding of hydroxymethyl glutaryl CoA is 4 micromolar. But the KD for binding of the inhibitors is about a nanomolar. So you're gaining a lot by sticking this hydrophobic mess on. And what happens-- so here's, again, this hydrophobic mess. And what you can see, this is one of the analogs I had before. Again, hydroxymethyl glutarate-- the glutarate and the two carboxylates are there. And then they stick on something hydrophobic in this part of the molecule. And if you look at the conformation of these helices in this region-- and you really have to look at the three-dimensional structure to see something-- that's the region where you see changes in binding. So it's an induced fit mechanism of binding. And in fact, this induced fit occurs in all of the analogs that are looked at. And so if you look here-- and you can look at this. I would say take it home and spend some time looking at it. Again, this part of the molecule in all of these analogs is exactly the same. And what you had different is this hydrophobic mess in this part of the molecule and changes in this region of binding. And people are still working on it, trying to make-- usually, it's the first couple of drugs that make all the money. And if you're third or fourth, you don't make enough money. But there are lots of problems that keep coming up. And so people are still really heavily focused on trying to lower cholesterol levels. So HMG CoA reductase-- I told you last time, it's a huge protein, 880 amino acids. Half of it's stuck in the ER. You can cut off the-- the other half that is soluble is in the cytosol. And we're going to come back to this because this rate-limiting step plays a key role in sensing of cholesterol levels. So the end of lecture three and into lecture four, we're going to come back to HMG CoA reductase because of its central role in cholesterol homeostasis. So we're not here yet. Remember, the goal of the terpenome was to get to the building blocks. We still haven't gotten to the building blocks yet. What were the building blocks? Isopentenyl and dimethylallyl pyrophosphate. Remember, the common building block is an isoprene, but the isoprene is not the reactive species we needed to get it into some form where you can actually do chemistry on it. So our goal has been to get to IPP, and we are here. So the rest of the biosynthetic pathway to get to IPP is pretty straightforward. I'm not going to draw out the details at all. But we go from mevalonic acid. And so we use ATP and a kinase. And we use a second ATP and a kinase. And then we use a third ATP. And so if you look at the pathway here, what happens is you're phosphorylating the alcohol that we just created with HMG CoA reductase. So we're sticking a phosphate on. Another ATP sticks a second phosphate on. And in the end, we need to get to isopentenyl pyrophosphate. And so we have a third enzyme that is going to phosphorylate to facilitate, finally, conversion of the C6, three acetyl CoAs, into the C5 isopentenyl pyrophosphate. So if you look at what's going on in that reaction, we have a third ATP. And the ATP is used to phosphorylate the alcohol. So what we're doing basically is making it into a good leaving group. And we've got the two phosphates on there by the first two kinases. And so now what we want to do-- I forgot a methyl group here. And so now what we've done by phosphorylating this is we've activated this for a decarboxylative elimination reaction. And so now, where are we? We're now finally at isopentenyl pyrophosphate. So we've gotten to our C5. And the key thing, remember, is we started with acetyl CoA. During this reaction, we lose CO2. And that's why we've gone from a C6 to a C5 during this reaction. We lose that. And again, we see ATP used over and over again. And GTP, you see the same thing. It's used to make things into better leaving groups and facilitate the overall chemistry. So I'm not going to talk again about the details of any of these steps. The steps are all straightforward. You've seen these steps in primary metabolism with the role of ATP over and over again. But the key thing we want to talk about is the terpenome. And to get to the terpenome, we needed to get to isopentenyl pyrophosphate and dimethylallyl pyrophosphate so that we can look at the new way of forming carbon-carbon bonds with C5 units. So we've gotten through the first few steps. That's what I call the "initiation process." We started with acetyl CoA and got to IPP, dimethylallyl pyrophosphate. If you look at these hydrogens, hopefully, you know allylic hydrogens are moderately acidic. And there's an isomerase that can convert this molecule into this molecule. And so this is dimethylallyl pyrophosphate. And we're into the second part of the biosynthetic pathway for cholesterol. So we're through the initiation. We've got our building blocks. Now, what we want to do is do the elongation step. And so now we're going to use this. So IPP-- we're now going to look at the elongation reactions. And I guess I'll use a second board over here. And so let me do it over here, and then I'll do the next one. And so what we want to do then is take C5 plus C5. And we're going to look at this reaction in detail. And the enzyme that's going to do this is called FPP synthase, farnesyl pyrophosphate synthase. I'll write that down in a minute. And you form C10. And C10-- then this is the same enzyme. So it's also FPP synthase. FPP is the product, farnesyl pyrophosphate. And IPP gives us C15. So this is a major elongation reaction. And what I want you to see is that this C15, three C5s stuck together, is linear. And if you go back and you look at your notes from last time, we talked about isopernoids and terpenoids. And you can make linear molecules that can go from a couple of units, like geranyl, geranyl, C10-- sorry-- geranyl pyrophosphate. That's C10. That's called a "monoterpene." And you can add another C5, isopentenyl pyrophosphate. That's a C15. That's called a "sesquiterpene." "Sesqui" comes from 1 and 1/2. So what you'll see in the next thing, if you add another five, you have a C20. That's a diterpene. And if you go to a C30, that's a triterpene. You can google it, but the nomenclature's complicated. But that's where they come from is the different C5 units. So really, this chemistry is the basis for all the reactions in the terpenome. So what we're going to do is go through that chemistry. How do you form a new carbon-carbon bond using these building blocks? What are the general principles? And I showed you the hundreds of different kinds of natural products that you can find in humans and plants and bacteria all over the place. They play an incredibly important role in primary and secondary metabolism. And what we're going to look at is the general way that these carbon-carbon bonds are made. All right. So again, let me stress that this is linear. And you'll see that when we actually look at this. So FPP-- let me write this down. So it's farnesyl PP, pyrophosphate. We talked about this last time. It's a central player in many, many, many reactions, and it's a C15. So the farnesyl pyrophosphate synthase was the first enzyme characterized for parental transfer reactions, for these C5-forming reactions. It's been studied extremely extensively by Dale Poulter's lab at Utah, and it's served as a paradigm, really, for thinking about all of the biochemistry. Did any of you guys ever hear of Saul Winstein? No. It shows how old I am. Anyhow, Saul Winstein was a faculty member at UCLA many years ago, probably in the 1970s. But if you've taken 5.43, hopefully, they still talk about-- or what's the advanced physical organic chemistry course you guys take? Any of you'd had that? Anyhow, you've had-- have you heard about Saul? You've never heard of Saul-- bad, bad. Anyhow, he's the one that figured out how to think about classical and non-classical carbocations. And Dale Poulter worked from him. Dale Poulter moved into enzymatic reaction systems and really sort of unravelled how these things work. And the paradigm I'm going to give you-- every enzyme's different. But the paradigm I'm going to give you I really think came partially founded on the physical organic chemistry and from Dale's lab. So these are pretty important contributions. And what we'll see is this is called a "type I synthase." And if you read the assigned reading, you'll see there are type II synthases. So there's more than one structure of the enzymes involved in these systems. We're going to specifically focus in class on the type I synthase. And it's basically an alpha helical bundle. And I think on the next-- if I could remember what I have. Yeah. So this was taken out of the article you are supposed to read. And so this is FPP synthase. This is a monomer, but it's a dimer. And all I want you to see if you take the 30,000-foot view, there are five helices here. They're in red. If you look at this long helix, it's everywhere. Everything's a little bit juggled around. You can see you have a couple blues here and a couple of blues here. So they're structurally homologous to each other. And I think what's most remarkable about this-- so FPP synthase takes two C5s, makes a C10-- this is a C15. So it's a linear. Squalene synthase, which we'll look at in a minute, takes two C15s and makes a C30, the precursor to making the ring structure for cholesterol. But these two guys in the middle, which look-- and this is linear, as well. These two guys in the middle, which look remarkably similar-- actually, structurally, if you superimpose the structures, they're really similar-- form cyclic terpenes. So they form cyclic sesquiterpenes. I'll show you this in a minute. And they use FPP. So here, all of these enzymes use FPP. And they all look alike sort of from the 30,000-foot point of view. And the question is then, how do you control what the chemistry is in the active site? So they have homologous structure. So these are all structurally homologous. Another thing that you need to remember about these systems is that they have similar metal binding motifs. Now, if you look at the reaction of IPP, if you think about this, this isn't what PP looks like. Does everybody know what PP looks like? Hopefully, you've seen this over and over again. What would the metal be or metals? AUDIENCE: Magnesium. JOANNE STUBBE: Yeah. So whenever you have pyrophosphates or ATPs or GTPs, you always have magnesium. Magnesium plays a central role in everything in biology. And we can never look at magnesium because the ligands are fast-changing. It moves around all over the place. So it's hard to freeze out and understand the function of magnesium. But it turns out most of these proteins require three magnesiums. And as with many metal-based reactions, if you line things up, you really don't find that much sequence homology. But if you know where to look, you find sequence homology around where the metals bind. So what you see in the case of the linear farnesyl pyrophosphate, you see a DDXXD motif. And you find that in almost all of these enzymes. And if you go to the terpenoids-- so the non-linear ones-- you see a D. Again, I don't expect you to remember something like this. But I do expect you to remember that these metal motifs, once you know how to think about something, are actually very helpful in trying to define the function of an unknown open reading frame, if you know how to look. And more than 50% of all annotated genes code for proteins we have no idea what they do. So looking at these kinds of motifs can actually be quite informative. So then what you need to do to really understand what they're doing is dive in and look at where the metals bind, if you're lucky enough to be able to get a structure with the metals bound. So we have alpha helical motifs and metal binding motifs. And then the other kind of motif that I think is really interesting for the linear system-- so that's farnesyl pyrophosphate-- is how you control chain length. So FPP synthase-- that's what we're talking about-- is a dimer. And the metal binding motifs sit up there. So the metal binding motifs sit in the top, the way I've drawn this here. So you have metals. There are thought to be three metals. And then we're building C5, C5, C5. Where does the chain go? And what you'll see is there is a cone shape which migrates towards the bottom of the structure. I'll show you a picture of this in a minute. And so then the question is, what controls chain length? So if you end up looking at the structure, what you see is a phenylalanine. And the phenylalanine is a molecular doorstop. So the chain is extending, because we're going from C5 to C10 to C15. Why don't we go to C50? And I showed you in the first lecture dolichol and lipid II have C20s, C55s. How do you control the chain length? So that's an interesting question in polymer biochemistry. Here, we control it by a molecular doorstop. So if I replace the phenylalanine with an alanine, what might happen? Go. AUDIENCE: You'd have longer. JOANNE STUBBE: Yeah. So they made up to-- I can't remember. I haven't read the paper in a long time. But they can make C50-mers. So they can actually see, and they're not uniform. So that's a key thing. You want them to be uniform. So if you look-- I think in the next has a picture of this. Again, this is graphics from really quite some time ago now, 1996. So the picture's not very good. But you can see sort of the tunnel, and the metal binding sites are actually up there. And if you look at the structure, you can see the phenylalanine. So that's what we know sort of about the type I synthases. They're involved in making the C15 farnesyl pyrophosphate. So now we want to look at the chemistry, what's going on in the chemistry. And can we make a generalization about how this chemistry is used to put all C5 units together? So that's what I want to focus on next. Whoops. So what I'm going to now look at are the proposed mechanisms. And I'm not really going to go into much detail. I'm going to give you a generic overview of the things you need to remember if you encounter something like this. The first guess would be a mechanism similar to the one I'm proposing now, but then you have to look at it in more detail to figure out what's really going on. So what do we have? We have dimethylallyl pyrophosphate. And I have a cartoon for you to look at there, but I'm going to draw it differently than this cartoon. But you can just watch me because, again, the key thing is thinking about how you form the carbon-carbon bond and what's going on in these reactions. So we just looked at the pyrophosphate. And if you look over there, what do you need? You need to have a bunch of metals bound. And recently-- this is a fairly old paper. They have better papers. I think I took all the pictures out, because it's hard to see things without looking at it in detail. But in fact, the magnesiums are interacting with the pyrophosphate and adjacent to the pyrophosphate. And it's clear they play a key role in catalysis. But whether they move during the transformation, again, I think we just don't know that much at this stage. It's hard to trap it in an informative state, like it is with all crystallography. So here's dimethylallyl pyrophosphate. Here's isopentenyl pyrophosphate, the two guys we were after. And the first step in all of these reactions is ionization. So this is an unusual reaction in biochemistry. There are almost no examples of carbocation in biological transformations. This is one of the few places where you see this. So this is the ionization step. And all of the reactions we are going to be looking at involve ionization, but other kinds of chemistry can also happen that we're not going to discuss. So what have we generated? We generated an allylic cation. And what we also have is we lost pyrophosphate. And I'm being sloppy. I'm not drawing out how these are interacting with metals, but the charges are pretty much neutralized in some form that we don't know the details of. So you can't forget about the charges. And so we can just put down magnesium 3+. And then what we want to do, we want to make a carbon-carbon bond. Whoops. Let me get this right. If I make a mistake on the board-- like sometimes, I always get mixed up with four or five carbons-- raise your hand and say, you've got the wrong number of carbons. You tell me. You be the cops. So what we're going to do now is we're ready to form a carbon-carbon bond. And we're going to be forming a new carbocation. Hopefully, you remember from introductory chemistry that carbocations that are tertiary are more stable. And when you look at terpene types of chemistry, you see tertiary carbocations used over and over and over again. That being said, I'm putting brackets around this because despite the fact that I draw this intermediate, no one's ever seen it in the enzymatic reaction using the normal substrates. So you have to play games to study mechanism, just like you have to do in organic chemistry. So what happens now is you're set up to form the carbon-carbon bond, which has been the goal of what we've been trying to do in the first couple of lectures. And so what do you generate? You generate the new carbon-carbon bond, which is the skeleton for geranyl pyrophosphate. You generated a new carbocation, and it's a tertiary carbocation. And our pyrophosphate is still sitting in the active site. And so now what we're ready to do is we're going to form our C10, geranyl pyrophosphate. And we'll see one of the types of reactions that you see over and over again when you make carbon-carbon bonds is loss of a proton. And that gives you the C10, which is these two things stuck together, which is a monoterpene, which is called "geranyl pyrophosphate." So what's interesting about this-- and I think this is sort of something that's pretty general-- is the pyrophosphate in the active site. If you look in the active sites, they're amazingly hydrophobic. And the pyrophosphate in some way stereospecifically-- I haven't drawn the stereochemistry here-- removes the HR proton to generate the olefin. So what you've now generated is geranyl pyrophosphate. So here's C10, and this is geranyl pyrophosphate. Let me also put brackets around this intermediate. Again, we haven't seen this intermediate. And how do we know this is true? Because we know a lot from Winstein and Brown about carbocation chemistry. And people have been really creative in figuring out how to show that this model is in fact correct. Hopefully, I have C10 there. And so this is an intermediate because we're still going to go on. The enzyme doesn't stop at C10. It adds another isopentenyl pyrophosphate. So if you want to think about how nature might design that, if you look at this molecule and you look at this part of the molecule and replace it with an R group-- so we have an R here. What does this look like? It looks just like dimethylallyl pyrophosphate. But we need to put the R group somewhere. So in the case of FPP synthase, we're going down the tunnel. So we're getting it out of the way. But we're going to do the same chemistry that we just did over again, and we just replaced a methyl with an R group. So that's the basic chemistry. It's pretty straightforward, the only chemistry that I'm aware of in biological systems that involves carbocations. These are special carbocations. That is, they're, in general, stabilized. They're allylic. Or in many cases, they can be tertiary. So let's emphasize that. Again, if you don't remember your organic chemistry, you should go back and look up the sections on carbocations. So I told you the farnesyl pyrophosphate is sort of central to many things. And farnesyl, in this case-- I'm not going to draw out farnesyl pyrophosphate. The chemistry's the same. You can repeat it yourself. But here is our farnesyl pyrophosphate, but look what it can form. Remember, you saw all those smells. If you break a pine needle, you have pinene. What you see is this one intermediate can form all of these compounds. So the question is-- with an enzyme that looks just like farnesyl pyrophosphate in three-dimensional structure. So that's sort of amazing. And what you're doing here is taking a linear molecule. And in this particular case-- and I'm not going to talk about this slide in detail, but I will talk about one case in detail-- what you're now doing is getting it to do alternative chemistry. So how would you design the active site of your enzyme to end up doing that, to use the same chemistry, ionization? And then you have to do cyclization and loss of a proton or whatever. How does nature design all of this? So once we get through this set of lectures, I would suggest this would be something you could go back and practice on. How do we get to all these guys? I'm going to show you one example of that. I'm not going to go through this slide. It's way too complicated, but I think it shows you sort of the amazing diversity of the terpenome, using farnesyl pyrophosphate. So what I want to do is give you an overview of the rules. And then I'll go through one specific example. So let's make general mechanistic comments. And in the original, version of the PowerPoint, this slide wasn't in there. Anyhow, the first thing is you've already seen up here, and this is going to be common, is you lose a proton. So the first step is ionization. So ionization happens in almost all these reactions. There are exceptions to this, but most first steps are ionization. The second step can involve proton loss. And I'm going to write down what the steps are, and then we'll come back and look at a specific example. And we're going to see this in cholesterol. One can have with carbocations-- if you go back and you think about what you learned if you've had the second semester of organic. With carbocations, you can do hydride transfers. So that's a hydrogen with a pair of electrons. We can have hydride transfers. We're also going to see-- and both of these are key in cholesterol biosynthesis. We can have methyl anion transfers. And the other thing is these reactions all go stereospecifically. And that's one thing. If you become an enzymologist, you realize that's what's cool. That's why you have such big huge enzymes, so they can control the stereochemistry of everything. So they do everything with 100% EE. And they don't have to worry about it like chemists worry about it, but they pay a price. They have a big huge protein. The third thing-- and this is going to become important. It was just important in the slide I showed you previously-- we're going to see cyclizations. And cyclizations require, in general, protonation of an olefin-- I'll give you an example of that-- or protonation of an epoxide. So in some way, you're going to have to do some more chemistry to get your olefin. Everybody know what an epoxide is? So we're converting an olefin into an epoxide. We're going to protonate it, and then we're going to do cyclizations. And the third general type of reactions is water addition. So if you have a carbocation sitting around. You add water, bang, you have a reaction and form an alcohol. So the other generalizations I want to make-- so that's the chemistry. We're going to see this chemistry play out over and over again because I've selected examples of this for you to look at. But it's quite common. The second thing besides these mechanistic issues is, how do you distinguish between linear versus cyclic? And you've already seen the strategy with farnesyl pyrophosphate. You really sort of have a tiny little cavity where the IPP and the dimethylallyl pyrophosphate bind, and then you have a long tunnel. What do you have in the case of cyclic terpenes, which you saw in the previous slide to this one? And the key thing is the shape of the active site. And what you will see if you look at a lot of these active sites is, in general, they're very hydrophobic. Why is that true? So somehow, you've got to take care of the pyrophosphates. But they're very hydrophobic because we're dealing with these hydrocarbons, which are hydrophobic. So the question then is, can you take this farnesyl pyrophosphate and fold it? And folding it in different ways-- if we go back to the last-- whoops. If we go back to the last slide, if you look at it here, for example, and you ionize here to form a carbocation, you can have a cis or a trans carbocation. And that then can lead to further types of chemistry, where you form different kinds of ring structures. So it really is all about folding in the active site of the enzyme. So the active site is the key to determine which of these many kinds of things that can happen that if you did this in solution, you might actually get a mixture of all of these kinds of things. So the key then is hydrophobic and the shape of the active site. And then another key thing is I'm going to show you that in many of these reactions, you go through-- like we saw up there-- these carbocation intermediates. Well, there might be three different carbocation intermediates you could go through. How do you decide? How do you decide-- how did enzymes evolve to give you specific carbocation intermediates? How might you stabilize a carbocation intermediate? Anybody got any ideas? What would you expect to find in the active site then? I'm going to show you on the next slide, which is sort of a generic active site of a terpene that can cyclize. Any guesses? How would you stabilize a carbocation? AUDIENCE: Negative mixtures. JOANNE STUBBE: Yeah. So one way-- you might have an aspartate. Nature doesn't do that. So that might-- well, the problem is if you do that and you form a covalent bond, that's the end of your reaction. So So how you do this is I don't think we really totally understand it. But how else could you stabilize it? Anybody else? What did you learn about weak non-covalent interactions in biochemistry that could help us? Everybody hates waiting on covalent interactions, the key to everything-- key to everything in how enzymes function. AUDIENCE: You could just have something [INAUDIBLE] in general. JOANNE STUBBE: But electron-rich-- but that would be doing-- that's what she was suggesting. You have a carboxylate, an aspartate or a glutamate. Then you would form a bond, and then you would be stuck. So the way nature actually does this is she uses aromatics. And it was discovered maybe about 15 years ago that you can have an aromatic whatever. And you have some kind of a cation. So this is called a "pi cation interaction." Usually, the pi cation interactions are with metals. But here, we have a carbocation. So the model is that you might find in the active site tryptophans, tyrosines, phenylalanines. And so these become really key. And in fact, if you look at an active site-- so I don't even remember which enzyme this is. And somebody was trying to study something, and they have a small inhibitor in the active site. But you notice you don't have a long site where this chain can extend. What you've done is constrained the active site much more, and that shape is going to be key to the many different reactions you could have. And then if you look carefully, you can't really think about this. But you have phenylalanine, tyrosine, tryptophan, and another tyrosine in the active site. And that's what you see in many of these protein structures all over. Again, we have FPP synthase, which has this thing. And then we have these terpene cyclases, which have this thing. And each one of them is different. And so the difference is related to the shape. And it's proposed that this stabilizes this interaction. It's been challenging to show this chemically, but these interactions are worth quite a bit. These are also hard to measure, but it's something that was discovered and now has been actually widely observed. And the other thing I want to mention about these enzymes, which I think is interesting and distinct from other enzymes that you've encountered, is that, in general, they're really not very specific. So if you start looking at these-- look at this. How could you make one cation here versus the three others? If you start looking at how to get to these cyclized products, you say, how the heck did nature ever do that? There's no way you could guess at what the product would be, in my opinion. So what happens is these enzymes actually when you start looking-- we have good analytical methods-- are really promiscuous. So they might produce a predominant product, but they always produce a bunch-- 1%, 5%, sometimes even more-- of other products. And I think if you look at the chemistry that we've been talking about, basically, all of this sort of makes sense. So what I want to do now is give you an example of all of these reactions in one case. And this case, I guess I didn't write down the references. But I took it out of the literature. It's from David Christianson's lab. So here, we have farnesyl pyrophosphate, and here's the product we want to get to. So you'll have something like this on a problem set that I'm going to ask you, and it will be simple. I won't give you something that's so hard to see. But for me, lots of times, when you look at these rearrangements, it's easier if you make models. I don't know if anybody ever uses models anymore. I still use models, because you have to bend things in the right way to see what's possible and if the orbital's overlapping in the right way. You've got to really think about the stereochemistry. So what do we have here? So the first step is ionization. So we would form an allylic cation here. That's what we just did over here, which I hid. So that's what we just did over here. Oh, we didn't do it over here. Here-- over here, we formed this allylic cation. And once you do this, then they didn't show you that intermediate. They went on to the next step. So once you generate a cation there, they drew the conformation such that this thing could cyclize. But when you cyclize, you have electron deficiency at this carbon. So you have a second carbocation. This is not allylic, but it's a tertiary carbocation. So now the question is, what can happen? And again, you've got to keep your eye on what your goal is way down at the end. And you could probably draw more than one mechanism to get from A to B. And then you have to figure out experiments of how you would test it if you really care about that. So what happens here? You're losing a proton. And again, the pyrophosphate is acting as a general base catalyst. So that's exactly what happens in the case or what's proposed to happen in the case of farnesyl pyrophosphate. So you generate this species. Well, this might be stable. You might actually be able to isolate that as an intermediate along the reaction pathway. But we know in the end, we end up with two six-membered rings with this stereochemistry and with methyl groups in certain places. And so then you have to think about how can we get there. So remember that I told you that terpenoids do cyclizations. And one way they can do it is to protonate the olefin. So here, there might be a group in the active site. Maybe it's the phosphate that would help facilitate. You've just used it as a general base catalyst. Now, it's got a proton. It could now function as a general acid catalyst. You could protonate this position and now form two six-membered rings and a new carbocation. So in general, the nomenclature is when you draw these things, if you have a stick like that, that means you've got a methyl group. If you want to put a hydrogen there, you put a hydrogen on it. So if there's nothing there because CH3 takes up more room and they become very complicated to draw, the methyl group has methane. And the hydrogen, you put on. So you can distinguish one from the other. So now what happens is remember, one of the mechanisms I told you is hydride transfer. And again, I think looking at the stereochemistry of these systems helps see how this could happen. But these are all stereospecific. So you have hydride transfer from this position to this position. And when you have hydride, a hydrogen with a pair of electrons, what you have left is a new tertiary carbocation. And this new tertiary carbocation-- let me see what's going on-- is now-- in the end, we get a methyl group here. We have no methyl group there. Now, we have a CH3- group migrating. And that's the third method that I described. So the CH3- group migrates, giving you a new carbocation. And then the last step is, again, loss of a proton. So here's an example. This is a complex example, but there are 70,000 of these guys. So these are sort of the general rules. Nature has figured out how to make all these different kinds of natural products. So what I want to do now-- so those are the general overview of how these systems work. What am I doing? Oh, I'm sorry. I get so lost. Anyhow, I wanted to get through cholesterol. But next time, we'll come back. And in the very beginning, we're going to see how we take C15s to go to C30s and then how you cyclize this in the most, in my opinion, amazing reaction in biology-- other than ribonucleotide reductase, anyhow. See you next-- see you Friday.
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MIT_508J_Biological_Chemistry_II_Spring_2016
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9_Protein_Folding_2.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high- quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ELIZABETH NOLAN: OK, so we're going to get started. And we're going to continue on with folding. So we had some introduction last time about this module and thinking about in vitro versus in vivo studies. And where we're going to move on today is discussing molecular chaperones. And effectively, there'll be three case studies over the next two to two and a half lectures-- so trigger factor, GroEL/GroES, and DnaK/DnaJ. And so first is some background. We need to talk about what are molecular chaperones. And so effectively, these are proteins that influence protein folding within the cell. And they can do this by a variety of ways. So they can help to prevent aggregation and intermolecular interactions between polypeptides. They can facilitate folding by limiting conformational space and preventing side reactions. An important point to keep in mind throughout this is that these chaperone proteins bind to proteins transiently here. What are the types of processes they can assist in? A variety are listed here. And we see that it's quite broad. So they can help in de novo folding, so for instance, folding of a nascent polypeptide chain emerging from the ribosome. They can assist in refolding. So for instance, if proteins have unfolded or become aggregated because of stress, they can help here. They can assist in the assembly of oligomeric proteins in protein transport, and they also assist in proteolytic degradation here. And so we can classify these chaperones into three main groups depending on how they act, and those are listed here. So we can have holdases, foldases, and unfoldases. So something you might want to ask yourself as you see these different chaperone systems is to ask, what is the role? Is it one or multiple? So holdases help to stabilize non-native confirmations. So effectively, the chaperone will bind a polypeptide and a non-native confirmation and stabilize that for some period of time. Foldases assist in folding of a polypeptide to its native state. And unfoldases, as the name indicates, can help with unfolding proteins, so for instance, if a protein has misfolded and that needs to be undone, or maybe a protein needs to be extracted from some aggregate that's formed in in multiple proteins that's a problem. And so we're going to think about the chaperones in the cytoplasm in two main groups for the ones that interact with newly synthesized polypeptides. So first, we can think about trigger factor, which is a chaperone that's associated with the ribosome, as we'll see. So trigger factor is involved in co-translational folding, meaning the polypeptide is still associated with the ribosome and de novo folding. And then we'll examine some downstream cytosolic chaperones. So these are chaperones that do not bind to ribosome-- GroEL/GroES, DnaK/DnaJ. And just as a general overview of this molecular chaperone concept here-- so this is taken from the required reading-- effectively, what's shown in this scheme is a variety of different states a polypeptide can find itself in. So here we see a partially folded protein. This protein may form from an unfolded protein or maybe a native protein. We have an aggregate. And if we look down here, we're seeing the effect of some generalized chaperone. OK, so one important point to make from this is that the chaperone's not part of this final structure. It's just helping the polypeptide get to its native state. OK, and we can think about different rate constants, whether it be for folding or aggregation, chaperone binding Kon, chaperone dissociation Koff. So for instance, if we look here, we have a partially folded polypeptide. And imagine the chaperone binds that. Or maybe the chaperone binds an unfolded polypeptide. OK, it's going to act as a holdase or a foldase. And what can we see down here-- or an unfoldase-- what we see down here is an indication of an event that's driven by ATP hydrolysis. And so what we'll see and what's known is that many of these chaperones switch between low and high affinity states for some substrate polypeptide. And these low and high affinity states are somehow regulated by the ATP binding and hydrolysis. So here, for instance, we see that step. And imagine a Koff getting us back into this direction here, right? So you can begin to ask yourself questions like, under what conditions and terms of these rate constants is folding efficient? When would a chaperone act as a holdase? When would aggregation occur? So aggregation would occur if this Kagg is much greater than, say, for instance, Kon here to work systematically through this scheme. And here are just some points and words related to that scheme and things to think about from a broader picture. So in terms of the systems we're going to examine in the cytoplasm, this is the overview slide. And where we're going to begin in this overview is with the ribosome. And we see that in red here we have a nascent polypeptide chain emerging. OK, so what does this scheme indicate? What we see is that here is the player trigger factor, which is involved in co-translational folding. And we see that about 70% of nascent polypeptides interact with trigger factor. And these can arrive in a native conformation. We see there's two other systems here. So on the right, we have GroES and GroEL. So GroEL provides post-translational folding. We look and see about 10% to 15% of peptides in the cell interact with GroEL/GroES. And as we'll see, it provides this folding chamber on a protected space. We also see here that this system uses ATP. OK, and here we have another two players, DnaK and its co-chaperone DnaJ. And we see they're binding to some sort of polypeptide in a manner that's different than GroEL/GroES. OK, about 5% to 18% of polypeptides come into contact with these two players. We also see this system as ATP-dependent, and there's another player, GrpE, which we'll see is the nucleotide exchange factor needed here. OK, so we see that maybe there's some crosstalk here. And here we have some needed native polypeptides. So just some things to keep in mind-- it's important to think about concentrations and some approximate concentrations are listed here. If we're thinking about the ribosome DnaK/DnaJ, GroEL, and trigger factor. Just to note that many chaperones are also called Heat Shock Proteins, and this is because their expression increases with increased temperature or stress. So Hsp70, Hsp60, that's for heat shock protein. So where we're going to begin is with an overview of trigger factor. Yes? AUDIENCE: I wanted to ask. What do you mean by native and this protein exist in several conformations in this slide? ELIZABETH NOLAN: So proteins are dynamic, right? We know that. So native means a native fold, so a native state of this protein as opposed to the protein being unfolded if it's supposed to be globular or being some undesirable oligomer aggregate. So when this polypeptide comes off the ribosome, that's a linear sequence of amino acids. And it needs to adopt its appropriate conformation to do its job, OK? And as I said last time, we're not discussing natively unfolded proteins in the context of this class. AUDIENCE: So proteins can have many different native receptors in this slide? ELIZABETH NOLAN: Yes, they're dynamic. But there is going to be, like if you need a beta sheet, a domain that has beta sheets, for instance, that needs to fold and form there. So we can discuss further if you have more questions about that. But think about ubiquitin, for instance, from recitation week one. That had a very defined shape, right? So it's native fold from looking at that PDB file. AUDIENCE: OK, I think I just need to understand in that previous slide. When I talk about one protein here, right? 70% percent of the proteins-- ELIZABETH NOLAN: Yeah, this is thinking about all the proteins and all the peptides in the cell. AUDIENCE: OK, I thought it was one type of protein. ELIZABETH NOLAN: No, this is looking at proteins in broad terms. And so what we'll see as we move forward is that certain types of proteins interact with GroEL where others don't, right? Trigger factor interacts with many, many of them because it's associated with the ribosome, and the ribosome's synthesizing all polypeptide chains there. OK, so we're going to start with trigger factor. And the first thing just to be aware of when thinking about trigger factor and where it acts-- so we saw it sitting on top of the exit tunnel of the 50S in the prior slide-- is that there is a lot of things happening near the exit tunnel of the ribosome. OK, so here we have our 70S ribosome. Here's the polypeptide coming out. A few proteins are indicated. In addition to trigger factor, just be aware that there's other players here. OK, one of these is Signal Recognition Protocol, which I mentioned briefly last time. This is involved in delivering membrane proteins to their destination. We also have enzymes that do work here, whether it's an enzyme for removing the N-terminal methionine, enzymes for deformulation of that N-terminal methionine, et cetera. So somehow, trigger factor needs to work in the presence of these other constituents there. OK, so when we think about trigger factor, what you want to think about is a protein of about 50-kDa that's shaped like a dragon. OK, this is ATP-independent. So what I said earlier about low and high affinity states and switching between these states being driven by ATP binding and hydrolysis, that does not apply for trigger factor. It's the exception in what will be presented in this course. And it's associated with the ribosome. And what trigger factor does is it provides a folding cavity or cradle over the exit tunnel. And by doing so, it gives this emerging polypeptide a protected space to begin to fold, so reduction of intermolecular interactions. So if we take a look at trigger factor, one depiction is shown here. And as I said, think about a dragon. And it's actually described as having a head, arms, and a tail. And so we can think about this also in terms of N and C-terminus. The region of trigger factor that interacts with the ribosome and binds the ribosome is down here in the tail region. And so what trigger factor does and what's indicated by the cartoon you've already seen is that it associates with the translating ribosome with a one-to-one stoichiometry. So think about having one trigger factor over that exit tunnel here. And as you can see, these different domains also have some additional activities. And the main chaperone activity is attributed to the C-terminal region here I've color-coded. So what are some characteristics of trigger factor? If we look at the surface and consider where different types of amino acids are found, so whether they have polar or non-polar residues, that's depicted here. OK, what do we see in this depiction? Where are these residues? Is a given type of residue clustered in any one spot? Yeah, I see some heads shaking no. No, right. They're distributed all about. We see non-polar and polar residues distributed across the surface of trigger factor here. And so why might this be? What's thought is that, effectively, trigger factor uses its entire inner cavity-- and you'll see how that forms in a minute-- for substrate accommodation. And you can imagine that all of these different polypeptides emerging from the ribosome have different amino acid compositions, right? So this allows it to be relatively versatile from that perspective. If we take another look at structure and think about how trigger factor binds to the ribosome, what's found here, so we're looking at structures of trigger factor bound in orange and unbound in green to the ribosome. And the ribosome's omitted for clarity. There's a helix-loop-helix region that is involved in that interaction. And what's found is that when trigger factor is bound, it's quite dynamic. And it can swivel around this ribosome binding site by about 10 degrees in every direction. So why might that be important? One, this flexibility may allow it to accommodate many different polypeptides that are emerging from the ribosome. And it may also facilitate its coexistence with those other proteins and enzymes that are acting by the exit tunnel that we saw on the prior slide. OK, so here is just a model attempting to show the different ways and degrees to which trigger factor can move from various structural studies when attached to this ribosome here. So this N-terminal domain, I have protein L23 listed here. It also contacts the 23S ribosomal RNA and the protein L29. And what's found is that some salt bridge interaction is important. And we'll see that in a moment. Here, if we look at a depiction of trigger factor actually bound to the 50S, so here we have the 50S. Here's the exit tunnel. And here's our dragon-shaped molecule sitting. I like to say on top. I guess it's on bottom here. But anyhow, the polypeptide's coming out, and it has this cavity where it's protected from all of the other constituents in the cell. And here's just a rotated view showing it on top-- so tail region, head, and arms. So we have this cradle over the exit tunnel. It's giving a protecting space for folding of that nascent polypeptide chain. If we just look at a little more detail here, what do we see? So there's a salt bridge between an arginine residue, arginine 45 of trigger factor, and glutamate, glutamate 13 of the ribosomal L23 that forms a salt bridge. So in this depiction, we have trigger factor in red. We have L23 in green. OK, and so you need to be thinking about the amino acid side chains here, and that's something, too, Joanne and I want to stress a bit after recitation last week is, really, in this course, the importance of thinking back to the chemical structures and properties of the molecules that come up within this course-- so positive charge, negative charge, that interaction here for that. So what happens when a polypeptide is emerging from the exit tunnel and it encounters trigger factor? There's many possibilities. And as I said, trigger factor is dynamic. And an interesting point about this protein is that trigger factor can differentiate between vacant and translating ribosomes, OK, and so what's found from in vitro studies is that the binding affinity, which I'll describe as a dissociation constant which is 1 over the Ka of trigger factor for the ribosome varies by several orders of magnitude depending on whether or not the ribosome's translating. So we have the Kd measured on the order of 1 to 2 micromolar if the ribosome is vacant, and a Kd of about 40 to 70 nanomolar for a translating ribosome. OK, and just in recitation 10, we'll talk about binding studies more. But just if needed for review, if we're thinking about A plus B going to AB, we have Kon and Koff. And Kd is Koff over Kon, and Kd is 1 over the Ka here. So let's look at some aspects of a model for a trigger factor dynamics during translation. So as I said, it can differentiate the vacant and translating ribosomes. What's been found from in vitro studies is that the mean residence time on the ribosome is about 10 seconds. So what are the possibilities? One, trigger factor can bind to a vacant ribosome, and that's shown here. And it can bind to a translating ribosome, and it does this with higher affinity, so greater Kon. So what happens after trigger factor binds to a translating ribosome? What we see is that the nascent polypeptide chain is coming out. And in this cartoon, what's depicted is that it's beginning to fold in this protected region here made by the trigger factor cradle. And what we see from this point is that there's three possibilities. So if we look first on the left, what happens? Trigger factor dissociated from that polypeptide that's emerging from the ribosome. So recall these chaperones bind and release the polypeptides. In this case, it's left. There's some folding that's happened, and this peptide is still associated with the ribosome. So what might happen next? Maybe this polypeptide has the ability to reach its native state without the help of trigger factor anymore. So that's shown here. It's released, and it's folded. Maybe some other chaperones in the cytoplasm helped with that, but it's not shown here. Alternatively, maybe trigger factor binds again. So maybe this is one domain, and then somewhere else, there's some other region that needs some help with folding. And we see that here. So it can bind and release the same polypeptide more than once. What are our other options? So maybe trigger factor, after being here, remains bound to the ribosome, and the polypeptide is released. Or look what happens here. We have trigger factor bound. We see release of the polypeptide with trigger factor bound, or here we see that there's even two trigger factors bound to the same polypeptide emerging from the ribosome. And just thinking about this from the perspective of the number of different polypeptides that are synthesized by an organism, all different lengths, all different levels of complexity, it's not too surprising that there's various possibilities here. So again, if you're presented with data, you need to ask, what does the data say? And what type of particular aspect of this model does it support? Yeah? AUDIENCE: How often is the ribosome actually vacant? ELIZABETH NOLAN: How often is the ribosome vacant? Yeah, I don't know how often the ribosome is vacant. So in vivo, in your test tube, you can completely control that, which is what's going to give some of these data here. Joanne, do you know? No. Yeah, anybody know? I don't know, right? So does it make sense to have many vacant ribosomes? AUDIENCE: Are there more vacant ribosomes maybe like floating around than there are membranes bound [INAUDIBLE]? ELIZABETH NOLAN: So I think that's a can of worms we're not going to go down right now in terms of where the ribosome is here. So let's look at a functional cycle. This is just another depiction of a potential functional cycle where we have the ribosome bound to mRNA. There's a nascent chain. Here we see several trigger factors bound, and we see options. So either the native fold, or maybe there needs to be some work of downstream chaperones, right? And at some point, triggered factor will be dissociated, and it can come around and rebind again. So there is some evidence the formation of a trigger factor dimer when it is not with the ribosome. We don't need to worry about that detail too much for our thinking about what's happening here, because this is a one-to-one stoichiometry. So how is trigger factor influencing the folding process? If we think about foldase, unfoldase, and holdase, so these cartoons show that some folding is happening in that cradle, especially the ones we saw before, right? And that's perfectly reasonable that somehow trigger factor is allowing or accelerating productive co-translational folding of that polypeptide. So from that perspective, it would be a foldase. Is it possible that it's also a holdase? And could trigger factor, in certain cases, keep nascent chains unfolded? Maybe to help prevent premature folding that would be an error during polypeptide synthesis, that's another possibility. And they're not mutually exclusive, right? So again, it's a question of an individual system and looking at the data. So the behavior may depend on the circumstance in the polypeptide chain. Rebecca? AUDIENCE: I'm just curious. So when we're talking about it acting as a foldase, mechanistically, is the trigger factor physically interacting with and promoting a certain conformation? Or is it just providing a space where everything else is isolated? Or do we even know? ELIZABETH NOLAN: Yeah, so do we even know? So this is something we'll talk more about in the context of the chamber GroEL. But what are the possibilities? One is that trigger factor is just providing a safe place for this polypeptide to fold to its native conformation. Because recall last time, we discussed the primary sequence and how primary sequence can dictate the fold and what's thermodynamically most stable, right? But in the cell, the cell is very crowded, right? So trigger factor can protect this polypeptide from all those other constituents in the cell that might cause unwanted intermolecular interactions, for instance, and cause a different folding trajectory. Is it possible that the cavity wall of trigger factor could influence that energy landscape? So that's the other aspect of your question. Is it an Anfinsen cage, so just allowing folding? Or is it actually affecting the landscape? I don't know if we're suggesting that it influences the landscape, the energy landscape. But that doesn't mean that literature is not out there for that. So I think of it typically as a cradle. And as I said, we'll come back to this idea with GroEL where there have been studies and people arguing one over the other. OK, so with that, we're going to leave trigger factor and move to the macromolecular machine, GroEL/GroES. And so GroEL falls into a subset of chaperones that are called chaperonins. OK, and these are chaperones that are essential for viability in all tested cases. OK, so that tells you this machinery is really important for the cell and must be involved in folding of some important players here. So in terms of GroEL/GroES, what do we have? I can describe this as bullet-shaped. a bullet-shaped folding machine. And so GroEL is the chaperone, and GroES is the co-chaperone. And they work together. And so what we have if we draw this in cartoon form is we have GroES, and we can describe GroES as the lid of the folding chamber. And here we have GroEL. OK, and what GroEL is, this gives us cavities for folding. OK, and we can think of it like a barrel. And as drawn, we see two pieces here. And as we'll look further, we'll see that these are two heptameric rings. The ring that has the lid attached is called the cis ring. Or sorry-- the chamber or heptamer with the lid attached is cis, and the one below is trans. And this is huge. So this whole thing is on the order of 184 angstroms just to give some scale. So EL is the chaperonin, and ES is the co here. So what we'll do is look at the structural characteristics of GroEL and GroES individually and then think about function here. So for GroEL, what we have are to have to heptameric rings. OK, and so if we look from the top here, what we have are the seven subunits arranged in this ring. OK, and what we see is that there's an inner cavity that's about 45 angstroms in diameter, OK? And each subunit is about 60 kilodaltons, which is why this is called Hsp70. And so if we take a look in this structural depiction here, in the middle, this is the top view. OK, and the different subunits have been color-coded. They're all the same polypeptide. They're just differentiating them here so it's easy to see each one. And here's that inner cavity. If we look at the side view again, we need to consider a little more detail. OK, so each one of these is a 7-mer. OK, and each subunit of GroEL three domains that are organized A, I, E-- so apical domain, intermediate domain, and equatorial domain. And then if we look at this bottom ring here, they're organized like that. OK, so effectively, what we have is a back-to-back arrangement. OK, so we have 14 subunits and two back-to-back rings. And so if we take a look again at this depiction, what's been done is that in this top 7-mer ring, for one of the subunits, the three domains have been colored. OK, so we see that the apical domain is an orange, the intermediate domain in yellow, and this equatorial domain in red shown here. And this is one isolated subunit again with this color-coding here. So what happens when the lid binds? So we're currently looking at this as just GroEL the to heptamers. But we need to begin to think about GroEL with its lid. What happens when the lid binds is that there's a conformational change. OK, and so I'll just draw this, and then we'll look at the structure. OK, so imagine that this is one GroEL subunit of the 7-mer ring, OK? When GroES binds to that ring, what happens is that the GroEL subunits change from a closed conformation, which I'm kind of showing as closed, to something that's open. OK, so effectively, it's like opening up a hinge. OK, and so the consequence of this is that when the lid binds to this cis cavity, the size of the central cavity expands dramatically. So it basically doubles. And that's something that's not clearly indicated here. So we can modify the cartoon. OK, so let's take a look and then talk about why that's important. So here are two depictions where we have GroEL/GroES, and we can look at a GroEL subunit that does not have GroES bound, so with the trans ring. Or we can look at a GroEL subunit where the GroES lid is bound, so in the cis. OK, and so this is actual structural depiction of what I tried to indicate on the board here where we have closed and open. And so this opening is making this cis ring much larger in terms of its central cavity, OK? So these are major conformational changes and details of which are described here. But effectively, the two points to keep in mind is, one, that diameter and size of this central cavity doubles. And we can think about why that might be important in terms of accommodating a larger polypeptide as we get towards the functional cycle of this. And also, what we'll see as we move forward is that the distribution of hydrophobic and hydrophilic residues on the interior of this cavity changes dramatically when GroES binds here. So briefly, to look at GroES, what does that look like from a structural perspective? So GroES is also a heptamer. OK, each subunit is only about 10 kilodaltons. It's about 30 angstroms in height and about 75 angstroms across here. And what we see if we look at the structure of an individual GroES, so again, here, what we see is that there is a beta sheet region. And there is this region here that's described as a mobile loop. And if you look, the beta sheet region's on top, and this mobile loop is down where it binds to GroEL. OK, so effectively, when GroES docks onto GroEL, these mobile loops bind to hydrophobic peptide binding pockets that are on the top of this heptamer there. OK, here's another depiction. So you're seeing the beta sheet parts on top, and here are the mobile loops that can bind to peptide binding grooves of GroEL. AUDIENCE: So does that mean the inner cavity of the cis part of GroEL is always open I guess after GroES binds? And trans is always closed? Because it looks like just from the way we've drawn it. Does GroES bind to the other side also? ELIZABETH NOLAN: Yeah, so right, this is how we've drawn it. We've drawn it like a bullet. And so does GroES bind to the other side? And how do these two chambers function? OK, and as we move forward getting to the functional cycle, what we'll see is that both rings are functional, but they're functional at different points in the cycle. OK, so GroES, yes, can bind to either one, but it's this bullet type of shape that is considered to be functional. So you might ask, what about a football? If we stick another GroES on the bottom, we get a football-shaped species. And there are some in vitro studies that show a formation of a football with two GroEL rings and two GroES rings, but those are found at very high ATP concentrations. And so it's thought that they may not be significant, that they're a transient species effectively of unknown significance that at least in the test tube, you can form at very high ATP. OK, yeah? AUDIENCE: And then is the cis and trans, it's not predefined, just it depends on wherever the GroES makes it? ELIZABETH NOLAN: Right. It's going to depend on wherever the GroES is. OK, and what we'll see as we move forward is we need to think about also how ATP binds. And ATP binding will also happen in one or the other, depending at the point in the functional cycle here, OK? So we just want to get the structural aspects under control before we look at the functional cycle. So this is one last slide on the structure. And so I find this to be a really beautiful machine. Here we have the bullet-shaped two GroELs and one GroES. And here we have the different domains colored. And here what we have is a cutaway view to look at the interior of the chambers. And so we have the cis chamber on top, the trans chamber on bottom. And in the color-coding here for this cross-section, what we have in yellow are hydrophobic residues, and in cyan, hydrophilic residues. OK, and so what's important to do is take a look at the cis chamber and the trans chamber and ask, what's going on in the interior? And why might that be important? So what do we see comparing the distribution of yellow and cyan, or hydrophobic and hydrophilic? Lindsay? AUDIENCE: It's much more hydrophobic in the trans chamber. ELIZABETH NOLAN: Yeah, right. The trans chamber is much more hydrophobic in terms of that lining than the cis. So the cis chamber, as we'll see in a minute, is where the polypeptide will be folding. So a polypeptide will end up in the cis chamber, and the lid will be on top. So why might this be an important feature-- not only that we need this cavity size to grow, but we need a change in the lining to be more hydrophilic? AUDIENCE: Because if it's assisting folding, it's likely that the hydrophobic residues are more likely to be buried in the center the protein in the polypeptide. So you'd want to facilitate favoring the hydrophilic residues to be interacting on the outside of the protein? ELIZABETH NOLAN: Yeah, so often, that's right to think about. Where do we find different types of residues, say, in a protein with a complex fold? And typically, we think about hydrophobic residues on the interior and hydrophilic residues on the exterior. So for instance, there is a model of folding called hydrophobic collapse. And effectively, you have hydrophobic interactions, and then the rest of folding occurs, right? So you'd imagine there's a benefit to having a hydrophilic exterior if you want the exterior of the protein to be hydrophilic here. So what is the functional cycle? And in thinking about this, we need to think about ATPs and ATP hydrolysis. And what you'll find as you read is that often the model is drawn a bit differently depending on the paper you read. And that's because they're just some uncertainties out there. So don't get hung up on that. I have two different examples within the lecture slides here. OK, but if we just think about the functional cycle-- and I'll just draw a little bit, and then we'll go to the board. So imagine we have one GroEL here, OK? And as drawn here, there's no GroES. There's no peptide, and there's no ATP. So imagine some peptide comes in that needs to be folded by this machinery and ATP. OK, and these end up inside of the chamber. And then we can have our lid come in. OK, so now this polypeptide is in this protected cavity, and ATP can bind in the equatorial domain of GroEL. So each GroEL monomer will bind one ATP. So there's seven ATP bound in one heptamer if it's in the ATP-bound form, OK? And so let's take a look in a little more detail. So what do we see? And the thing to keep in mind, as I said before, is that both chambers are active and functional. They're just functional at different points within this overall cycle. OK, so if we begin here, what do we see? This top GroEL heptamer has no cap. What we see here is that we have the bottom GroEL bound to ADP and GroES. Some unfolded polypeptide comes along. It binds, so maybe there's some hydrophobic interaction between the top of GroEL and some region of this polypeptide. What do we see happening? The ATPs come in, so I indicated them together, there's some timing where there's questions. These ADPs from the bottom chamber are ejected. We see ATP binding, so there's seven-- one per subunit. The polypeptide binds, and here comes GroES. OK, and so once this polypeptide is encapsulated in this chamber, there's some residency time. And this is often quoted on the order of 10 seconds. Also note here. Look what happened at the bottom ring. GroES got ejected. OK, so with GroES binding here, there was ejection of GroES from the bottom and loss of these ADPs. OK, there's ATP hydrolysis during this time. The polypeptide is trying to find its fold. And then look what happens here. We see GroES coming into the bottom. Again, we have release of ADPs, release of GroES, and this polypeptide kicked out, which may or may not be in its native fold, OK? If we take a look showing this as a complete cycle here-- and again, I said before there can be some differences from depiction to depiction-- but here, we are seeing GroEL. We have the top one and the bottom heptamer. Here's some polypeptide that needs to be folded. It's initially grabbed by the top part of GroEL. ATP comes in. We have this ATP-bound form. Here comes GroES. The polypeptide gets pushed into this chamber, and now it's closed. We have ATPase activity, so ATP hydrolysis to give the ADP-bound form. OK, and then what happens here? OK, what we're seeing now, this bottom ring is becoming functional. ATP binds another polypeptide. OK, and then we have release of GroES in the polypeptide from the top chamber. OK, and then you can flip this and work around the cycle again. OK, so this is a case where we can think about the affinities of the ATP and the ADP-bound forms of GroEL and what that means GroES binding here, OK? And so the ADP-bound form of GroEL has a lower affinity for GroES than the ATP-bound form here for that. So each GroEL heptamer acts as a single functional unit, and both rings are active as shown here but in different points of the cycle. OK, and so the thinking is that ATP binding and hydrolysis drives uni-directional progression through this cycle. With that said, there's a lot of questions as to how. So what is it about this ATP binding and hydrolysis event that allows this work to happen? That's a question that I see as still pretty open. And so I'll close with that here now. I suggest to review this cycle before next time. And what we'll address on Wednesday is experiments that have been done to sort out what are the polypeptide substrates for GroEL/GroES. So we know they must be some important players given that these are essential for viability. What are they? And how is that determined?
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MIT_508J_Biological_Chemistry_II_Spring_2016
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7_Protein_Synthesis_6.txt
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The following content is provided under a Creative Commons license. Your support will help MIT Open CourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ELIZABETH NOLAN: Where we left off yesterday was beginning to discuss methods for unnatural amino acid incorporation into proteins using the ribosome. And the methodology that was introduced and where we need to continue today is the Schultz method of using the native ribosome to play some tricks and get unnatural amino acids into proteins. So we'll work through this further to show how the rest of the machinery was generated and then we'll consider some of the limitations and some of that came up in questions last time. And then we'll close with a discussion of one strategy that's a different strategy that uses actually an orthogonal ribosome, which is really, really neat here. So where we left off last time in terms of the Schultz Method was that we needed a unique codon for the unnatural amino acid, right? And a stop codon was reassigned. So TAG or Amber stop. And the other thing that we need that we'll discuss now involves the requirement of an orthogonal tRNA and aminoacyl-tRNA synthetase pair that can be used in this method. So the question is, where does this come from? So where do we get a tRNA and an aaRS that can be used for this unnatural amino acid of interest. And one way to think about this in terms of a search is to think about different tRNAs and aaRS from different organisms. And so what's found if tRNAs are compared between bacteria, eukaryotes, [INAUDIBLE],, there's evolutionary divergence. And so can that evolutionary divergence be taken advantage of? And effectively, is it possible to find some tRNA and its aminoacyl-tRNA synthetase from one organism that's orthogonal to the corresponding tRNA and aminoacyl-tRNA synthetase in the organism of interest. So effectively, if we want to use E. coli, we want to find a pair from another organism that's completely independent of the endogenous E. coli machinery. So what does this mean? A lot of trial and error was done to identify a pair from another organism. And where they ended up finding one is from a methanogen. So methanococcus jannaschii here. OK, and this initial pair was for tyrosine. And so, there's some features of this pair that are noteworthy to bring up. So first if we think about the aminoacyl-tRNA synthetase here. This one has an unusual feature. So when we discuss these aaRS, remember we discussed the mechanism and we also discussed what happens if a wrong amino acid is selected. And we learned that they have editing function and that there's editing domains. What also came up in those discussions is that we need to take every one of these enzymes as a case-by-case basis. And as it turns out, this particular enzyme does not have an editing domain. So then the thing to think about is, why would that be useful from the standpoint of incorporating an unnatural amino acid? So what's the benefit there? So what does this editing domain do? AUDIENCE: It's one less thing to have to fix if you're assuming that the editing domain would recognize and hydrolyze an unnatural amino acid that you put in even if you got the binding site to recognize it, or the first binding site to recognize it. ELIZABETH NOLAN: Exactly right. There's no deacylation happening, so no hydrolosis. So it's just more likely this unnatural amino acid can be a successful substrate and there's less engineering that has to be done in terms of modifying the enzyme here. So, another point in terms of-- they found that it does not acylate E. coli tRNA. OK, and that's important for trying to use this in E. coli here. What's the potential problem? So is this going to be specific for the unnatural amino acid of interest? No way, right? Unlikely at least, and depending on what type of a natural amino acid you're thinking about, it may definitely be a no way. So there's some experimental work to do to make this specific for the unnatural amino acid of interest, which means there has to be some mutagenesis and selection, which we're not going to talk about in detail here. So what about the tRNA? So we need to think about the tRNA structure right and how tRNAs interact with aaRS, right? And recall, we had in an earlier lecture one example of a crystal structure of this complex. And we saw there's many positions where they interact. What was known in this system here is that they figured-- OK, and also keep in mind, just backing up a minute, this tRNA, as we know, based on this nomenclature has an anticodon for tyrosine. So that's going to have to be mutated to be the anticodon on the Amber stop in order to use in this method. Right? So this is going to have to be mutated to give us tRNA(CUA) where this is indicating the anticodon here. So that mutation, we don't want that mutation to disrupt the interaction between the tRNA and aminoacyl-tRNA synthetase, right? And it turns out there were minimal interactions in that area for the native system. So the thinking was that these mutations could be tolerated here. So with that said, what is the potential problem with this tRNA? So if we want to put this tRNA into E. coli, it can't be recognized by any of the E. coli aaRS. So all of these recognition issues come up. So, here again with another example, where there needed to be some mutagenesis and selection to prevent interactions between this tRNA from the methanogen and the aaRS of E. coli. And effectively what they did is to pick 11 positions on the tRNA, which I'll just chart out. OK, so here's our tRNA. Here's our CUA anticodon here. And effectively these ends are positions where they randomized and did mutagenesis here. So they identified these 11 positions. OK? And these 11 positions do not interact with the aaRS here of the pair. So the idea is to maintain this interaction, but prevent any interaction of this tRNA with E. coli machinery here. So effectively, they used a method called directed evolution to do selection. And what might happen out of that, imagine you have some large pool of mutant tRNA, what might happen? So here OK, so the end result is that the tRNA might be non-functional. Right? So the mutation was not helpful. OK? It might be non-orthogonal, meaning that it's recognized by the endogenous E. coli machinery or it may be orthogonal here. OK, so recognized only OK? And so this is what needs to be selected for here. And so assays need to be done that allows these to be differentiated here. So the end result is an orthogonal pair. But the point is, you can't just take this pair from the other organism. It needs to be further modified. So where does that put us in terms of the cartoon we saw yesterday without some of these details? So here we have the tRNA that has this amber anticodon. So that's our orthogonal tRNA. We have an unnatural amino acid that's able to get into the organism of interest. And we have the orthogonal tRNA synthetase. So these give us this aminoacyl-tRNA with the unnatural amino acid. And then that can be incorporated into the A site of the ribosome. Right so this is a case where we have a plasma DNA. Here's the gene of interest in red. And somewhere in that gene, a stop codon has been placed to allow for incorporation of this unnatural amino acid somewhere within the polypeptide chain as shown here. So if we think about the scope of this methodology, where does this take us? So, it's quite broad. This type of work has been applied beyond E. coli, so in yeast and mammalian cells. At present, there is many, many different unnatural amino acids that can be incorporated and it's used by many labs. So that's something to keep in mind. If you're developing a new method, you'd really like other folks in other labs to be able to use your method. There's a lot of troubleshooting to do experimentally to get it up and running. And Joanne's a wonderful person to talk about that if you're curious for details. Just some amino acid scope, and you know what maybe we could do. So these are some earlier examples of unnatural amino acids that can be incorporated. And what are some of the neat things? If we look just here for example, there's an azide. Why might we want an azide? AUDIENCE: Click chemistry. ELIZABETH NOLAN: Yeah, click chemistry, right. Some chemistry that could be done after protein expression or maybe in a cell. Here we have a benzophenone. So they're useful for cross linking experiments and we'll likely talk about benzophenone cross linking in recitation five in detail. We see some sugars here. This is the damsel group. That's a fluorophore. So there's many possibilities here. Just looking at these molecules, what's something similar about all of them? We think about them compared to a native amino acid. AUDIENCE: I was just that they're small. ELIZABETH NOLAN: OK, they're quite small. AUDIENCE: It's a kind of modified tyrosine. It will have some sort of benzo group that's modified. ELIZABETH NOLAN: So they're sort of phenylalinine or tyrosine like, right? And does that make sense from the standpoint of using this machinery initially? Yes, and you can imagine looking for other pairs to put in other types of unnatural amino acids. So that's reflective there. Just as some further examples, So this is another example of using an unnatural amino acid that can be useful for click chemistry. And I picked this in part, for one, this unnatural amino acid looks very different than the ones we saw on the prior slide. But there's aminoacyl-tRNA synthetase and the tRNA for this alkyne. And so you can imagine expressing a protein with this at a specific location. And then after the fact, clicking on a molecule like this fluorophore here. So just thinking about this process, why maybe was this put on later rather than in the cell? AUDIENCE: Do you mean clicking it on or synthesizing-- or putting that whole thing on? ELIZABETH NOLAN: Yeah, as you can imagine someone could have thought, rather than clicking this on after the fact, why not just use this whole moiety here as the unnatural amino acid? So this fluorophore. AUDIENCE: It would be hard to find a synthetase to accommodate that fluorophore. ELIZABETH NOLAN: It might be hard to find a synthetase. AUDIENCE: Might just be too much [INAUDIBLE].. It might not fit physically within the ribosome machinery. ELIZABETH NOLAN: That could be. AUDIENCE: Are you asking why we would not put it in? ELIZABETH NOLAN: Yeah, I'm just asking you to think about this, right? So you know, what needs to be thought about, right? So here, there's still a chemical step after this unnatural amino acid was put in. And in this case, why might that be? Maybe it's a permeability issue. We don't know if that molecule readily taken up by the organism. Is it a size issue, that it's hard to get machinery to accommodate this type of molecule here. AUDIENCE: Is it folding? ELIZABETH NOLAN: Folding of-- AUDIENCE: If you had it, is the question like, you put it on the floor, which is after like it's been processed-- ELIZABETH NOLAN: Yeah, maybe it messed up. AUDIENCE: If it's a floppy thing, it might interfere with folding, or folding might interfere with its, like-- ELIZABETH NOLAN: Right, so can the the polypeptide breach its native confirmation with this perturbation. Just to think about. And here are just some examples of unnatural amino acids that can be used for fluorine NMR as was mentioned last time. OK. So this is all really exciting, but what is the limitation? And there is a major limitation of this methodology as it was first described. So the major limitation is that the efficiency is low. OK? And if we consider wanting to incorporate one unnatural amino acid into a polypeptide, so there is one amber stop codon put in, what was found is that about 20% to 30% efficiency for incorporation of one unnatural amino acid. OK? And then this value plummeted to less than 1% for incorporation of two unnatural amino acids. So imagine there's two amber stop codons put within the gene. So why is this? This is because what's observed is that only a small amount of the protein or polypeptide synthesized reaches completion. And so, how can we think about this? Imagine here, I'm just going to draw some polypeptide chain going from end to C terminus. Let's imagine this is 20 kilodaltons in size. And maybe this unnatural amino acid is being placed right in the middle. OK? So we want to put an unnatural amino acid here. OK? So, imagine you make your plasma DNA to do this. You have the tRNA and aaRS and the unnatural amino acid, and you do your expression, and then you take a look by SDS page, so gel electrophoresis, what you see? So imagine here we have 20, 10, five, so kilodaltons here. Right? So we have some molecular weight markers let's just say here. If you do this, say for the native sequence. So you haven't put in the stop codon. Imagine there's your protein. If we have the unnatural amino acid, what do we see? Something like this. So what does this tell you? First of all, why do you look at the native one? Effectively, you want some positive control because if you can't express your polypeptide with the native sequence, you're not going to want to go try to stick in an unnatural amino acid, right? There's a problem. So that's your positive control. So we see in this make believe gel, there's one band at 20 kilodaltons, which is the size of that. If that ever happens to you, you've had an instant gratification protein trap. So, what about this lane with the unnatural amino acid? What do we see and what does this data tell us? Lindsey. AUDIENCE: It's like early truncation. ELIZABETH NOLAN: Yeah, something happened. So early truncation, and why are you saying that? We see two bands. There's one band with the expected migration to about 20 kilodaltons. And then there's the second band that's coming up around 10 kilodaltons. And based on what I sketched out here, that unnatural amino acid is roughly around the 10 kilodalton mark. OK? What about the relative intensity of these bands? What do we see more of? AUDIENCE: The truncated one. ELIZABETH NOLAN: We see more of the truncated form. So what's going on? we? Need to think about our ribosome. And there's some polypeptide being made. And then what's coming here? We either have our tRNA with the unnatural amino acid or the release factor, right? So there's going to be competition for binding in the A site between the tRNA and the release factor. And so this is getting back to, I believe, Max's question from last time about using the stop codon, right? There's fundamentally a problem here. So, yeah. AUDIENCE: How does the release time test different for different stop codons? ELIZABETH NOLAN: Yes, so we discussed that I think in lecture four. So there's a release factor one and release factor two, and there's three different stop codons. So they both recognize one of the same and two different. And in this case release factor one recognizes the amber stop codon here. So we're not worrying about release factor two competing with this stop codon because it doesn't recognize this stop codon here. Right? So if release factor one goes in, we get premature termination. And that results in truncated protein. So is this a problem? And how much of a problem is it? AUDIENCE: So you're saying that the release factors comes in because it's recognizing the codon that's trying to-- or that originally was a stop codon? ELIZABETH NOLAN: Yeah, because the codon is still a stop codon. AUDIENCE: So in the wild type, it wasn't that we replaced-- sorry. So we replaced it with a stop. But the stop wasn't there originally. And so that's why you get the full 20 length, right? ELIZABETH NOLAN: Yes. So you have-- AUDIENCE: Yeah, ELIZABETH NOLAN: OK, continue. AUDIENCE: There was no stop before. Now there's a stop, but it's not supposed to act like a stop, right? ELIZABETH NOLAN: Right. AUDIENCE: So here it is acting like a stop kind of? ELIZABETH NOLAN: It depends what enters the A-site. So a stop codon is a stop codon. But the idea is that this tRNA has been tweaked to allow a tRNA to recognize the stop. But there's going to be competition because you have the tRNA that's going to deliver the unnatural amino acid. But you also have release factor around. So this release factor one is in the endogenous pool. So the question is, which one gets there and does the job? Right? And so what that gel is telling you is that there's a mixture. Right? Sometimes the tRNA will get there and translation continues until you get to the desired stop where you want translation to stop, in terms of stopping. Or if the release factor gets there, you get termination. So you get some truncated protein. AUDIENCE: How do you know, though, that you've got in the end-- that you actually got the unnatural amino acid in the 20 [INAUDIBLE] and not just the original? Is that fluorescing? ELIZABETH NOLAN: No. I mean, just imagine we're just looking at protein here-- I mean, where this came from. AUDIENCE: So it would look the same? ELIZABETH NOLAN: If you had a fluorescent amino acid, you'd see something-- no. Because if you didn't have the unnatural amino acid there, what else could be there? AUDIENCE: Just like the native. ELIZABETH NOLAN: But what native amino acid can be incorporated if there is a stop? AUDIENCE: Oh, because you also put in the mRNA. ELIZABETH NOLAN: Yeah. Right. So there has to be a stop. Now, that's also backtracking why you need to make sure everything's orthogonal. Because you don't want one of the endogenous amino aminoacyl-tRNA synthetases to put some endogenous amino acid on this tRNA. OK? So either full length with the unnatural amino acid or truncated because RF1 came along here. Right? So in terms of how much of a problem this is, in some respects, it depends on what you need and what you want to do. If you're over expressing protein and you can deal with this mixture and get enough full length, maybe that's OK. If you're doing an experiment in cells, you have to ask, what is the consequence of also having some truncated protein around? What does that mean for the cell? What does that mean for your measurement there for that? So how can we get around this problem of RF1? So effectively, we want to diminish RF1 mediated chain termination. What are some possibilities? Is that feasible? So we could do that and we could get a better yield. That would be great for protein overexpression. If we could minimize truncated phenotypes, that would be great for an experiment in cells. You don't need to worry about what this truncated protein might do. So what are possibilities? So can we knock down or knock out our RF1? AUDIENCE: [INAUDIBLE] ELIZABETH NOLAN: So this is a wonderful little story. I'll just tell a little bit about, we're not going to go into huge detail. But for quite some time, it was thought that RF1 was essential in E. coli. So a lot of experiments were done with E. coli K12 and even if you go look on a website about all the genes in E. coli K12, it will tell you RF1 is essential. But then in 2012, a paper came out in ACS Chemical Biology, where they were doing some work in a different strain of E. coli. So there's many different E. coli's. And K12 is a laboratory workhorse. And there's also strains, E. coli B. And they're also laboratory workhorses. So maybe many of you have used BL 21DE3 cells for protein expression. So this lab was working in E. coli B strain, and found that RF1 could be knocked out; that it's not essential. So then the question is, what's going on? And as it turns out, the essentiality of RF1 in E. coli turned out to be due to an issue with RF2. And in the K12 release factor 2 has a single point mutation that makes it less able to stop at certain stop codons. So when you had both of those together, it was deleterious. So RF1 can be knocked out. Would you want to do that? AUDIENCE: So, RF1 can be knocked out without RF2 or RF3, I don't remember. ELIZABETH NOLAN: Yeah, there are three release factors. RF3 is a GTPase. It's a little different. AUDIENCE: There's redundant kind of behavior. ELIZABETH NOLAN: There's some redundancy. And I mean, something too just to ask is, if you can knock it out and the cell is viable, viability is different than normal healthy cell. So those E. coli B, without RF1 will grow, but are they growing and replicating as well as the wild type? No. No. But is it good enough? And I think again, it comes down to asking what is it that you want to do? So maybe if you're over expressing protein and you're going to purify that, it's not such a big deal. But again, if you're looking at some cellular process, you're going to need to think about what's happening if RF1 can't terminate translation for, you know, its repertoire of proteins and genes there. There will be some consequence of that perturbation just to keep in mind. But there's certainly work going on with that now that it was found not to be essential. So in vitro translation, just something to think about. If you're going to work in a test tube, could you just do this outside of the cell? And then, the possibility we're going to discuss in closing is this one of a new ribosome, which I think is pretty cool. So, is it possible to have an orthogonal ribosome here to get around this problem? So effectively, can we make a new ribosome that only translates the message encoded in a plasmid that has the gene of interest where you want the unnatural amino acid to go? And so thinking about this in cartoon form, imagine we have E. Coli or some organism, and there's the native ribosome, and this native ribosome translates all of the native wild type mRNAs and gives synthesis of the proteome. But then imagine we can put in an orthogonal ribosome into this organism. And this orthogonal ribosome only recognizes an orthogonal mRNA, which means it only translates off of this orthogonal mRNA and only gives you synthesis of the protein you want with the unnatural amino acid. So how to think about doing this? Need to think back about the initiation process, and that mRNAs have a ribosome binding site. So effectively, it's necessary to engineer an mRNA that contains a ribosome binding site that will not direct translation by the endogenous ribosome, so some new ribosome binding site. OK? And then this orthogonal ribosome needs to be engineered such that it's specifically binding to the orthogonal mRNA. And it doesn't bind to the wild type mRNAs there. So no translation of the cellular message because this ribosome binding site and orthogonal ribosome are a match. OK? So a unique binding site. So in thinking about how to do this, you want to think about the ribosome structure. And we know that the 16S rRNA is involved in binding to the mRNA at the beginning of the initiation step. So what was done was to mutate the 16S and come up with an orthogonal ribosome. So this has been done. That's not a solution to the problem of RF1 terminating translation on its own. So then the next question is, if we can have just this orthogonal ribosome and orthogonal mRNA, can we improve that system to minimize RF1 mediated chain termination? So effectively what we want to do is prevent RF1 from binding to the A-site of the orthogonal ribosome. But it's still going to do its job for the endogenous ribosome here. So what needs to happen? And we'll go through the steps. This is just the schematic in cartoon form. So imagine we're starting with native ribosomes and orthogonal ribosomes. And we have tRNAs and RF1. And nothing has been done to this orthogonal ribosome so RF1 can still bind there. And so we want to have some evolution. So mutagenesis and selection of the orthogonal ribosome such that only the tRNA goes to the A-site. And RF1 only goes to the wild type ribosome. So there's other possibilities. One possibility I'll just throw out there is using rather than a triplet, a quadruplet codon there, which we won't talk about. There's more than one solution to the problem. But where we're going to focus on is work done to minimize RF1 and how to think about doing that from the standpoint of what we know of ribosome structure and the interactions. OK? So the name of this new O ribosome is Ribo-X and so what did they do? They started with the orthogonal ribosome. OK? And so, the first is that there needs to be some mutation to the ribosome, so libraries of mutants. There needs to be some selection process. So effectively, there is a requirement for of activity from the ribosome. and. When there's this, there needs to be some sequencing. Or identity determination, so where is the mutation? Here. And then with some mutant in hand, that looks like it's a good option, there needs to be assays to study it. And we're pretty much going to focus on step four. I'll briefly say something about steps one, two, and three here. So the first thing is, if we want to mutate this O ribosome, how do we think about designing a mutant library? And so, what we need to think about in this case because the goal is to minimize RF1 mediated chain termination and enhance tRNA getting into this A-site, we want to look at how the ribosome interacts with RF1, how it interacts with the tRNA, and also think about the mRNA there. And so, there's crystal structures available. There might be biochemical information available. But really to ask, where does that make sense to make mutations? And so if we think about the stop codon being recognized by the tRNA and RF1 in the A-site, somehow we want to mutate the ribosomal RNA in that region to give us the desired outcome. So what they did is mutate 16S rRNA to favor suppression of the amber stop codon by the tRNA. And crystal structures guided the library design. And so they looked at crystal structures where tRNAs are bound to the A-site or where RF1 is bound to the A-site. And from these, they selected seven different positions of the RNA and randomly mutated them. So that gives you some new mutants to study. Then there needs to be a selection process. So the mutant needs to be active. Some of these mutations might cause the ribosome to be inactive and that won't be very helpful. And so they developed an assay based on antibiotic resistance to select. And effectively, an enzyme that provides resistance to chloranfenicol, which is an antibiotic that blocks translation and was put under the control of the O ribosome. So you can imagine using antibiotic resistance as a selection there. And then the sequencing, once we've selected first some mutants, we have to ask where is the mutation? And so what they found after going through this work is that for Ribo-X it's only a double mutant in the 16 S rRNA. So two positions, U3531G and U534A. So these mutations in proved suppression of the amber stop codon, and I also point out these mutations are very unusual. So, at least at the time of this work, no sequenced natural ribosome had these two mutations here. And they're found in very few examples of sequenced RNase here. So, I mean, just to think about the ribosome's so huge and just two point mutations can make this change here. So what's seen in terms of some characterization. What do we need to ask in terms of characterization. Bless you. So something we want to ask about is fidelity here. So if we think about fidelity, one, we can ask, if we're using this to express some protein, what is the protein yield and how does that compare to the native ribosome? We want it to incorporate amino acids correctly with high fidelity and incorporation of the unnatural amino acid. So doesn't this incorporate amino acids? So that's the question we need to ask. OK? And then of course, we need to ask about amber stop codon suppression efficiency. And so, in thinking about this what is the point of comparison? So we can imagine in all of these comparing this new orthogonal ribosome Ribo-X to the starting orthogonal ribosome. Right? Here. So what are the experiments? So first let's think about protein yield. And I'll just say, I have a pet peeve when people don't report their protein yields in experimental. So if you're doing biochemistry, always think about doing that there. So what they did is an experiment where they made a plasmid. So we have an orthogonal DNA that will give orthogonal mRNA. So this gets transcribed... to give the orthogonal mRNA and then it gets translated by either the O ribosome... or Ribo-X. And the result of this is a fusion protein where we have a protein called GST, glutathione S-transferase, and then MBP, which is maltose-binding protein. And as we move forward, it will become clearer why they use this fusion. OK so just the first question is, how does the yield of protein compare? Are they doing a similar job or were these two mutations detrimental? So here's the result from this experiment one looking at protein yield. OK, so again we're looking at an SDS page gel that's being stained for the protein. And we see that this GST and BP fusion has a molecular weight of 71 kilodaltons, right? And what we see up here are the components that were in each of the experiments for each of the lanes. So here in this lane, we have no O ribosome, no Ribo-X but the plasmid was included. Here, we have the orthogonal ribosome in the plasmed, here Ribo-X in the plasmid. So what do we see? Pardon? AUDIENCE: Are they all the same yield? ELIZABETH NOLAN: Are they all the same yield? There's three lanes. AUDIENCE: [INAUDIBLE] ELIZABETH NOLAN: Yeah, so no orthogonal ribosome, no translation. And that's a good thing to see, right? That tells you that this orthogonal mRNA is not being translated by the endogenous ribosome. That's an important observation. And then I think what you meant to say, is that in these two lanes where we have either the starting orthogonal ribosome or Ribo-X, what we see is what appears to be a very similar amount of protein. So here, you know you assume and you look at the experimental, the same volumes were loaded, all of these things. We're getting the same amount of protein yield. So that's a great result here. So that's good news. What's the next experiment? And we'll close, I think on this experiment. So the next experiment is amino acid misincorporation. So again, what they did is they used this GST MBP fusion protein. And there's a linker region here. And in this linker region, they engineered a protease cleavage site here. So for thrombin here. And why did they do this to look at amino acid misincorporation, whether that's happening. Effectively, they took advantage of the fact that GST contains cystine, whereas maltose binding protein has no cystine. So their idea was let's use radio labeled cystine as a probe and monitor for radioactive cystine incorporation. So effectively, what can be done is that this can be expressed and purified in the presence of the radio labeled cystine. Thrombin can be used to cleave. And then you can look and ask is there radioactivity associated with GST? And we hope the answer is yes. And is there radioactive activity associated with maltose binding protein. And so where we'll begin on Friday is looking at the data from this assay. But until then, what I'd like you to think about is in terms of amino acid misincorporation, kind of strengths and limitations of this assay. Right so the choice of using one amino acid to take a look there. OK? So I'll see you Friday.
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MIT_508J_Biological_Chemistry_II_Spring_2016
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1_Introduction_to_Biological_Chemistry_II.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ELIZABETH NOLAN: Welcome to the class. We're going to discuss the themes that are going to basically permeate every topic and module we'll talk about here. And one of the central themes of this class is that we're interested in studying the cellular processes of life at a molecular level, right? So as biochemists and chemists, we're interested in this level of understanding. And what we see here is a cartoon depiction of the cell. And we see that there's many types of biomolecules in this environment. So what are our core themes for this year? First, we believe that life must be studied on a molecular level to truly understand it. And so we need to think about the cellular environment, both on a macroscopic scale, and on the molecular level. And this environment is complex, and it always needs to be considered, right? So as experimentalists in biochemistry, often we're doing experiments in aqueous buffer with proteins or some other biomolecule. How does that relate to a context like this one here where the environment is very different and much more complex? Something we'll see, especially in the first half, the first four modules of this course, is that in cells, complex processes are carried out by macromolecular machines and elaborate systems. And these systems are fascinating. You'll see that we know a lot, but as we learn more, there's more and more questions that come up, and more questions we need to address with that. In addition to these macromolecular machines, some additional themes for this course involve homeostasis and signaling. And these will be especially emphasized in the second half of the course when Professor Stubbe takes over there. So how do we think about homeostasis and signaling in these contexts? Something that will come up again and again is how, basically, understanding cellular processes at a molecular level, or the molecular features, can help explain mechanisms of human disease, as well as therapeutics. So an example we'll see in the early part of this lecture involves the ribosome. So many antibiotics target the ribosome. And by understanding ribosome structure and function, we can understand how these small molecule therapeutics work. Another example involves the proteasome which we'll hear about in the second half of the course. So there's therapeutics that target the proteasome, for instance, for cancer. And cholesterol biosynthesis will come up, and how does our understanding of cholesterol biosynthesis lead to ways to treat coronary disease? Something that JoAnne and I really like to think about day-to-day and convey to you in this course is the importance of experimental design, and choice of methods. So as scientists and experimentalists, how do we think about designing an experiment, because that design is really critical to the outcome, and what we can make of the data? And so throughout lectures and recitations, things to keep in mind, and that we'll reiterate, are that all techniques have inherent strengths and limitations. And so it's something we all need to keep in mind when we analyze data and think about how an experiment was done. And these systems we're going to look at in 5.08 are very complex. And what that means is that many different types of experimental method are needed in order to answer complex-- and sometimes not so complex-- questions. So one method alone just often isn't enough. We need insights from many different techniques and types of expertise. And so we look forward to informing you about different types of methods-- whether they be established and quite old or new-- that are important today. And as I alluded to before, something we have to keep in mind when doing biochemistry in the lab is that the test tube is very different from the cell. These environments are vastly different, and so we always need to think about how to relate data back to a cellular or physiological context. If you measure a dissociation constant of one micromolar, what does that mean in a cell versus one picomolar, for instance. Another point to make is that the hypothesis is a moving target. So we have the hypothesis, experiments are designed to test this hypothesis, and there's some outcome. Maybe that supports the hypothesis, maybe not. Or maybe there's some new insight from a related field that really changes how we think about something. So in many cases we're integrating data and insights that are quite new, and Professor Stubbe and I won't have all of the answers. And so that type of uncertainty is something that we aim for you all to gain some level of comfort with. So there's many complexities in primary data, often uncertainties. And that's just an aspect of this course. And scientists, it's something we grapple with every day in our own work. So we're introducing that to you here. And along those lines, just keep in mind, we know so much. And I think it's amazing, and-- if I step back and think about this for some of the systems we'll see-- actually overwhelming. And it's really due to dedicated efforts of many, many people over many, many years. But with that said, there are so many remaining unanswered questions, and we hope that you'll find inspiration in some of these questions as looking forward within this field. There. OK, so what about the cell and macromolecular crowding? Just to emphasize this point a bit more, here we have an E. coli. OK, so E. coli are laboratory workhorses for biochemists. They're fascinating, I love E. coli. But I just show you this simple E. coli cartoon and this depiction here to emphasize how crowded the cellular environment is. So we have an equal E. coli of about two microns long, and maybe half a micron wide, a volume of about a femtolitre. And if we think about the E. coli genome for a minute, it encodes about 4,000 proteins. That's a lot of proteins. And if we think about one E. coli cell of this small size, can just ask a simple question, how many ribosomes are there? So we all know the ribosomes are needed for polypeptide biosynthesis. How many ribosomes are packaged in one E. coli? Any guess? So, 10, 100, a million. AUDIENCE: Order of 1,000? ELIZABETH NOLAN: Pardon? AUDIENCE: Order of like, 1,000? ELIZABETH NOLAN: Yeah, let's say 1,000 times 15 or 20. So there's about 15,000 to 20,000 ribosomes in one E. coli cell. And as we'll see in Friday's lecture, the ribosome is very large. How did they all fit? And there's not only the ribosomes, but there's many, many other players, just as noted here in this cartoon. So you can think about what does that mean in terms of concentrations. We'll bring up concentrations of biomolecules in the cell throughout this course, and what does it mean having them packaged together so much here? So, very different than the test tube. Our goals, some of which I think have been communicated by me so far. But just to emphasize, we're interested in these macromolecular machines and chemical processes responsible for life. We hope by the end of this course, everyone gains an appreciation for the complexity of life, and our current understanding of the topics we present to you this spring. There's close links between basic fundamental research and medicine, and technology development as well. Understanding the experimental basis for understanding, methods and hypotheses. And what we think is something that we hope to achieve, and that you can bring to other places after this course is really to be able to knowledgeably and critically evaluate methods and results, especially primary data. And we also hope that we convince you that biological chemistry is really thought provoking and fun, and hope you all think that right now as well. So what are the actual topics we're going to cover? We organized this course into modules, and these modules are listed here. And different modules will have different numbers of lectures dedicated to them. But where we'll go between now and spring break-- I'll present to you during these weeks-- is that we're going to focus on the lifecycle of a protein for the first three modules. And many of you are familiar with aspects of this. We're going to present these topics, I think, a bit differently than what you've seen before. Again, very much from the standpoint of experimental methods and hypothesis testing. So we'll cover protein synthesis, doing a careful case study of the ribosome. We'll continue with protein folding. So asking the question, after the ribosome synthesizes a polypeptide chain, how does that polypeptide assemble into its native form? What happens when proteins are misfolded? And then we'll move into protein degradation, and we'll look at proteases and machines that are involved in proteolytic degradation. And where we'll close the first half is with module four, which is on synthases, or often called assembly-line enzymology. And this is a different type of template-driven polymerization that's involved in the synthesis of natural products. And then after spring break, Professor Stubbe will take over, and the focus will be on cellular processes that involve homeostasis, metabolism, and signaling. And so these topics will involve cholesterol biosynthesis, and a type of molecule called terpene. And so a third way to make a carbon-carbon bond will be introduced in this section. So you've heard about Claisen and Aldol condensations in prior biochemistry courses, this will be another route. And then, we both love metals and biology, so there's a whole field of bioinorganic chemistry, and it will be introduced to you here with iron homeostasis as a case study. And moving from here, and something quite related, involves reactive oxygen species. So I'm sure you've all heard about these somewhere, maybe in the news, maybe from your lab work. What are these reactive oxygen species? Are they all reactive? What kind of chemistry do they do in a cell? How do we study that here? And then, of course, we'll close with a section, a module on nucleotide and deoxynucleotide metabolism-- excuse me-- as well as regulation. And then an integration of course concepts. So we have a lot of exciting topics and exciting things to tell you about. In terms of level of understanding for this course, as I said, many of these systems are complex. We're going to look at huge macromolecular machines, and multi-step processes. This is a biochemistry course, and we are interested in molecular level, in addition to this big picture. And so things to keep in mind when thinking about structure. You need to think about the amino acids, and please review these if you're a bit rusty. So to know the side chains, PKAs, et cetera, that's all important to have in mind. What are the protein folds? What are the arrangements of these macromolecular assemblies, and how do we study that? In terms of reactivity, we'll see bond-breaking and bond-forming reactions. So again, we need to think about things like PKAs, nucleophiles, and electrophiles. If you need to brush up, organic chemistry textbook or biochemistry textbook is a good place to go. And then something to keep in mind is dynamics. So the macromolecular structures and enzymes and proteins we'll look at are dynamic. Often we only have a static picture or some number of static pictures. But there's conformational change, transient binding occurs, and we always need to think about kinetics. So these are things just to keep in mind when you're reading and questioning to yourself about any given system here. So what about experimental methods? This is just another topic to go over in this course overview. So there's many methods that come up in 5.08. And we don't expect that you have knowledge of any or all of these at the stage of starting the course. The difficulty that comes up is that we can't introduce all of these methods to you at once in a level of detail that's needed for everything we do. OK, so what will happen is that if methods come up in problem sets that haven't yet been addressed, we'll give you enough background information in the problems that material, such that you can think about the questions and answer them. And we'll let you know when a method comes up. You know, you'll hear this in recitation x, or we'll talk about it more in class. So right now, what I'd like to do is just go over a few of the methods that you're going to see multiple times. And the thing to keep in mind is that the context in which these methods are being used may differ, but the underlying principles are the same. And we choose methods that are being used today, and are important. Some of these were developed decades ago, some of these are very, very new, and hot off the press. So if it's an older paper, please don't brush it off as, like, oh, this is old. And so, you know, it's not new. We're all really excited by technology and everything here, but many of these older methods are robust, and used all the time here. So what are some methods and tools that we'll have under our belt? The first to point out are methods involved in macromolecular structure. So we care a lot about structure, because we need structural understanding to be able to comprehend how these systems work. And so one method we'll see a lot-- and you'll discuss in recitation this week-- is x-ray crystallography. And in addition, a method that will come up quite a bit-- and we'll see both of these in the initial discussions of the ribosome-- is electron microscopy. And another method to be aware of-- and if you're curious, talk to your TA, Shiva-- is NMR. OK, so NMR has a lot of applications here within biological chemistry, but we won't discuss that. What can go along with methods is bioinformatics. So how many of you have used BLAST? How many of you know what BLAST stands for? AUDIENCE: Basic Local Alignment Search Tool. ELIZABETH NOLAN: Yeah, Basic Local Alignment Search Tool. So what does this let you do? It lets you find regions of similarity between sequences, whether that's amino acid sequence, a nucleotide sequence. And you can use that information to make hypotheses and design experiments there. So that will come up. I have additional methods and possibilities. What about fluorescence? So how many of you have done an experiment that involves fluorescence, either in lab, or in your research? How many, did that involve a fluorescent protein? What about a small molecule? Yeah. That's fluorescent. So have you thought about why the protein was used, versus maybe why a small molecule, and what are inherent strengths and limitations or one or the other, depending what you want to do? So fluorescence is used in many, many different contexts. We can think about proteins like green fluorescent protein, we can think about using small molecules. And we like fluorescence because it allows us to see. We can get visual information. And so, where fluorescence will first come up in this class is with the ribosome. And in recitation week two, there'll be some discussion about using small molecule fluorophores to label tRNAs, and using fluorescence as a readout of steps in the translation process. And there's a lot of considerations and caveats to that. Do we have a pizza delivery? Thank goodness no. Often in this class, we get pizza deliveries for someone else. I didn't know if that's already starting. Yeah, yeah. We'll also see GFP being used in the proteasome section for degradation. So a folded protein has fluorescence, a degraded protein does not. What about kinetics? So what different types of kinetic studies can be done? So what do we all hear about in introductory biochemistry class? Pardon? AUDIENCE: [INAUDIBLE]. ELIZABETH NOLAN: Yeah, steady state kinetics, right? Turnover. So we have steady state, which I encourage you to review Michaelis-Mentin Kinetics here. And you'll also be introduced in the first weeks of this course, and especially recitation three-- so recitation two is going to build up to this-- pre-steady state kinetics. So here, you've heard about this in 5.07 or another course, introductory course. And we're looking at multiple turnover of an enzyme. And these experiments are set up with an excess of substrate, right, in order to afford conditions that allow multiple turnover. So there's formation of an enzyme substrate complex, and then there's product formation. So review as needed. So what about pre-steady state kinetics? How many of you are familiar with this method? Not so much. So what does the name suggest? Pardon? JOANNE STUBBE: So I'm deaf, you have to speak louder. ELIZABETH NOLAN: Yeah, we're both deaf. JOANNE STUBBE: I'm really deaf. So if you want to say something, so I can hear it. Speak up. AUDIENCE: Yeah, maybe observing single molecule by some spectroscopy. ELIZABETH NOLAN: Yeah, a single turnover, maybe, I think is what. If we're having multiple turnovers here in the steady state, right? If we're before the steady state, what does that mean, right? It means we're in the initial, really initial part of this reaction, where we're looking at a single turnover here. And how would you do this? Basically, you look with subs having limiting substrate rather than excess substrate. And this is just to give a little prelude in terms of thinking about experimental design. So here, look at the first moments of a reaction. So what type of time scale is that? AUDIENCE: Small. ELIZABETH NOLAN: Yeah, small. Maybe a millisecond time scale, compared to a timescale of seconds or minutes. So what does that mean? It means you need some different experimental setup. You can't do pre-steady state kinetics in the way we've done steady state kinetics, say in a lab class for instance. So you need a special apparatus. And what does it let you see? Here you're looking at multiple turnover, products forming. You know, here in the early stages, what can you see? Maybe intermediate formation. And why might that be important for thinking about mechanism? So those will come up in the first weeks of recitation. Another topic that will come up, and is something that you always need to think about, and relates to integrity of materials, is that of purification. So how are proteins purified. For studying the ribosome, how do we get ribosomes that are pure and are correct? Or what if you'd like to use a mutant ribosome? How does that get generated? So here, you can talk about ribosome or protein purification. And so, I'll present to you on ribosomes and mutant ribosomes in week four of recitation. And this topic more generally of proteins will come up in passing again and again. So how many of you have purified a protein? Many. How many of you used an affinity tag? So are they the answer to all problems? No. They can be a huge help, but they can also be problematic in one way or another, right? So with the ribosome we'll look at a case where there was really some elegant work done using an affinity tag approach to allow researchers to obtain new ribosomes. We'll also, though, talk about the limitations of that type of methodology, and the things you need to think about if you're doing protein biochemistry, and how a tag may affect your experiments and data there. In addition, to think about is assay development, and analytical methods. And so there will be many different types of assays that are presented in this course. And something just to think about-- how do you develop the right assay, and what are all the considerations? How do you know your assay is a good one for the question you want to address there? This is actually really complicated. And so there'll be some case studies that come up in the course, but just more broadly to think about. So often in lab classes, you may have an assay, but you might not be aware of all of the considerations that went into actually developing that assay such that it works. And then there's the analytical methods that are used, either for analyzing assay data or other data. And again, these have strengths and limitations. Just some that will come up, to present western blots and immunoprecipitation. So these methods involve antibodies, and so we need to think about the antibodies themselves here. Radioactivity. OK, how does this work? Why do biochemists like to use radioactivity and assay development? And how to think about this productively and correctly. So should you be afraid of iron-55, yes or no? How does that exposure compare to being in an airplane, for instance. Seriously, because there's a lot of fear associated with radioactivity that may or may not be well-founded, depending on what you're doing. And so this gives us a lot of sensitivity. And JoAnne will talk in week two of recitation about radioactivity, and designing experiments that use this as a read out. What else? So affinity measurements. OK, so dissociation constants, or affinity constants, how are these measured? When reading the literature, is the value a good one, or a not-so-good one, and how can you make that distinction? Mass spec and proteomics. So these will be in the later half of the class-- I believe recitations 11 and 12-- and many others. And we're introducing CRISPR this year, in the context of the cholesterol unit as well. So as I said, we can't take care of all of these methods immediately. We'll let you know when they're coming up, when you need to know more details about them as we go through the course here for that. So we can get started. And in the last few minutes, what I'll do is just give you a brief overview of the macromolecular machines we'll look at through modules one through three. And basically, what is the big picture? And then we're going to break that down into looking at individual components. So if we think about the lifecycle of a protein, basically, we'll fast forward to having mRNA from transcription of the genetic code. And then we have the macromolecular machine, the ribosome that allows for translation of this method message to give us a polypeptide chain. So some linear sequence of amino acids. And then what happens? We need to get from a polypeptide chain to some functional unit. And so there's a whole number of interesting players that are involved in protein folding. So we have folding, which is enabled by chaperones, is what we call these proteins that facilitate folding. And that's going to give us some structure that has some function here. And this protein has some lifetime in the cell. So at some time, for some reason, it will be time for this protein to get degraded. In which case, we need machinery that will facilitate the process to break down this folded protein into smaller fragments-- whether that be individual amino acids, or short polypeptide chains of seven to eight amino acids. So from here, we have degradation to give us small fragments. And the players here are proteases and chambers of doom, one of which is the proteasome. And actually, I forgot to mention there will be a second guest lecturer in recitation this year, Reuben Saunders, who is a senior, and does research in the Sauer Lab on one of these chambers of doom, called ClpXP. And so he'll present on single molecule methods, and fluorescence methods to study how this degradation chamber works. So that will be really exciting. He was a student in our course two years ago. So let's just take a look. We have the ribosome here. What are the structural features of this macromolecular machine, and how does it do its job? We'll look at a number of seminal studies that were done. And it is truly fascinating and incredible. What about protein folding? So look at this macromolecular machine here, GroEL, GroES, look at how big this is. So how does this chaperone allow some nascent polypeptide that's unfolded or partially folded to obtain its native structure? And there is many details in this depiction here that probably aren't apparent yet. But by the time we're done with module two, it will be there. Protein degradation. So here is just a cartoon-type depiction of a chamber of doom and its accessory protein from E. coli, ClpZ, ClpP. So look, we have a folded protein here, it's a beta barrel, our friend GFP that emits green light. And somehow, this protein gets threaded through ClpX, enters this chamber-- which has multiple protease active sites-- and that protein gets all degraded. So how does this work? How did ClpX and P work together to allow degradation of this condemned protein? And then finally, where I'll close is on something I think a little bit different for most everyone, and it's a type of template-driven polymerization involved in the synthesis of small molecules like penicillins and erythromycins. So these are antibiotics. So how do we get at molecules like these from simple amino acid precursors, or precursors like those you've seen in fatty acid biosynthesis here? And often, these are described as assembly lines. And something we'll just need to keep in mind in this unit is, are these proteins really acting like an assembly line, or is this just a way to help us think about the templates and what's going on here? So that's where we'll close. OK, so with that I'll finish up, and on Friday we'll begin with looking at the structure of the prokaryotic ribosome.
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MIT_508J_Biological_Chemistry_II_Spring_2016
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33_Reactive_Oxygen_Species_3.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JOANNE STUBBE: The last time, we were finishing the first part of the reactive oxygen species module, focused on how we as humans fight bacterial or viral infections using neutrophils, the white blood cells. And I introduced you to this cartoon. So here's our neutrophil, all of this blue stuff. It has an unusual-looking nucleus. Neutrophils have weird nuclei. They somehow sense the bacteria you saw, the bacteria getting chased by the neutrophil. And then somebody asked me this question. Somebody asked me this question last time about where the NOX2 proteins end up. And let me reiterate again that they can be found in multiple membranes. The predominant membrane in neutrophils is in these little vesicles within the cell, OK, but the bacteria out here, so they need to somehow engulf the bacteria. So they can also be found in the plasma membrane. And they need to engulf the bacteria to form phagosomes. And so here you see them again. So this is the phagosome with the NOX2 protein, with this predisposition where the NADPH is in the cytosol. OK? So you see that the location here, the NADPH is also in the cytosol. And so you need to think about where you're located. And we'll see in a few minutes that there are lots of different kinds of NOX proteins, and they all have different locations and all have different factors that control the regulation of all of these things. OK, so what I want to do today is start out by looking, talking about the NOX complex, what the chemistry is, because it's not only involved in killing bacteria, but we'll see-- and you've already seen in last week's recitation-- in the signaling process. It's the same protein. And so once the bacteria are engulfed, you now have a phagosome within the neutrophil. And here's the NOX complex. Remember, it's composed of two proteins, the 91-kilodalton glycoprotein and a 22-kilodalton protein, both membrane-bound. I drew those on the board last time. And the NADPH/NADP is in the cytosol. But all the chemistry is going to happen in the lumen of the phagosome. OK, so somehow the reducing equivalents from the NADPH need to be transferred into the lumen, where oxygen is going to be reduced to superoxide. So I want to talk a little bit about how that happens first, given what we know about the cofactors in NOX2 that are essential for this process. And then what we'll do is look a little bit at how the superoxide that's generated in the phagosome gets converted to hydrogen peroxide, and ultimately with another protein we're briefly going to discuss, a heme-dependent protein, myeloperoxidase, which uses chloride, can form hypochlorous acid. OK? So that's where we're going. This is a cartoon I drew on the board the last time, where you can see that we have a flavin domain, a flavin domain that's located in the cytosol. And then you have two hemes. And the unusual part about this system is that these hemes-- we've seen heme before with reversible binding of oxygen in hemoglobin-- they're both hexacoordinate. So there's no binding site for oxygen. So the chemistry is not happening by reversible binding of oxygen to the heme. In fact, you know, how close this is located to the surface, we don't have any structure of this, but somehow this reduction has to occur by an electron transfer process that probably occurs through the edge of the perforin system. So what you have here-- and I think this is an important teaching point, because I think there are only two ways inside the cell that you control all the redox balance. The redox balance within NADPH and flavins really play a central role in everything, so you really need to understand how these cofactors work, and how they're controlled. And so if you have a flavin-- and this is also written-- you don't have to write this down. This is written on the next handout, so you don't have to draw all this stuff out. OK, so I'm going to draw the business end of the flavin. OK, so this can be flavin adenine dinucleotide. So this is the oxidized state. And the two important places where redox chem-- redox chemistry, but look at flavin. If you don't know anything about heterocyclic chemistry, it's confusing. But in fact, we know a huge amount about flavin chemistry from studying model reactions in organic chemistry. Decades ago, Tom Bruce did that. And so the two places where the chemistry in general happen are either the inside-- so this is 1, 2, 3, 4. This is the N5-- or the C4A position, and this is the C4A position here. And I'm not going to go through this in detail but this is the oxidized form, and what we have is the reaction of this oxidized form with the reduced form of NADPH. And I'm not going to draw out the whole structure here either. But hopefully you all know now that NADPH and general works by hydride transfer, so it's almost always a two electron transfer. The one electron chemistry is really outside the realm where it would normally happen inside the cell. So basically the chemistry involves transfer of a hydride, a hydrogen with a pair of electrons to the N5 position. And so you go from the oxidized state to the reduced state. And so again I'm not going to draw out the whole flavin. The rest of the-- I guess I don't have a picture there but-- well, let me just show you where we're going. So the key is, which is interesting about this, we need to get across a membrane. So how do you get across a membrane? So the flavin domain is way over here, and we need to have two hemes. Remember you can do electron transfer over 10 to 15 axioms with very fast rate constants. Somehow these reducing equivalents from NADPH need to be transferred to the flavin. And the major function of the flavin inside the cell is to mediate two electron one electron chemistry. And here's an example that. The two-electron chemistry is being provided by the NADPH ass a hydride. But what we have to do is the hemes in this system are in the plus 3 oxidation state, so what we need to do is be able to convert-- ultimately we want to reduce oxygen to super oxides, so we need an electron. So that electron is coming from NADPH. So we need to have an electron transfer-- a single electron transfer to the heme because it only can-- iron can only be reduced by one electron. So we end up then with this system. So this is the reduced state of the flavin. And you can draw a resonance structure. This is deep-- you can draw all kinds of resonance structures with flavins. That's why I can they can do one-electron chemistry. So one-electron chemistry, you can make the one electron oxidized a reduced state depending on which state you're starting in and the electrons are delocalized. They turn out to be blue, or they turn out to be red depending on the prognation states. So what you can then do is-- let me just write that over here. So I'm just drawing a resonance structure of that so you can see-- let me not. And again let me just show you that that's there so you don't need to write that down. You don't need to write this down. It's all there. So just pay attention to me. So you're going to do an electron transfer to reduce an iron 3 to an iron 2. Then you've got another iron 3 because we got to get through this membrane. So that heme-- and again, this is where the redox potentials become critical-- can transfer an electron because now in the reduced state to the other heme so that becomes in the iron 2 state. So the one out here is in the iron 2 the state. Now it can transfer electron from oxygen to form super oxide. But at the same time during this process, we're transferring the two electrons that the Flavin received from the NADPH one at a time, so then we can repeat that process. And that's why you get the stochiometry of the overall reaction. You get two super oxides produced. So you have a resonance form of this. Let me see. I think I need to write over here. So anyhow you have a resonance form of this where you-- And now we're ready to do-- so this is the same as-- so this is a resonance form. These are the same structures. In the flavin, this is attached either to adenine or to a ribose biphosphate FMN versus FAD. And now what you're ready to do is you have the heme. And so the heme again is embedded in the membrane. And so now what you're doing is electron transfer. So you do an electron transfer reaction and you get this structure. So make a big dot for the radical. And now we've reduced one of the hemes to iron 2. And if you look at the redoxx potentials-- I haven't ever read these original papers-- but they're close to being matched in terms of redox potentials. I haven't read how they measured these kinds of things, but the system needs to be set up so that you can do transfer the electrons across the membrane and ultimately reduce the super oxide over here. And so now what happens so you've gotten to this stage. So now you have a semi quinone form of the flavin. this guy can then be re-oxidized by the next heme, generating the iron 2 form and regenerating the iron 3 form. So this guy then-- so let's-- to distinguish between them-- again I don't think this-- hopefully most of you were seeing this, but you have another one, so it donates an electron to covert the iron 3 form so we have an iron 3 form to the iron to form. And it itself becomes re-oxidized. And so now what's happening is you're set up to do another electron transfer where this is going to go to the iron 3, transfer it again, and in the end this guy is now in the lumen-- adjacent to the lumen. I don't know where it's located. We don't have a structure but this guy is probably through the hemage going to convert oxygen into super oxide. So is everybody following that? The main point here is you all hear about the flavin being the major mediator between one-electron chemistry and two-electron chemistry. Most of the time people don't draw out the details of this, but these things can all be observed spectroscopically because they're colored. Yeah. The second iron reduction happens from the semi [INAUDIBLE]? JOANNE STUBBE: The second iron-- yeah. Happens from the semi-- AUDIENCE: So you generate your process. JOANNE STUBBE: Right. Right. So you regenerate the oxidized form, so in the end over here, you go all the way through this. And again I haven't looked at the kinetics in the paper very carefully, but it very efficiently does this and shuttles the electron across. So you're doing the same thing. You're reorganizing the reduced form of the flavin, but you're doing it one electron at a time. So here's the key take home message. So all of this then happens in the phagosome. And so what you're generating then is superoxide, OK? Now what happens to superoxide? So superoxide could potentially do chemistry, but we talked about, last time, what are the properties of superoxide? It's not all that reactive, and frankly having read a lot of papers, I think we don't really understand all the details of how the bacteria die when they're engulfed by the phagosome-- but a key player in all in this overall process. And it's certainly not the only player, because you can actually wipe out my myloperoxidase, and you can still kill bacteria. So it's much more complicated than what I'm telling you, but a key player in when most people describe this is myloperoxidase, which is a heme protein. I'll show you that in a minute. But it turns out that these myloperoxidases exist in little granules. Just like you saw the little vesicle with the NOX2 it, that was predominantly sitting inside the neutrophil, you also have little vesicles. And the vesicles are stuffed with myloperoxidase. And somehow there's a signal, and the myloperoxidase then fuses with the phagosome. So this is a phagosome, and you have a huge amount of protein in there. It gets dumped into the phagosome, so you have a heme protein. And that one's dumped in here. Inside the cell, you generated a gradient, so there's some complicated independent reactions. You need to sort of neutralize the pH, which happens. But once you get inside the cell of the myloperoxidase, the protonation state is such that you can rapidly protinate superoxide to hydrogen peroxide, and we'll see that hydrogen peroxide reacts with myloperoxidase, which then reacts with chloride which is also present. In the hypochlorus acid, we'll see is a key player, and how can you tell that? Because if you isolate the proteins that come out of the phagosome, they're all chlorinated. So you generate-- if you go back, and you look at the little sheet I showed you about reactivity-- hypochlorus acid is really reactive. It's reactive, very reactive kinetically and also thermodynamically. OK, so the myloperoxidase-- so once we get-- so we've gotten our superoxide, so now we're in the phagosome. And now we want to look at myloperoxidase, and so that catalyzes the reaction of hydrogen peroxide and chloride to form hypochlorus acid. And this is myloperoxidase, and it's a heme dependent protein. And so the question is how does this work? And so what do we know about myloperoxidase? People have been studying this for decades, and you're going to see the chemistry is actually quite complicated. It's very important, so people are always trying to figure out the details of the chemistry. But the devil is in the kinetics in the environment of the phagosome, so it's not so easy to sort all this out. But if you look at the structure, number one, you see this is-- this is from an X-ray structure. It's bent, so it has an unusual structure. It has an axial ligand that's a histidine, and there's no second axial ligand. And it's covalently bound in two places. There are parts in the heme that are hanging off the protoporphyrin IX, where it's covalently bound to the protein, and the covalent attachment's distinct from most of the heme. So that's all you need to know, in terms of what we're going to be talking about. So you have a heme protein, and most of the time I don't talk about the heme systems. But I think the heme systems-- you guys ought to know something about hemes. We've talked about hemes with reversible oxygen binding. You've seen hemes in cholesterol biosynthesis. In many of the natural products, biosynthetic pathways, you have hemes that do hydroxylation reactions or epoxidation reactions. Hemes play a central role in many reactions inside the cell. And this one, the general reaction, I think is pretty straightforward to understand. So what you have-- and so again this straight line is protoporphyrin IX, so I'll just write protoporphyrin IX. And again, it's ligated to histidine, so this is part of the protein. And there's no second axial ligand. You take the superoxide which rapidly disproportionates, so this can be rapidly disproportionated in the presence of protons to form hydrogen peroxide and oxygen gas. And we've already gone through that reaction. And so now what happens is the peroxide is going to bind to the heme, and so this is the key to the reaction. So you lose a proton, and the oxygen binds to the heme. And you generate that species. Now this species-- and again, this is dependent-- this is where it becomes distinct when trying to think about all the chemistry that hemes can do. You need to look at the two axial ligands, and what the environment is around the ligand. So that's another thing. I've tried to stress how important these ligands are-- and the second coordination sphere around the system. So what happens now, in this system, is in some way the enzyme catalyzes heterolytic cleavage of the oxygen oxygen bond, and forms what is formerly an iron IV species. But it's not an iron five species, so we've lost a molecule water. Somehow this leaves as water, so we have some groups in the active site that can facilitate that cleavage. And this is formerly an iron V species, so we're using electrons from the iron porphyrin system to facilitate cleavage of that bond. Now how do we know this? We know this because we know a lot about the spectroscopy of hemes and of the iron in the hemes, and we can actually look at all of these intermediates. And so what does this mean here? What happens is-- remember your porphyrin. Well, you can see the porphyrin, but you have this. You have all these pyrroles. And so iron V is a hot oxygen. Nobody's ever seen the iron V in any of these systems, so what you see-- spectroscopically, you see an iron IV and one electron oxidized porphyrin ring. OK, so that's what this is. This is a one electron oxidized porphyrin ring. So now what do you want to do? In the normal reaction, what you want to do-- the one that forms hypochlorus acid is a two-electron reaction, and the chloride can come in-- and you're going to form hypochlorus acid. So the chloride comes in and attacks, and one of the electrons goes back to the iron. And the other goes back to the porphyrin. So these one-headed arrows means you're doing one-electron transfers, and so what you've generated then is hypochlorus acid. The pKa of this is 7.4. And then you've generated back your iron III porphyrin, so you're ready to start again. So you're doing a two-electron process, and we know that the driving force for this reaction is large. It's 1.16 volts, so this is a very favorable reaction. Now if any of you have thought about heme-dependent systems-- before, if you have an iron oxo, this is a hot oxidant. It's dying to be reduced. It can be two-electron reduced, but it can also be one-electron reduced, depending on what small molecules are around here. And myloperoxidase does both kinds of chemistry. The predominant chemistry-- so this is what you need to look at the rate constants for the reaction. The predominant chemistry is thought to be this, but I will show you a slide where we know it can catalyze a lot of other chemistries by one-electron transfers, as well. And what's happening in the phagosome with the cell, if you want to look at that, the [INAUDIBLE] review article I gave you spent a lot of time thinking about these kinds of reactions. And sort of think it's beyond the scope of what we need to talk about. So here what we have happening-- so here now I'm just going to-- I'll tell you what R is in a minute, but what we're going to do is have one-electron transfer. And so instead of reducing this two electrons at a time, we're going to do two one-electron transfers, OK? So the system has to be set up. You have to have the right RH. You need to know what the redox potentials of these are-- determined by the ligands. All of that stuff, you need to think about. So now we're doing one-electron chemistry, so this is another possibility. And this again depends on how much chloride you have around. It's a potent oxidant, so it can be rapidly reduced, depending on whether you have an RH around that's going to actually do the reduction. And so what you generate then-- is you reduce the cation radical, and you form an R dot. And then the next step, which you can do, is a second one-electron reduction, so you're doing two one-electron reductions. And this driving force is not as large-- 0.97 volts-- but again it's one-electron. And you're back then to iron III in water. I'm being sloppy about where the protons have come from. And you produced another R dot. So what could these R dots be? One of these R dots is ascorbate, vitamin C. So one of the RH's, which can-- the ascorbate can form radicals. I'm not going to go through this in any detail. Another RH could be tyrosines. So this is the amino acid tyrosine, and you form tyrosyl radicals. Has anybody ever seen that before in our department? Anybody seen use of this before? AUDIENCE: Apex? JOANNE STUBBE: Yeah. So, Apex. So this is the technology that is the basis. She doesn't use myloperoxidase, but King's Lab-- it uses a ascorbate peroxidase, which catalyzes a similar reaction. So your R dot then becomes a phenoxide radical. And that can then do for the chemistry. Anyhow, so what you're generating, most people believe the key bad player in all of this, but I'm just telling you it's more complicated than that, is the hypochlorus acid, which clearly gets formed, and can be evidenced by the chlorination reactions you see of all your proteins. So you can isolate chlorinated aromatics out of the phagosome. So if we go through here-- let me just give you this one first. So I've written this out for you in some detail. Again, it's two electrons, one electron, HOCl is two electrons. And that's thought to be the predominant species, but let me just tell you that-- so this is the one we're talking about in the handout. This is the major reaction, but you can do a lot of other reactions. And so what you need to look at to see if these are important-- just like with superoxide, you need to look at the kinetics of the reaction under the conditions where these molecules find themselves-- to figure out what's really going on, and how much these other pathways contribute. [INAUDIBLE] actually just-- there's a paper online in the annual reviews of biochemistry, where she talks about the neutrophils in a lot of detail, and the complexity of all this radical chemistry. So what I want you guys to take home from this is that we're working pretty hard with the NADPH oxidases, to engulf a bacteria. We're using reactive oxygen species, superoxide and hypochlorus acid to try to do in the bacteria inside the phagosome. So that's all I want to say about this section of reactive oxygen species, and now what I want to do is talk about something we've already talked about, because we've gone over this-- because we've gone over this in recitation. We spent a couple of recitations on the Carrol paper. And so I told you-- when I was introducing this, I gave you an outline of where we're going. We're going to reactive out of control versus controlled hydrogen peroxide superoxide production and signaling. So again, it's this thing all about-- it's all about homeostasis. Just need to get my act together here. So where are we going with this? This is the outline of where we're going to go, and so I'm not going to write the outline, because it will take me too long. And I really want to get-- you can read the outline on my PowerPoint. But what I want to do very briefly is give you an overview of how these reactive oxygen species are thought, in general, to play a role in signaling. A lot of people working on this-- there are many, many proteins involved. We're only looking at one of these proteins, the epidermal growth factor receptor, which we talked about in recitation. I also want to sort of give you an overview of the importance of cysteines in general in the proteome, and the role they play in this signaling process. We'll see. We are looking at one small modification, sulfenylation, but we'll see that there are many other modifications. And so I think one of the things for the future is figuring out, like the Carrol paper did, how biologically important are all these modifications that we can now identify because of the amazing power of Mass Spec and the creativity of chemists to figure out how to generate reagents. And so then I'll specifically introduce you very quickly to NOX and growth factors in NOX2-- the big family of NOX2. And then I'm going to talk about the general principles of signaling, what's required. And this is true of all signaling, not just with NOX, but I'll use NOX as an example. And then I will probably spend no time on this at all. The last part is how do we know all of this? We spent two recitations on this, so I'll tell you the key things I want you to remember. But you've now read papers. You've thought about this. And hopefully you can go back and think about it again, and it will all sort of now make more sense to you. OK, so where are we going? And so what I want you to see is sort of the big picture-- so again, the overview. And the overview now is not of the bad radicals that we're talking about, but they are still the bad radicals-- but controlling them in a way. So in the radical systems we're going to be looking at-- we're not going to be looking at all of them. But the signaling agents that we're going to be looking at-- so this is an overview of signaling. The signaling agents we're going to be looking at are superoxide, hydrogen peroxide, and NO. And we've already talked about the fact that we're not discussing NO, but in our department in biomedical engineering, Tannibaum's lab has been a major player in figuring out how to look at the modifications of cysteines by NO. It's not by NO. It's by a metabolite of NO that then reacts with the cysteines. So this is a very-- and he does that by Mass Spec, so this is a very active area of research. So now we're looking at signaling, not bad stuff. And one of the things that-- where have we seen this before? We've seen, although I don't think we realized it-- we were looking at iron homeostasis. This is what happens when you get up at 4:00 in the morning. OK, homeostasis. Homeostasis. And we have two proteins, the iron responsive binding protein one and two. And what do we know about iron responsive binding protein one? It has an iron sulfur cluster. And remember it has to go from the apostate to the iron cluster state, so we have IRP 1, and we go from the apo to the 4 iron 4 sulfur cluster. And it's believed, but it has not been very well studied, that this can be a sensor of oxidative stress, and so both NO and reactive oxygen species, such as hydrogen peroxide, can cause the metal center to be destroyed. And somehow you get to the apo state, and that's an active area of research. And then we know what the signals are. You have translational control by iron responsive elements at the five prime or the three prime end. The same thing happens with IRP 2. So I'm just trying to tie things together, but does anybody remember what the sensor was with IRP 2? Anybody remember what it did? AUDIENCE: Ubiquitin ligase? JOANNE STUBBE: The what? AUDIENCE: Is it ubiquitin ligase? JOANNE STUBBE: Yeah, so we had a little ubiquitin ligase remain with the sensor that binds in ways we still don't understand, so again this is something that's very much an active area of research. You have iron in oxygen-- sorry, LBX L5, leucine-5, domain of the ubiquitin ligase. I'm not going to draw all of that out. So again, these are all tied-- with what happens with iron and oxygen, it's all tied to these reactive oxygen species. So what we've already talked about-- so this is the major focus, our growth factors. And I'll show you using PowerPoints, but this kind of signal is signaling is also involved in cell proliferation and cell differentiation. So it's widely used in the example that we chose to use, because it was one of the ones that's been most carefully and recently characterized as EGF receptor. EGF receptor is also of great interest, because it's a target. It's the target for successful drugs used clinically in the treatment of cancer. So we then have another. So there are two other important signaling pathways that are proposed to be involved. One is called the antioxidant pathway, and there is a transcription factor. Some of you might have heard of it. Has anybody heard of NRF2? No. So if you're more biological or-- have you heard of the antioxidant? You're biological. Have you heard of NRF2? AUDIENCE: No. JOANNE STUBBE: No. OK. So anyhow, we have an antioxidant pathway, and NRF2 is a transcription factor. And it turns out-- I'm not going to go through the details of this, but for those of you who want to read about the details of this, this is a cell signaling review article published, where they go into all of the proposed mechanisms of how these reactive oxygen species connect to signaling. We're going to focus on epidermal growth factor receptor, but I think you need to know the picture is much bigger. It turns out that NRF2 is in a complex with an E3 ubiquitin ligase, and part of that ligase-- it's a multi enzyme complex-- is keap. Keap has a huge number of cysteines on it. These cysteines get oxidized by some kind of reactive oxygen species. In that one, you want to turn on your antioxidant defense. And so what happens when Keap cysteines get oxidized. It dissociates from NRF2, and NRF2 can go into the nucleus, and it turns on a whole bunch of genes. So I mean you're just getting the idea. I'm not going to go through any details, but it plays a central role. And this system, if you google it, you'll find there are hundreds of papers on this system. This is a very interesting system. I keep waiting for them to get to some stage where I can really talk about the biochemistry, but we aren't at that stage yet, in my opinion. And then the other thing is DNA damage and repair. And if you have DNA damage, or you're starting to get oxidative stress, and these things are out of control-- hydroxide radical is reacting with your nucleic acids-- you need to do something. So you turn on a signaling pathway, and this is controlled by a kinase. I'm not going to go into this in detail, but some of you might have heard of this. This is called the ATM pathway, and what you see-- actually, you'll see, I think, in the next five years if you remain biochemist-- all these acronyms for all these pathways, you're going to get it, because now we're seeing them over, and over, and over again. And at first, it was hard to see how this all fits together, but we're getting there. It's all fitting together, I think, in an interesting way. So all I want you to get out of this-- we're going to be talking about growth matters, but this is also a huge area, just like we just saw with oxidative stress trying to kill our bacteria. OK, so the next slide is one I took out of this article. I'm not going to go through this in detail, but we've already been through the iron responsive binding protein. So this is a summary of that, and we spent a lot of time talking about this. This is what we're going to talk about now-- the role of sulfenylation, in controlling kinase activity, and phosphatase activity. That's what we spent the last two recitation sections talking about. This is a generic approach to that, and I'm going to show you there are many, many growth factors that are thought to go through the same pathway. And so I just want you to remember that. So this is a big area, and so now the next thing is-- so there's another sort of overview picture I want you to get. So there's a second overview picture that I think is also important, and that cysteines really are playing a major role in all of these modifications. They are the easiest to oxidize, and so I think cysteine modifications are important. And there are many, many modifications. The question is, do they happen inside the cell? Do they happen inside the cell in a way that we can connect them to some interesting biology? And then what triggers off? Ultimately, what triggers off these modifications? So we've been talking about the kinases, and we've been talking about sulfenylation. That's what we spent two recitations on, so that's one important thing. But somehow, we're going to see that one of the important things that I've tried to stress in recitations was these reactions need to be reversible, so ultimately sulfenylation-- some way, come to this later. But it's going to be able to be converted back into the reduced state. We've seen that you can form sulfonic acids. This is also-- if you look at the Carrol paper carefully-- we didn't talk about this very much-- it's also reversible. There are sets of enzymes. Hydrogen peroxide can do the oxidation-- the back reaction that people have discovered an enzyme. They can do the back reaction. And then there's an irreversible step. So people don't know, but because it's irreversible this is likely not physiologically important. So in addition to these states, if you start reading the literature, or you read any literature now, we have glutathione that we've talked about. Glutathione is this tripeptide with glutamylcysteine glysine. It is able to convert the sulfenic acid into a glutathionlyated protein. We're seeing these all over the place. Is this the signaling pathway? How is it controlled? What's going on with these? I think we don't know the answer to that. So you can actually have glutathione react to give SSG. You can also have other kinds of proteins that I think-- so here they have RSH. This could be a thioredoxin protein, which if we get to deoxy nucleotide formation, there are hundreds of thioredoxins inside the cell that do the same kind of thing. And so one can also go from here, so you can have a little protein called thioredoxin, and it has two cysteines. And it can convert this back into the SH, and it itself can become oxidized. And there's a way of cycling the thioredoxin. So you're getting the idea, OK. Over here in this model, we're not going to talk about this, because I decided not to talk about reactive-- not on nitrogen species, but nitrous oxide can get converted into peroxynitrite. Peroxynitrite is able to actually catalyze formation of molecules like this, which is thought to be involved in signaling. They're also controlled reversibly by thioredoxins, and again this is what the Tenenbaum Lab studies. And these things can also cyclize to form these kinds of structures. So these are called sulfenamides. People have found them. They have X-ray structures of them. Are they important in signaling? I don't know, but you can see that you have many, many kinds of modifications. That's the takehome lesson from this, and then the big question is how important are these in terms of controlling homeostasis? So what I want to do now is briefly look at the players. I think I'm going to raise this. But briefly look at the players we've already started to look at, and make a few points, and make the general points about the signaling process, using epidermal growth factor receptor as an example. And so I'm hitting you over the head with this again, because we have already looked at this a couple of times. So we have an overview of EGFR and NOX. So we just looked at NOX. This is also NOX2. It's the same protein. They're found in different places. And so remember with NOX2 we had all these factors that I told you, if we're involved in the phagosome, you had a GTPase, you had phagosome oxidase. Now you have, in some cases, similar factors-- in other cases, additional factors-- that play a role in these multi enzyme complexes that allow it to do something else, OK? So nature reuses, over and over again, these different factors. So this is the cartoon picture you guys have seen before. We use this in recitation. This is where I started to have you think about recitation 11, when we started this, to try to introduce you to the system again. And so I just want to make a couple of points about this, but here's our epidermal growth factor receptor, which you all know now is a tyrosine kinase. Here is the NOX protein, and here you can see we have Rac1. If you go back, and you look at your notes from last time Rac1 is a GTPase. You can control its activity with little proteins that can bind to it and inhibit it. And the funny thing about this, and people were asking me questions about this, is that the chemistry, the tyrosine kinase, is in the inside of the cell. NOX1, the NADPH-- and going to NADP is in the inside of the cell just like we just talked about-- but where is-- because of the predisposition of the flavin and the two hemes, where is superoxide produced? On the outside of the cell. That's rather bizarre. This is still rather bizarre to me. This is the model people have in the literature, but you're reducing equivalents from NADPH to convert oxygen to superoxide, which rapidly disproportionates to form hydrogen peroxide. And so then the question is we're saying hydrogen peroxide is the key signaler that's doing sulfenylation. How does it get into the cell? So the model then is it gets into the cell through an aquaporin. And is this aquaporin just moving around in the membrane, or is this some organization within the membrane? So this is going to be useful. We've already talked about the fact that hydrogen peroxide is not very reactive. So one way you can get something to be more reactive is by increasing its concentration. So nature does this all the time. So if you can somehow stick things together and generate something, and it's generated right adjacent to where you're going to react, it has a greater probability of reacting here than over here. And where have you seen that before? Any of you thought about that? Graduate students should know this. AUDIENCE: DNA templated synthesis? JOANNE STUBBE: DNA what? AUDIENCE: DNA templated synthesis? JOANNE STUBBE: No. So, yeah. No. So you do, but I mean in terms of all of these reactive oxygen species. The Ting Lab, that's what she does. She generates these things in the middle of the cell, and it's all dependent-- how long can this go? Remember we talked about this diffusion question. How far does it go before it actually reacts? So the idea, and the question you need to ask yourself is, if you generate this, are these organized? Do you remember from the recitation? Are these guys organized? What did we learn the last time in recitation? So we talked about-- we had this cartoon, and we talked about this. You looked at the data. What did the data tell you? Are these guys organized in some way, so that this hydrogen peroxide can actually do sulfenylation reactions? So what was the evidence for that? Does anybody remember? I mean, so if you don't remember this, you need to go back, and you need to read the paper again, OK? And I have all of this stuff on a PowerPoint, but I'm not going to go through it again. Yeah? AUDIENCE: There was colocalization between, like, the NOX. JOANNE STUBBE: Right. So there was colocalization between the NOX2 in the growth factor, epidermal growth factor. And what else was there colocalization from? It's not shown here, but there was also colocalization of a phosphatase-- which that's not shown in here, but it's going to be shown in the next slide-- plays a key role in controlling the phosphorylation state. OK, so this idea of-- I'm going to write this down because I think this is a central idea in biology-- is how you localize things to make them more reactive. Whether this makes it reactive enough-- it does make it reactive enough, because we can clearly sulfenylate, but are we missing something on top of it, to make it reactive enough to be able to do what we need to do? So what we see with this system is EGF, the Growth Factor, causes EGF dimerization. That's what I've shown you in the cartoon over there. I'm not going to draw out the cartoon, because you've seen this cartoon a bunch of times. And what does that do? The tyrosine kinase activates itself by phosphorylation, and we're going to come back to this. So one way again, everybody has seen phosphorylations. Whether they activate or inactivate, you need to study. So here's the tyrosine kinase domain. When they come together, it has activity that it can phosphorylate itself, so you get into this form which then triggers signaling cascades. So the other thing we need to think about in this paper is-- so here we are going from the tyrosine to the phosphorylated tyrosine. You should write down that this is active. How do you activate a phosphorylated tyrosine if this is active? You use a phosphatase. So this, also, we saw in this paper. And in this paper, this Carrol paper, she identified, or she claims to have identified-- you can make your own judgment on that now-- the phosphatase, in the cell type that she looked at, that played a key role in the tyrosine kinase activity. So you're converting it from an active form to an inactive form, which is what you see by phosphorylation, dephosphorylation all the time. I'm going to come back to this in a minute. So here's a protein tyrosine phosphatase. So we have a protein tyrosine phosphatase. And so while I didn't write that out, these ends, there are lots of different kinds of phosphatases. But they have a thiolate in their active site, and these things is attached to a tyrosine on a protein. So here's our tyrosine kinase. So what happens again is you use covalent catalysis in two steps. So you phosphorylate, and then you hydrolyze. So you phosphorylate, and then you hydrolyze. So you end up then with your tyrosine and the kinase, and you end up back with ES minus. So again, there are many different kinds of phosphatases, but all the ones involved apparently in these signaling processes-- if you go back and look at the Carrol paper-- all have cysteines in their active site. You've seen this, covalent catalysis with cysteine, over, and over, and over again now at this stage. So this is going to be a key control. This is the active state, so this form is the active state. And over here-- sorry. So if you take this now, and you treat this with hydrogen peroxide, this becomes sulfenylated, and this becomes the inactive state. So sulfenylation, just like with the tyrosine kinase that we talked about in recitation, can become activated. The sulfenylation in this case becomes inactivated, so that's what it's all about in these post translational modifications. And the question is-- I'm sorry, I'm over again. But the question is are these models correct? So what we'll do next time is spend a little bit of time talking about the six general principles of post translational modification, in general, and what the expectations are using what you've already-- which we've already seen in recitations 11 and 12. And then we're going to move on to the last module on nucleotide metabolism.
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MIT_508J_Biological_Chemistry_II_Spring_2016
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R7_Application_of_Single_Molecule_Methods.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. REUBEN: I'm Reuben. I am currently a senior in Chemistry. And I am a UROP in Bob Sauer's lab over in the Department of Biology. As you guys know from this class, one of the modules that you cover is ClpXP, the bacterial analog of the proteasome. And today I hope to tell you about some of the cool work using modern biophysical techniques that people in Bob's lab have done to gain some really, in my opinion, interesting insight into the actual mechanical mechanism of this motor protein ClpXP as it unfolds and degrades protein substrates. Before I get started, I want to say, please stop me at any time to ask questions. If I say something that doesn't make sense, please let me know, and I will try to clarify. So today I'm mostly going to be talking about modern single molecule methods, which are the cutting edge in biophysics. And so the first question I wanted to ask you guys is, what are the possible advantages of looking at molecules one at a time rather than performing some sort of bulk assay? Do any of you have any ideas? AUDIENCE: Maybe the average of a bunch of different molecules isn't truly representative of what a single molecule looks like. REUBEN: Yeah. So the average is the average. But the average as a statistic has some sort of weakness. It obscures a lot of the vagaries of behavior that may be lost in that average. So it's worth saying that in classical biochemistry you are looking at a lot of molecules simultaneously. In one microliter of one micromolar solution, that's 6 times 10 to the 11th molecules. So taking some sort of bulk measurement averages across all of these molecules. And as you said, you can lose a great deal of information about the variation and the dynamics of these molecules. So another thing that you can lose, in addition to information about variation, is you can lose information about the procession through some sort of complicated biological process. So let's say that you are studying ClpXP as it recognizes, unfolds, and degrades protein substrates. If you were looking at, for example, 6 times 10 to the 11th molecules of ClpXP sitting in a test tube, then you're smearing across all of these different time points of ClpXP, where you have some ClpXPs that are unfolding substrates, some ClpXPs that are not bound to substrate at all, some ClpXPs that are translocating substrate. So any sort of readout that you have loses this information about the different states that ClpXP can inhabit. And also, it's very difficult to gain detailed kinetic information about how ClpXP would, for example, transit between these different states if all you can see is a bulk average. So another way of putting this is that a bulk measurement like fluorescence, you're looking at the unsynchronized activity of these molecules. But by studying single molecules at a time, you can do some sort of post hoc synchronization to actually gain insight into the kinetics, or the order of the different states during some sort of biophysical process. JOANNE STUBBE: You can do better than that, even in bulk you can synchronize, right? REUBEN: Yeah. So there are certainly tools like stopped-flow. JOANNE STUBBE: Which is what we talked about in the first couple of recitations. That's why I brought it up. REUBEN: Yeah. So stopped-flow is a fantastic tool, which has been used really successfully to study a lot of systems, particularly, I think, the Rodnina lab, which I think you guys covered, used stopped-flow very effectively to study the early stages of translation. But I think that stopped-flow is good for looking at some early stages of processes, but for trying to track a long, complicated process across a significant period of time, stopped-flow is not really an ideal technique. Because what happens is that the rates of transitions between different processes-- let's say that there is a single rate constant. The actual time that an individual molecule spends in a state before switching to the next state is stochastic. It's an exponential decay process based on the rate constant. So if you are looking at a bunch of rates all put together, and you start a stopped-flow experiment, it's true that for the first couple of seconds all of the molecules are synchronized. But over the rest of the experiment, they blur out all the way across the time space. And so you can really lose a lot of detailed information about the kinetics. Whereas if you're looking at molecules one molecule at a time, you can track an entire process. JOANNE STUBBE: But again, the issue where you can only look at them in one way-- REUBEN: Oh, absolutely. JOANNE STUBBE: --ways. And so, if you don't know anything about the bulk system-- I hate it when people are touting single-molecule, which I think is very powerful, but the fact is that you have many ways to look at reactions that can't be carried over into the single-molecule venue. So that's the caveat. REUBEN: I completely agree. I was actually just about to, after I finished this slide, ask what were the disadvantages of single-molecule. JOANNE STUBBE: All right, sure. REUBEN: And you raise a very good point. No, I always appreciate comments from the audience-- [LAUGHTER] --and from the faculty in the audience. I don't know. It's true that there are some measurements you can't make in single-molecule. But it's also true that there are some measurements, such as the measurements in this paper, that you can't make in bulk. So you can measure some unique properties of molecules-- particularly molecular machines such as force generation and processive motion-- using measurement techniques such as the optical tweezers, which I'm going to talk about a lot in this lecture, that there's no bulk analog of a measurement like this. So JoAnne mentioned a couple of the possible disadvantages of single-molecule studies. Do any of you have any other ideas of potential disadvantages that might make sure that you want to do some bulk experiments? So the really big one is that single molecules, there aren't great ways to look at them. There's single-molecule fluorescence, there's optical tweezers. After that, you know, I'm not really sure what else there is. And both of these techniques study from a significant noise issue. You can really only look at one thing at a time. And a single fluorophore, for example, in a single-molecule fluorescence study, is not very bright. You have enormous noise coming from your instrumentation. And it can be very difficult to tangle out the signal that you're trying to go after from your noise. Another major issue is that many of these experiments are actually very difficult to run, and it can take, for example, months and months and months to do single-molecule experiments, which, you know, a bulk protein degradation experiment, you can mix some proteins and run a gel, and actually learn a great deal about the behavior of the proteins. But all the same, there are some measurements, such as the measurements in this paper, that you cannot take without single-molecule measurements. So I hope that today I will be able to tell you about some modern biophysical techniques, particularly optical tweezers, so that you can all learn about some of the latest tools in the biophysicist's toolkit. So I am going to talk about optical tweezers today, but I did want to mention that there is another single-molecule method, single-molecule fluorescence. And this is actually what I do in my experiments. That's also very useful for looking at the dynamical properties of complicated biological processes. So single-molecule fluorescence basically involves tracking the fluorescence of single fluorophores over time, and quantitating it, and then studying the dynamics of the switching between the different states of the fluorophore. Almost all of these experiments use a physical property called FRET, which is energy transfer between two nearby fluorophores, the efficiency of which is distance dependent. I believe that you guys have learned about it. It's not that difficult to understand how you could go from doing FRET in bulk to FRET in single-molecule. If you haven't heard about FRET, I'm sure you-- JOANNE STUBBE: We'll hear it in the last recitation. REUBEN: I'm sure you will hear about it very soon. JOANNE STUBBE: [INAUDIBLE]. REUBEN: Yeah. So this is a very useful technique for answering some questions, but in some ways it's a little bit easier to understand than optical tweezers. So I figured I should talk about optical tweezers today, because I think it's also kind of cool that you can do the experiments that are in the paper today. So optical tweezers-- the main technique used in the Olivares and Sauer paper that I'm talking about today. And the general idea of optical tweezers is that you can use the momentum of light. Because as you guys know from quantum mechanics, light has momentum to trap certain types of particles within a beam of light and directly apply forces to them and directly make unique measurements about distance using this tool. So with optical tweezers, you can measure nanometer motions at sub-millisecond time resolution. And in a couple of slides I'm going to tell you how that works. And you can also directly apply force to probe mechanics and biochemistry. So you guys have probably all heard about optical tweezers, even if you haven't heard about them in the case of studying single molecules of protein. So this is a really cool video that I found. These are 12 beads, all trapped in their own optical trap. And some graduate student wrote a program to steer one of the beads around the other beads. So the system that we used in this experiment, that I'm going to tell you about today, is basically the same as this system, except you're not looking at 12 beads at a time. And each of the beads is conjugated to some sort of molecule of protein. So the general setup in single-molecule optical tweezers experiment in biophysics is to take a bead-- usually the bead is made of polystyrene, because it happens to have some nice properties-- and to conjugate onto it a single molecule of protein. In this case, they conjugated onto it a single molecule of kinesin, which is a motor protein. JOANNE STUBBE: [INAUDIBLE] like polystyrene shrinks and it traps that use polystyrene columns. Is that-- REUBEN: These beads are functionalized on the surface, and then covered with streptavidin, which I'll talk about in just a second. The beads don't really shrink very much, because you're not really putting them under pressure in any of these experiments. And so what they did in these experiments is that they took kinesin, and they put it on the surface of a bead. And then they took some microtubules, and they put their microtubules on the surface of a cover slip. And they brought the kinesin on the bead close to the microtubules on the cover slip, and they added ATP. And kinesin is a molecular motor which walks along microtubules. So what happened is that kinesin grabbed onto the microtubules, and it began to bind and hydrolyze ATP. And it began walking. And it dragged the bead with it as it walked along the microtubule. And with an optical trap, you can actually detect how far from the center of the trap a bead has been dragged with extremely high, less than one nanometer, precision. So they were able to tell that this kinesin was taking very discrete eight-nanometer steps as it walked along the microtubules, which could be related very effectively to the size of these tubulin domains that the kinesin was actually making contact with. So using this they could, for example, study the average time in between kinesin steps as a function of the concentration of ATP. And they could begin to ask some questions about how the binding and hydrolysis of ATP actually led to kinesin's processive mechanical motion. So before I tell you more about the biophysics, I just want to tell you a little bit about how optical traps work. Because I think that it's not particularly intuitive before you hear about it. Although, in the scheme of things, they're actually not that complicated. So I'm going to tell you three ways of thinking about optical tweezers, all of which are basically the same, but use different approach to look at the behavior of the system. So I'm just going to start by pretending that light is just a ray, not a wave, because that's a pretty good approximation in a lot of circumstances, including this one. So the way to think about optical tweezers is to imagine two rays of light, one coming from each side of a lens. And both of these rays, they have momentum, because they're light. And to think about these rays of light interacting with a dielectric particle such as a polystyrene bead, which has a higher index of refraction than the surrounding media, which is usually water. So what that means is that when the light hits the bead, the angle of the light will change. Basically the bead will deflect the light. And so what that means is that the bead is actually changing the momentum of the light. So what we know from Newton is that, if the bead is changing the momentum of the light, then the light must be exerting an equal and opposite force on the bead itself, opposing the momentum change of the light itself. So we can see here that this ray A coming from the left is being deflected up and to the right. And this ray B coming from the right is being deflected down and to the right. So we can think about the force on the bead from each of these rays being equal and opposite to the direction of the momentum change of these rays. And say that ray A is imparting a force on the bead which is pointing down and to the left, and ray B is imparting a force on the bead which is pointing up and to the left. So we can basically sum both of these forces and find that, in this system here, where the center of the bead is moved to the right of the true focus of these two rays of light, then the light is imposing a force on the bead which pushes it back toward the center. And it turns out that, whenever the center of the bead is not in the same spot as the center of the focused light, the light will impart a force on the bead, pushing it toward the direction of that center, no matter how you move it around within the trap. So do you all understand the ray approach to understanding how optical traps work? Cool. So I'm going to give a slightly more realistic picture. So as you guys know, light is not a ray. Light is a wave. And what that means is that you can't focus light into just a point. Instead, you can focus light into a small volume which is diffraction-limited in size to approximately a diameter of one-half the wavelength of the light, which is often called the confocal volume. So to think about the shape of the confocal volume, it's not quite a Gaussian, but it's very similar to a Gaussian. So it's useful to think about Gaussians, to say that, you know, it's easy to imagine a one-dimensional Gaussian or a two-dimensional Gaussian. At a confocal volume of light, what you actually have is a three-dimensional Gaussian in which each point in space has a scalar property, which is the intensity of the light at that point in space. And if you move from the center out in any direction, it basically decays according to this Gaussian curvature. So there's a gradient, a Gaussian-shaped gradient of light intensity going out in every direction from the center, if this is like the optical axis, and say this is like the x direction, and this is the y direction. I don't know if my laser printing is particularly helpful in this situation. So what you can think about is, say that from your beam of light in this Gaussian shape you have rays of different intensity. I've only shown two here, but try to imagine these rays coming from every part of this light beam, all being focused here so that rays from the center have a lot of intensity, therefore they have a lot of momentum. There's a lot of photons there. So they impart more force per deflection than rays coming over from the side. So if you imagine rays coming from this whole Gaussian volume here, then it's not too difficult to see that whenever you move the bead from the center, there's going to be a restoring force pushing it back to the center. And the last explanation for how optical traps work is my favorite, but it's a little bit more difficult to understand, even though it's actually classical rather than quantum. So what we know-- a dielectric particle such as a bead, what a dielectric actually is, a dielectric is a material which is polarized by an electric field. So polystyrene is polarized quite a lot by an electric field. And what that does is that, if you have an electric field, say, that has a negative charge here and a positive charge here, then the dielectric will be polarized in such a direction that it is opposing that electric field. And the electric field within the dielectric is lower than the electric field outside of the dielectric. And it turns out that, for very small particles, in this case particles that are actually smaller than the wavelength of light that we are actually using to trap it, you can approximate a dielectric material very well as just a simple dipole whose dipole moment is facing in the opposite direction of the electric field that it's actually sitting in. So think about what light actually is. Light is a rapidly oscillating electric field that also happens to have a magnetic component. So the particle that you're studying basically has a rapidly oscillating dipole, which is always opposing the direction of the magnetic field. And so we know that there is a cost in potential energy to separating two charges. And putting a dielectric material in between that charge separation decreases the potential cost of this charge separation. So if you have a gradient of electric fields, as you do at the center of a confocal volume, the dielectric will be most favorable if it is in the place where it is opposing the strongest electric field. And if you move it from the center, then it will oppose a weaker electric field. And we know that force is the gradient of potential. So the dielectric particle will feel a force driving it toward the strongest area of electric field, which takes place at the very center of the confocal volume. And so it will always be restored toward that spot. So this is kind of an intuitive explanation, but it takes a little bit of thinking about. If you have any more questions about this, I'm totally happy to answer them either now or after the talk. So do you all sort of feel like you understand how you can use light to trap a dielectric particle such as a polystyrene bead? AUDIENCE: What wavelength of light are you talking about? Like, is this monochromatic? REUBEN: Yeah. So in this case we're using monochromatic infrared light. The wavelength which is chosen for these studies, it turns out to not damage proteins very much compared to other wavelengths of light, and you're using it at very high intensity. And it's pretty easy to make very high intensity infrared lasers. But you could use, you know, the thing that really matters is what the index of refraction is at the wavelength that you're studying. So as long as you have a big difference, as long as you have a dielectric particle which has a higher index of refraction than the surrounding media at the wavelength that you're studying, then you can use any wavelength of light for this sort of experiment. So I'm not going to go too much in depth into the actual apparatus for an optical trap, but I do want to just give you the schematic and talk about the very basics. So the general idea is that you have a trapping laser, which is being focused right here on the specimen stage, basically just a microscope where you actually have the bead trapped. And then you have a second detection laser, which is much weaker, 100-fold weaker than the trapping laser, and is at a different wavelength. And this is being focused onto the same volume. And you have a quadrant photodiode, which basically gives off a different voltage in response to a different location of the trapping laser on it. And this is how you actually measure the where the bead actually is within the center of the trap. So if the bead is moved out of the center of the trap, then this detection laser is deflected slightly, because it's also refracts through the bead. And so that shifts the centroid of the detection laser as it's read out on this position-sensitive detector. And so you can take this reading 100,000 times a second. And that's how you can get this very detailed, one-nanometer level of resolution, which describes where the bead has been pulled by some biological molecule out of the center of the trap. So one other thing I should talk about, I want to give a couple of examples of things that people have used optical traps for. But it's worth mentioning that most experiments with optical traps are no longer done with just one trap. They're now done with two traps in which you have two different beads, each trapped within their own optical trap. And you do this because it basically mechanically isolates the system. So normally you can have vibrations just in the room that would shake your specimen stage. And it turns out that the distances that you're measuring are so small that these vibrations dramatically increase the noise of the system. But if you have both beads trapped within their own optical trap, that provides significant damping from any sort of mechanical vibration. And you can measure the bead-to-bead distance using this apparatus. And so here, this is from Stephen Block's group. They actually were able to measure single base pair incorporation into a growing mRNA chain as RNA polymerase walked along a template. So it's not perfect. It doesn't look totally linear. But it's pretty incredible to see two-angstroms resolution. And from this they could learn a great deal about the kinetics of RNA polymerase stepping along a template. And they could even keep it in registers so that they could know which base is being incorporated, which change in distance here represents which base incorporation, so they could study the kinetics of how sequence effects the rate of insertion. Another use of optical traps which has proved very fruitful for some people is to, you know, you can use them to measure forces that molecules apply on a bead, but you can also directly apply forces on molecules by turning the system around. If you move the center of the trap a little bit, then you can apply a force on a molecule which is, for example, bound in the trap, and see how a change in the force on the molecule will change the bead-to-bead distance, basically get a force extension curve. So here they trapped a protein domain in between two beads with a long DNA handle, which basically is designed to be a relatively rigid linker that just keeps the two beads relatively separated so that the optical traps don't overlay at all. And they applied force. And what you can see is that as you increase the force by-- in this case, they moved one of these beads just a little bit to the right. You get a little bit of stretching, which is partially in the DNA and partially in the protein domain itself. But then once you reach a certain threshold force, which in this case is 17 piconewtons, they actually unfold this protein domain. And you get a sudden jump in bead-to-bead distance as this now unfolded polypeptide no longer provides any sort of force pulling it back together. And then if they release the force and allow the molecule to relax, they actually get a different relaxation curve than their original extension curve, because this represents the relaxation of this unfolded polypeptide rather than the refolding of this domain. And another cool thing that they did in this experiment is they brought it right up to this unfolding force, and then they left it there. And it turns out that at this unfolding force, this protein rapidly unfolds and refolds, because it's a very small protein domain, so the protein folding is extremely quick. So this is the rapid transition between folding and unfolding which is taking place at this transition force. So they could use that to study the kinetics of the protein folding under force for this system. So today, the paper that you guys read for this recitation is about ClpXP, which is this protein that, as you guys studied in class, it recognizes protein substrates that have been tagged for degradation. And it uses mechanical energy released by ATP binding and hydrolysis to unfold folded protein domains and translocate the unfolded protein substrate through an axial pore into an associated sequestered peptidase, where the protein is rapidly chewed up into small peptides that are then recycled into their constituent amino acids. So I have another figure, which someone else in my lab made more recently, that illustrates a little bit better the actual unfolding and translocation steps, which were the steps that we studied in this paper. So the general idea is that ClpX will translocate a folded protein domain until it's reached a point where the folded domain is too big to fit through the axial pore. And then it will apply what we call power strokes, which are somehow related to ATP hydrolysis and binding, which yank on the ssrA tag and pull the protein down in a repeated attempt to unfold it. And you can say that there's some sort of strained protein structure. It turns out that most of these attempts do not actually successfully unfold the protein. But some proportion of these attempts-- maybe ClpX gets a particularly good grip. Maybe there's some sort of transient thermal destabilization in the protein itself. The protein successfully unfolded, and then it continues to take these small steps, these small power strokes along the unfolded polypeptide, translocating it into the peptidase for degradation. So in the experiments that Adrian actually did for this system, what he did is he attached ClpXP to a bead using streptavidin and biotin, which is, I believe, an interaction you guys are familiar with. And he attached a multi-domain protein substrate to a DNA linker so that the ssrA tag was at the end of this substrate. And he was able to move the two beads together so that, at some point, the ClpXP bound to the ssrA tag. And then he could record the bead-to-bead distance during the unfolding and translocation of this substrate. So what he saw is that, when ClpXP was successful in unfolding a protein domain, there was a sudden jump in bead-to-bead distance as the folded-- the whole experiment is taking place under a small amount of force. So as the polypeptide has now unfolded for that small amount of force, it's quickly pulled pretty taut. And then ClpXP translocates that substrate, now unfolded, through its axial pore, decreasing the bead-to-bead distance. And it continues this translocation until it reaches the next folded protein substrate, at which point it stops. It can't translocate any further. And it begins to attempt to unfold the substrate again. And eventually, after some sort of dwell that we call a pre-unfolding dwell, it will be successful at unfolding the substrate. There will be another jump in bead-to-bead distance, and the process will repeat. Go ahead. AUDIENCE: So you said that it was held taut. Is there an additional force that it experiences because it's being held taut, or is that accounted for? REUBEN: It's being held a little bit taut. I shouldn't say "taut." There is a small amount of force on the system. You know, often we record in the range of about five piconewtons, which is just required for the optical trapping apparatus to actually work, but should not have a significant effect on the behavior of the system. So I should have mentioned it back at the kinesin. You can understand the stall force for a molecular motor, which is the force that it takes to restrain it from taking additional steps. And it turns out that the stall force for kinesin is about seven piconewtons. The stall force for ClpXP is about 25 piconewtons. So it's dramatically higher than the small amount of force exerted on it by the trap. So it shouldn't have a significant impact on the results. What it just means is that, once you go from folded to unfolded, since there is nothing holding the unfolded structure in a coil, even a small amount of force will pull it out a little bit and cause this increase in bead-to-bead distance. JOANNE STUBBE: So I have a question. When you're developing these methods, like when we used to teach [INAUDIBLE] DNA polymerases, how you put the complex onto anything. REUBEN: Oh, I have a slide on that next. JOANNE STUBBE: OK. Because to me, that's the key thing. And the question is, in this particular experiment, how many iterations did they have to go through before they figured out length, attachment, all of that stuff? REUBEN: So fortunately, other groups have worked out many of these issues for other protein systems, so we were able to adapt those-- JOANNE STUBBE: And you would be able extrapolate-- REUBEN: --relatively easily. JOANNE STUBBE: from one? Because with the polymerases, you couldn't do that. REUBEN: Other people had studied these AAA's, particularly helicases, as well as various nucleotide translocases that are actually, in mechanical activity, somewhat similar to ClpXP. So we could basically adapt their systems without significant trial and error, which meant that we could get closer to the biology, or closer to the biophysics quickly, which was very nice. But other groups certainly spent 20 years getting these methods to actually work well. JOANNE STUBBE: Are there any generalizations that came out of that optimization? REUBEN: Yes. So there are a couple of generalizations which came out of that optimization. The first is that biotin-streptavidin is a really useful interaction. Basically all modern biophysical techniques use biotin-streptavidin or other basically binary, basically permanent interactions for these sorts of biophysical studies. JOANNE STUBBE: But biotin-enhanced streptavidin has four binding sites. Are they using a streptavidin with one binding site? REUBEN: No. You give the-- so when you're actually taking a bead and adding ClpXP to it, what you do is, you add a very small amount of ClpXP. And then once you've added it, you add just straight biotin to fill the rest of the binding sites. Because one thing that you never want is, you never want two ClpXPs on one bead to engage two substrates on another bead, because that would just totally screw up the data that you're recording. So you make sure that it's very, very sparse labeling of molecules of ClpXP on the surface. You know, I think we probably use femtomolar ClpXP during the actual labeling of the bead process. Another major takeaway from these experiments is that, for these dual bead experiments, it's very important to have a DNA linker. So the distance here is not really representative, but this DNA linker actually goes over here. Because you really want to maintain bead-to-bead separation on the order of more than a micron, so that your two traps don't overlay at all. Because this leads to much clearer readouts on your position-sensitive display, allowing much better data acquisition. JOANNE STUBBE: So again, just technically, I'm interested in this-- REUBEN: Oh, sure. JOANNE STUBBE: So when you have very low concentration of anything, usually you have a lot of problems, because the stuff sticks to everything. So how do you avoid, you know, just inherently, especially proteins. I mean, I've dealt a lot with proteins. The more glue you get, the worse-behaved they are, because they stick to everything, even when you modify the containers with different kinds of polymers and stuff like that. So-- REUBEN: Yeah, so that is a major issue. In this case, it's not a particularly big issue, because you're really-- all you need is one ClpXP on the bead and one molecule of substrate on the other bead. So it doesn't really matter what happened to most of the molecules that you add to your mixture as long as you have one active molecule here and one accessible molecule there. So it's probably true that there are ClpXPs which, at this very low concentration, are bound non-specifically to the surface of the bead, and probably bound non-specifically into the actual cover slip. But it turns out that those molecules have no signal in this process. So it's not as big of an issue. And there's no concentration dependence that you're trying to measure using these very low concentrations. So even if 99% of the stuff you add binds non-specifically, it's just not that big of a deal. For single-molecule fluorescence it can be a really major deal, and so you have to work really hard to basically [INAUDIBLE] all of your surfaces. So-- no, go ahead. AUDIENCE: I guess it doesn't matter as much for this system since you have two optical traps you're only measuring a one-dimensional distance. But for optical traps, do you just measure displacement, or can you know two-dimensional, like which direction it was displaced? REUBEN: You can tell which direction it's displaced, because actually that little picture-- basically it will deflect the beam of light onto this position-sensitive display. AUDIENCE: But the voltage is dependent upon both x and y directions when working with that? REUBEN: I can tell you more about the apparatus later. It's dependent on both x and y, yeah. It's called a quadrupole detector. So it basically has four different diodes, and you look at the relative ratio from these four different detectors. That tells you where it is-- basically a four-pixel camera, I guess. So the actual substrate, the attachment onto the DNA, it's via a sort of chemical biology which is worth knowing about, which is something called the HaloTag. So it turns out that there is this enzyme which, if you make a couple of mutations to ruin it, it will make a covalent bond with haloalkanes. Basically any haloalkane that you add to your reaction mixture, this enzyme will actually form a covalent bond with the haloalkane in the active site. I forget what the original function of the enzyme is. It's some sort of halogenase or something. So in this case, we synthesized this long DNA linker such that it had a haloalkane at one of its ends. And then we added this halo domain at the N-terminus of our long substrate such that it would form a covalent bond with the DNA which had a biotin at the other end. And then we added that to a bead. And it turns out that these beads are slightly smaller than these beads. So when we actually start these experiments, we trap a big bead in one beam and a small bead in another beam, and we basically move them very slowly with respect to each other until you eventually get an attachment between a ClpXP and a halo domain. And then you begin the actual recording of the experiment. So I have a nice animation of this actual unfolding process. So this illustrates the effects of an unfolding under force. So once ClpXP successfully unfolds a domain, you get this big jump in bead-to-bead distance, which decreases during translocation and then increases again during the next unfolding step. Alex? ALEX: Why do you use a DNA linker? REUBEN: We use a DNA linker to keep the Y-DNA instead of something else. It's really easy to make a long piece of DNA. You know, this is like a micron long. There are very few proteins that are well-behaved at that sort of length. DNA is very-- ALEX: But why not work with PEG or something? REUBEN: --well behaved. Stiffness. PEG is very floppy. DNA is relatively stiff. There's still some sort of floppiness, which you account for using what they call a worm-like chain model, which if you have any questions about that, ask me afterward. But DNA is much stiffer than just a single PEG, and also very strong compared to a single PEG. So again, what you see is unfolding translocation, and then unfolding again at the next substrate. So do any of you have questions about the readout that we get when we're actually looking at acquiring the data for this paper? Does this make sense to you guys? Cool. So I'm going to get into a couple of the figures of the paper. This paper is very difficult, and has a lot of data in it. But some of the figures I think are relatively easy to understand. So I believe you covered these different mutations of titin, which change the mechanical stability and change the rate of ClpXP unfolding in degradation. So using these mutations, we were actually able to directly investigate the unfolding strength of ClpXP. So this titin I27 domain, there are a couple of mutations you can make right by the C-terminus of the domain, where you basically switch a valine or something for a proline, which significantly decreases the mechanical stability. And we made these multi-domain substrates with these various mutants of titin inserted. And then we investigated the time of the pre-unfolding dwell before this big bead-to-bead distance for each of these unfoldings events before a domain. And what we saw is that, for a wild-type titin, the most stable, we saw relatively long pre-unfolding dwells. Whereas for the V15P titin, which has the intermediate mechanical strength, we saw shorter pre-unfolding dwells. And for the V13P titin, we saw the shortest pre-unfolding dwells, which basically are the flat periods in between these jumps. So it turns out that-- so you can quantitate the length of these dwells by basically plotting the number of dwells of a certain length, the frequency of dwells of a certain length or shorter, versus that length on the x-axis. So this is basically making a cumulative distribution plot of the dwells. And it's pretty easy to show that, if you're looking at a single exponential decay process such as a single kinetic step, then this plot should have an exponential shape where it's fit very well by a single exponential decay process. And so you can look at the rate of that exponential to actually gain a lot of insight into the rate of that particular kinetic step. So for the wild type titin, we found that the half-life for this unfolding event, which you can extract from this exponential decay rate, was about 55 seconds. Whereas for this V15P titin, which we knew is less mechanically stable, it was about 17 seconds. And for this V13P titin it was about six seconds. And we saw that these plots are not perfectly fit by a single exponential, but they are fit very well. And this sort of indicates that there is one rate limiting step, one major kinetic step in the actual process of unfolding, which is what we assumed, because we believed that these domains unfolded very cooperatively. And so basically this suggests that a single successful power stroke by ClpXP is responsible for the unfolding of these domains. So do any of you have any questions about how we measured unfolding in this circumstance? AUDIENCE: Well, just looking at the fits, it almost looks like it decays slower than a monoexponential decay. Right? Do you have an explanation for why that is like that? REUBEN: No. There aren't that many data points here. And some of these data points out at these high distances are a little bit hairy. I would say there's quite a bit of error in biochemistry, and my guess is that the quality of these fits is like 99.9. And without dramatically more data, it's not very useful to speculate about what this other exponential might be. So I'm not going to say. I honestly have no idea. JOANNE STUBBE: But I think that when you're looking at data like that, sometimes it's better to look way down here so you can see the quality of the fit, so you can look at the quality of the fit. It's hard to see it on that graph. REUBEN: Oh, sorry. Yeah. JOANNE STUBBE: [INAUDIBLE] would mean a lot of the data points. And then you look at the fit. REUBEN: Yeah. JOANNE STUBBE: I think the data looks [INAUDIBLE].. REUBEN: Yeah. Given-- I mean, this data took Adrian like a year to acquire, because you get so few data points per experiment. With the error, this is pretty darn good for this type of setup. It may be that sometimes it takes two very rapid power strokes, and that might add some longer exponential decay process. But it's very difficult to say. So we know from the crystal structures, as well as from biochemical studies, that ClpXP has these pore loops along its axial pore, which are actually directly involved in the mechanical activity of the protein. So we believe that there are these RKH loops, which are on the top face of ClpXP. And these loops make this initial interaction with the ssrA tag. And then there are pore-1 loops within the pore, and we believe that the pore-1 loops are the loops that are basically the levers of mechanical force application to a substrate during unfolding and translocation. So these pore loops, you can imagine them undergoing a conformational change, where they grab onto an unfolded polypeptide and basically are translocated downward in some sort of nucleotide-dependent manner, dragging the substrate with it. And if the substrate is unfolded, then it comes along. If it's folded, then this can be either a successful or an unsuccessful unfolding event. And then there are also these pore-2 loops, which we think are involved in holding the substrate at the bottom of the pore as the pore-1 loops are reset for the next power stroking event. And we've done a lot of experiments which try to explain how these power stroking events are related to the actual process of ATP hydrolysis to ADP and phosphate, and then phosphate release, and then ADP release. And we have some evidence which suggests that phosphate release after ATP hydrolysis is the step which is most intimately coupled with this particular conformational change that applies to mechanical force. But it's-- JOANNE STUBBE: So how do you look at that kind of measurement? REUBEN: The way that we've measured this most effectively is to do single-molecule experiments of the sort I'm going to show on the next plot with either excess phosphate or excess vanadate. So the idea is that after phosphate leaves, there is some sort of a rate constant, which describes the time it takes for this conformational change to occur. So if you have phosphate or a phosphate analog that rebinds before that conformational change can occur, then that should increase the dwell time. JOANNE STUBBE: That's only one phosphate binding site? I mean, you have six-- how many-- you have multiple subunits. And depending on whether the other nucleotide is around, you only have one binding site if you have a huge excess of Pi, or is it-- REUBEN: We think that the important phosphate-leaving step is taking place when there is ADP bound. So in this subunit where you have ADP bound and also phosphate bound, and then that phosphate leaves, we think that that is the step that's related to the actual mechanical motion. But we're not going to claim that that is the step, we're going to say we think that's the step. And our evidence supports that versus any of the other steps such as ATP binding or ADP release, because you can show that adding a lot of competitor ADP or changing the rate of ATP does not have the effect on this dwell time in between steps that you would expect if those were the steps that had this central mechanical role. But the evidence is not totally compelling. But just imagine that something related to this process of ATP hydrolysis is basically causing this pore loop to be translocated downward. So what we could do is-- I showed these plots where you see this unfolding and then the slow translocation. You can actually, because these optical traps have such high resolution, you can actually monitor the individual steps that ClpXP takes as it walks along a substrate, which I actually think is pretty awesome. So you can fit these steps to a model to get cleaner quantitation. And then you can quantitate the length of these steps. And you can see that ClpXP takes steps that are a range between about one nanometer and about four nanometers. And we quantize this. We say that ClpXP can take steps between one and four in multiples of one, because we know from the crystal structures that this particular pore loop translocation downward, it moves about one nanometer. And so what we've found is that you can look at the order of these step sizes. And you find that ClpXP, it rarely takes-- so the order of steps, which you can see A, it's taking a one-nanometer step, and then a two-nanometer step, and then a one-nanometer step, and then a couple other one-nanometer steps, and then some twos and then some threes. And you can see that for B and for C, as well. We found that the order of these steps is relatively random. It's very difficult to use one step to predict what the next step is going to be. So from that, we think that there is a significant degree of stochasticity in some aspect of the mechanism of this enzyme, determining possibly which subunits are hydrolyzing ATP to power a power stroke. But we say that it's not completely stochastic. The events are not all equally likely to occur after each other event. So for example, after a four-nanometer step, we very rarely see a second four-nanometer step or a three-nanometer step. We are much more likely to see a one-nanometer step or a two-nanometer step. Whereas it's much more likely that after a one-nanometer step, we might see a longer step. So we have come up with a couple of complicated kinetic models that can explain some of this data. And we're not saying that it's correct. What we're more saying is that this is the sort of thing that you have to think about when really asking deep questions about the mechanical activity of this machine. So a model which explains this behavior where you don't see several long steps in a row is to say that these steps, which are quantized according to this one nanometer, what they actually represent is cascades of steps, possibly a two-nanometer step being two very quick steps in a row, or a four-nanometer step being four very quick steps in a row, which occurred too quickly for our instrument to actually catch them. And you can say that each step is basically controlled by nucleotide hydrolysis or by phosphate release in a single subunit. And say that possibly a four-nanometer step involves phosphate release in four subunits, sort of one after the next. Maybe they're all contiguous, or maybe they're not. We don't really have any evidence going either way. And so you have boom, boom, boom, boom, in a cascade of four very quick steps. Whereas a two-nanometer step involves hydrolysis in two subunits, and a one-nanometer step involves hydrolysis in just a single subunit. So the reason that we're attracted to models such as this is that after you have one of these four-nanometer steps, you have basically lost your phosphate, or lost your ATP, or whatever, in four different subunits. And so it takes time for ATP to bind and possibly to be hydrolyzed again in these four different subunits. So it's unlikely that you're going to have the time for four ATPs to bind and be hydrolyzed before your next power stroke. Whereas it only requires binding in one subunit to power a single-nanometer step. So this model, we don't actually have much evidence that suggests that it has to be sequential orders of subunits going around the ring like that. We've chosen this counterclockwise direction completely arbitrarily. The reason that we think that it's limited to four-nanometer steps rather than five-nanometer or six-nanometer steps is that we know that ClpXP, even at saturating concentrations of ATP, basically never binds more than about four equivalents of ATP per hexamer. And we also know that there are these U subunits, these unloadable subunits, which are actually not competent for ATP binding. So possibly a four-nanometer step could involve phosphate release from all four of the subunits that have ATP bound. But then the step stops when this cascade event, this cascade of conformational changes, reaches an unloadable subunit, which has nothing bound. So we are trying to add more to this mechanism, but the behavior of this protein is very complicated. So I should just say, I didn't actually record any of the data in this paper. I don't work on optical traps at all. If you have any questions about the experiments that I do, which more directly investigate the model I just showed on the last page using single-molecule fluorescence, feel free to ask me anytime. These experiments were done by Adrian, who's a postdoc in the lab, who's probably going to start as a professor at UC San Francisco next year. This is the Sauer lab. It's a great place to Year Up. If any of you are looking for a new Year Up, come say hi. So if you have any other questions about optical traps or ClpXP, feel free to ask me.
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MIT_508J_Biological_Chemistry_II_Spring_2016
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13_Protein_Degradation_2.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: We're going to get started. And today we're going to move forward with the protein degradation module, and looking at the degradation chamber of E. coli, ClpXP. So this is a degradation machine. And effectively, what we see in this case, and as mentioned in lecture last time, is that there are gigantic chambers that isolate protease active sites. And so we're going to examine this particular machinery as a paradigm here. So the Clp system was first identified in E. coli, and it's highly conserved. And what we'll see is that there's encapsulation of an active site in a large degradation chamber. So there's two components, ClpX and ClpP. And so we're going to look at both of these components individually, and then see how this machine works. And so ClpP is the proteasome. And it's a serine protease. So we talked about serine proteases last time. So there's the catalytic triad of serine, histidine, and aspartate. So there's formation of a covalent acyl-enzyme intermediate. We learned that the serine residue is the active site nucleophile. And what we're going to see is that this degradation chamber has 14 active sites. And what this does is degrade proteins into short polypeptides. So those are peptides of about seven to eight amino acids. And so if we consider a cartoon of the structure, what we find is that we have two back-to-back rings here. And each of these rings is a 7-mer. So we have two heptamers. In terms of size, they're approximately 90 angstroms here and approximately 90 angstroms across. And then this region between the two rings is sometimes referred to as the equator. So here we have two back-to-back rings. And these rings generate a chamber. And proteins upwards of about 70 kilodaltons can fit. And so if we take this, looking at from the side, and just rotate 90 degrees to ask, what does it look like from the top? So this is a side view. OK. What we see is that there is a very small pore here. So we have the seven subunits. And then in the center there is an axial pore. And this pore is small, about 10 angstroms in diameter. So when thinking about that size, we need to think about the size of some large protein, right? If we have something on the order of 70 kilodaltons with a fold, that protein's not going to fit through this hole in that state here. OK? So it's too small for a big, folded protein. But then basically, if we take this and, rather than just looking at the top, we cut through and ask what's going on in the interior, what do we see? OK, so cut through. What we see now is that there's a chamber of about 51 angstroms. OK, so this is the interior degradation chamber. OK, so one question we're going to address as we move forward is that, how is it that a polypeptide gets through this hole into the degradation chamber that can accommodate a protein up to 17 kilodaltons? So small axial pore versus large degradation chamber here. So we'll look at some structures of ClpP and then go on to ClpX. So what we're looking at here are effectively what I've drawn out in cartoon form on the board. So here we have the side view of ClpP. We have the top ring, the bottom ring, and here's the region between the two, the equator. If we look at the top, here they're describing the axial pore as a portal. And here's the cutaway view. So this hole is very small. And if we look at the cutaway view, what we see is the degradation chamber here. And basically, the seven different serine protease active sites are shown here. And this is a side view cutting through the middle here. So we see that these serine protease active sites are down in this region here. If we take another view and look at-- again, this is cutaway, so cut through the side view-- what we can look at here is the machinery in the active site. So we learned last time about the catalytic triad, with the aspartate, histidine, and serine. And in this particular structure there's a serine protease inhibitor bound to the serine side chain here. So that can serve machinery. If we look at the structure of an individual ClpP subunit, those are shown here from several different organisms. What we see-- if we can think about this as the top part, and this is the bottom part of one ring-- we see there's a region with axial loops. There's a head domain and what's called a handle region. And the catalytic triad is located at the juncture of the head and handle region. And you can, again, look back to the cutaway views to orient that within the whole chamber. These axial loops, we'll see, are important for interaction with ClpX. And we'll talk about that component of the machine in a moment. Here, again, just structures of ClpP from various organisms, E. coli, Streptococcus, human. We see that they're all very similar here. So what is ClpX? Moving forward. OK. So ClpX is effectively an accessory protein. And in some respects we can think about it as a lid to the proteasome. OK. It's a hexamer. So there's a mismatch here, in terms of the number of subunits in ClpP and ClpX. This is different from what we saw with GroEL, GroES, where they are both heptamers. ClpX is a hexamer. And it's an AAA-- so triple-A-plus unfoldase. It's an ATPase here. And effectively, what we'll see is that ClpX has an important role as an accessory protein that unfolds the polypeptide that's destined for degradation by ClpP. So it unfolds the polypeptide, and we're going to have to ask how as we go through. OK. And in addition to unfolding, it also threads that polypeptide that's being unfolded in through the axial pores such that it can reach the degradation chamber, and threads it into the degradation chamber. And so if we look at ClpX from a top view, again, we have a hole. And we have a 6-mer here. So if we take a look, in this particular depiction what we're seeing are the top views and the side views. So here we have ClpP. Here we have ClpX. And what we want to ask is, how is it that ClpX binds to ClpP? And how is this mismatch in terms of the number of subunits accommodated, right? So it's not that one subunit is precisely going to act with one, because we have six in-- throughout. Because we have six in ClpX and seven in ClpP. And so if we think about how ClpX binds to ClpP, what we see is that we have the 6-mer. So we're looking from the side view. And there's some loops that are called IGF loops. So these are tripeptide motifs. And they're flexible. And these loops interact with hydrophobic regions of ClpP. And this flexibility helps accommodate the six versus seven subunits. So if we look here, here we see these tripeptide loops. And see, we're only seeing three. But there's one per subunit, so 1, 2, 3, 4, 5, 6 here. And then what's shown here in red are the hydrophobic regions of ClpP where these can bind. And where do we see these regions on ClpP here? They're a bit removed from the axial pore here. So how many of these loops are needed? Just to note, there's been studies done where these residues are deleted. And the question is, how many of these motifs are important for this protein-protein interaction? And what's been found in test-tube studies is that a minimum of two are required to get interaction between ClpX and ClpP here. Yeah. AUDIENCE: Is it known how many actually interact in vivo? Like, do all six interact at any given time? PROFESSOR: I would presume so, but I don't know. Right? So we very much think of this as coming together as shown there. But don't know. Joanne, do you know? No. No. I would say it needs to be pretty stable. Like, there's always dynamics. But as we see how this machine works, this hole is going to have to allow the polypeptide to thread through and get through that axial pore for the polypeptide to get in the degradation chamber. So you'd imagine you want that to be lined up quite well in order for it to be efficient there. OK. So what are these triple-A-plus ATPases? This is a very important group of ATPases. So what the name means, ATPases associated with various cellular activities. And they're super-duper, given this. Hopefully everyone gets a triple-A-plus on the exam tonight. The superfamily is involved in many cellular functions, and-- take a look. So, many diverse functions. Cell membrane fusion, trafficking of vesicles, cytoskeleton regulation, transport, organelle biogenesis, DNA replication, transcription regulation. And what we're really interested in here is protein degradation. So they come up in a variety of processes. And although these processes are very different, all of these triple-A-plus ATPases share a common protein architecture. And I'll just point out that there's an ATP binding module. And some details are given here in terms of the motifs. And then really what we'll focus on, in terms of aspects of this course, is that they form oligomers that are ring- or cylinder-shaped. And they're all hexamers here. And so, of importance to ClpXP, these ATPases have the ability to remodel conformation of macromolecules. And so here we're focused on unfolding. Yeah. AUDIENCE: [INAUDIBLE] question, but what are ATPases that aren't associated with cellular activities? PROFESSOR: Well-- [LAUGHS] [INAUDIBLE] AUDIENCE: Is this definition based on the architecture? PROFESSOR: Strictly, yeah. I mean, there's-- GUEST SPEAKER: If you look at tRNA synthetases, they have ATPase activity. Hundreds of proteins have ATPase activities. PROFESSOR: Right. So-- GUEST SPEAKER: They hydrolyze ATP to ADP and Pi. AUDIENCE: Right, right, right. But that's a cellular activity, right? AUDIENCE: So, like, what aren't AAA-plus ATPases? PROFESSOR: Well, aminoacyl-tRNA synthetases are not triple-A-plus ATPase. What we'll see in terms of the non-ribosomal peptide synthetases, they're not these triple-A-plus ATPases. Right? So these-- I mean, yes, OK. All ATPases, the enzymes in a cell, it's going to have some role in a cellular activity, right? So maybe this name isn't very helpful. But what's common about all of these is that they share this common structural motif. They form these hexamers. But within that, there's quite a bit of diversity, because they have all these different functions. So we can just see that here, to some degree. So-- oops. And there's a typo, which I'll fix before posting. But if we take a look just at two examples here of different hexameric rings-- so these are two different triple-A-plus ATPases-- what do we see? So in common, they're both hexamers. In common, they both have an axial pore here. But we see different elements of secondary structure. And granted, these are both depicted in a bit of a different way. But if we look here-- I mean, look. We have these alpha helical regions around the exterior that we don't see here. And in this view-- I show this particular one because it's depicted where the ATP is binding. So you can see the ATP binding to each subunit here. So, as shown, six ATPs bound. So the structural diversity is quite tremendous. And here's just another example. So these are three different triple-A-plus ATPases of the Clp system. So we're going to focus on ClpX, but it's not the only one. And so what we're looking at here is ClpX. We have another family member, ClpA, and here, ClpB. And so what we see is, subunit to subunit, whether it's X, A, or B, there's quite a bit of difference, right? So ClpX is the most simple, in terms of the architecture here, for that. And so one thing people think about is, in terms of the different activities that have been associated with these different family members, how is it that these different structural features play a role? OK. Here. So, coming back to ClpX and the depiction we saw before, ClpX is an unfoldase. And what's really a key point here is that ATP hydrolysis by ClpX is going to power conformational changes that allow for mechanical unfolding of this protein that's condemned for degradation by ClpP. And that's what's going to also allow for translocation of the resulting unfolded protein into the degradation chamber. So the action of ClpX is allowing that protein to fit through this axial pore and be threaded into the chamber. So with that, what are the questions we need to address in thinking about how this macromolecular machine works? One, how are substrates recognized? So there's some certain group of proteins that are going to be degraded by this machinery. What is the mechanism? How is it that ATP-dependent conformational changes of ClpXP drive unfolding and translocation? And what is the substrate selectivity? So that's where we're going to move forward with. And so the first question we need to ask is, how are the substrates recognized by ClpX? OK? Here. And so, what are possibilities? OK. First, what we'll consider is a degradation tag. So when I draw these cartoons, I'm only going to show one of the two rings for ClpP. It's not that it's only one. This is just for simplicity. But imagine that here we have X, here we have P. And we have some condemned protein, which I'll just draw as a circle. So the cell no longer wants this protein. It needs to go away. And we can imagine, as one possibility, is that a degradation tag can be attached to this polypeptide. And what we find is that there's a particular tag called ssrA tag that is used to tag proteins for degradation by ClpXP. So we can think of this tag as a zip code. If a polypeptide gets modified such that this tag is appended, it's going to end up going to this degradation machine such that it gets degraded. The tag is 11 amino acids. It's attached to the C-terminus. OK. And the sequence is A, A, N, D, E, N, Y, A, L, A, A. And so what happens in this case, as shown-- we can imagine that this tag binds to the pore of ClpX directly. And the tag, when binding, enables translocation. So here-- OK. And this pore has what are termed pore loops that are involved in tag binding. And in particular, there's a region, GYVG-- so, a four-amino-acid sequence-- that is thought to grip and drag the substrate. OK? Here. And of course it's not gripping like we would, but there's some interaction there happening that allows that to occur. So you'll see there's a lot of mechanical-type cartoons and language used in describing these machines. OK. So what is another possibility? So another possibility for how an ATPase could interact with a degradation chamber is that the protein substrate binds to an extra domain attached to the ATPase. OK. And I point out, this possibility is not for ClpX, but it's one to be aware of, because it can occur. We saw some of those other ATPase are quite complicated. So in this case, imagine we have our ATPase, we have the degradation chamber. And this ATPase has some extra domain that effectively can bind the condemned protein and help deliver it to the pore here. And just as a third possibility, and something that we'll see moving forward, is that there is involvement of an adaptor protein. So in addition to the ATPase and the degradation chamber, there's an adaptor protein that comes into play. So in this case, the protein-- OK, adaptor-- OK. And this protein helps direct it to the pore-- so, the condemned protein to the ATPase. OK. So for instance, here we have the ATPase. Here we have the degradation chamber. And maybe there's some additional protein that facilitates getting the condemned protein to the ATPase. And so something to keep in mind, and what we'll see with ClpXP, is that one and three are not mutually exclusive. And there's an adaptor protein named SspB that can help deliver ssrA-tagged polypeptides or proteins to the degradation chamber here. So we're going to think about these ssrA tags quite a bit. And something else to be aware of is just that these ssrA tags are not the only ways of tagging proteins for degradation. We're not going to talk about it in detail in this course, but you should be aware of something called the N-end rule. And this is really cool. So this rule basically states that a half-life of a protein is determined by its N-terminal residue here. And this can be called an N-degron. And these N-degrons are recognized by proteins such as ClpS and E. coli. And as a result, these proteins can get delivered to degradation machines. So for instance, in addition to ClpXP, there's an ATPase, ClpA, that can associate with ClpP and be involved in degradation of polypeptides via this N-end rule. And in terms of the rule, depending on the identity of this N-terminal amino acid, it may be stabilizing or destabilizing, in terms of protein lifetime. If you're curious to know more about that, we can refer you to some literature. So here we have a cartoon looking at a native protein substrate that needs to be degraded. It's been modified with a tag. We have ClpX here. We have ClpP. Here's the tag. And in addition, we can have this adaptor protein SspB and the adaptor ClpS. So let's think about this tag for a minute. And we need to think about this tag from the standpoint, one, of in vitro experiments, because we're going to begin to look at some experiments that were done to understand how this machine works. And we also need to think about this tag from the standpoint of the cell. So if we think about an in vitro experiment where we want to study how ClpXP degrades some protein substrate, we can use this ssrA tag. And it's quite easy to attach 11 amino acids to some protein or polypeptide at the C-terminus. We can do that by protein expression, we can do that by chemical synthesis here. And so we're going to look at a number of experiments where this ssrA tag has been appended to certain model substrates, moving forward. So what about in the cell? So when is this ssrA tag attached to a protein? So are all proteins that need to be degraded destined to ClpXP? Just intuitively, what do you think? I see shaking heads, no. Right? There's many, many proteases around. So what proteins are destined for degradation by ClpXP? That's what we're going to look at, and how this tag is appended. And so effectively, this ssrA tag, say, in E. coli, is used-- one, because protein degradation needs to be tightly regulated. But two, it's used for dealing with proteins that exhibited stalled translation. So this discussion is going to bring us back to the ribosome here. So we want to ask what proteins in the cell are tagged with ssrA. How is the tag attached to the [INAUDIBLE] protein as well here? This is just a cartoon showing an adaptor protein helping direct this tag to the substrate here. So we're going to just move forward to this slide. This tag is specifically added to proteins that are experiencing stalled translation. So it's estimated that on the order of 0.5% of E. coli translations result in ssrA tagging. And so this is thought to be one largely of quality control. So you can imagine, if the ribosome stalled, there could be buildup of peptide products that aren't wanted. And the translation machinery could get blocked, and we don't really want that to happen here. So here's our friend, the ribosome. And here's looking at the 50S ribosomal subunit. And we have a polypeptide emerging from the exit tunnel. So these should all be familiar at this stage. And so what happens when this ribosome is trying to synthesize a polypeptide and it just gets stuck? So this ssrA tag is attached to the C-terminus of proteins. And as we're going to see, this occurs cotranslationally. And it's very, very interesting machinery. So what we see here is that there's a new player we haven't yet seen. And this is called ssrA, or tmRNA, for transfer messenger RNA. And it's involved in attachment of this ssrA tag to polypeptides that are having stalled biosynthesis on the ribosome. And so this player acts as both a tRNA and an mRNA. And we can take a look at the structure shown here. So here we have tRNA in the cloverleaf depiction, just an Ala-tRNA Ala. And if we take a look here, what do we see? At this end we have a region of the tmRNA that looks like a tRNA. Right? Quite similar here to the [INAUDIBLE] prime end. And then we have this additional region. And then if we look down in here, what do we see? We see a region that, with a little imagination, we can think looks like mRNA. And if we take a look at the various codons, what we see is that the ssrA tag is encoded there, along with a stop codon. So effectively we have a tRNA look-alike. We have an mRNA look-alike that is encoding this ssrA tag. So what happens? There's a partner protein called smpB just to be aware of. And in complex with smpB, it's actually EF-Tu that delivers this tmRNA to the ribosome here. So this is pretty interesting, just from the standpoint of what we know about EF-Tu. We don't have a typical anticodon here. So how does that happen? We're not going to go into details, but it's something-- you know, curiosity should beg those questions. So what happens? We can look at this cartoon overview here. And so the color coding within this is helpful, in terms of keeping track of pieces. But here we start with our stalled ribosome. So the mRNA is bound. We see there's a peptidyl tRNA in the P-site. You know the polypeptide has a number of amino acids, and the A-site is empty. And for some reason, no new aminoacyl tRNA is coming in. So the ribosome stalls. And as a result, this ssrA, or tmRNA, is recruited to this stalled ribosome. And so here we see the tRNA end in yellow, with the alanine attached. And here we have that region that's mRNA-like encoding the tag. So this biomolecule gets recruited. And what do we see? It enters the A-site. So here we see the tRNA end in A-site, and we have the rest of the molecule here. Then what? There's formation of a new peptide bond, so we have peptidyl transfer. We see that alanine is here. Look. That looks quite a bit like the hybrid states we talked about, where we're seeing these ends shift into the E-site, not shown, and the P-site here. And then what happens? There's translocation and there's message switching. So the original mRNA gets kicked out. And what do we see? Now that mRNA-like region of the tRNA is in A-site here. Then what happens after replacement of the mRNA? Translation can occur, which results in synthesis of the ssrA tag. So that's how this tag is attached to the C-terminus of the polypeptide. And elongation occurs until that stop codon in the tmRNA enters the A-site. And then peptide release occurs here. So the result is a protein that has the ssrA tag attached to its C-terminus. And that protein will be directed to ClpXP. So, pretty cool. Yeah. I think so. There. We don't ever leave the ribosome too much within these units. Just to point out, this tag is universal in bacteria. So here's just a table of phylogenetic distribution. You're not responsible for these details. Yeah. AUDIENCE: About the last slide-- so is the tag attached after it's stalled? Like, is the original protein completed? Or just the original mRNA removed and detached the tag? Or it's both? PROFESSOR: Yeah. So what does the cartoon suggest? AUDIENCE: It feels like it's already on the C-end here. PROFESSOR: Well, the ribosome can stall at various points. So imagine you have a 100-amino-acid polypeptide that needs to be synthesized. The ribosome could stall after amino acid 20 or 40 or 60. It's not that the whole polypeptide has been synthesized and then this gets put on. It may be some fragment there for that. So, yes. So in terms of this adaptor protein, I just want to make a note in terms of the adaptor. So these adaptor proteins can help with regulating the substrate specificity of triple-A-plus ATPases. And effectively, depending on the system and depending on the adaptor, it may enhance or it may inhibit degradation here. So it's a case-by-case basis. This SspB shown in the cartoon here is a dimeric adaptor for XP, and it promotes degradation of certain substrates. And effectively, it enhances recognition of this tag by the machinery such that the degradation rates are enhanced. So it's not that it's required. It's just helpful, and accelerates the process. So just an interesting observation regarding SspB-- it can be co-purified with ribosomes here. And in terms of its structure, it has some resemblance to known RNA-binding proteins. And this resemblance has begged a question, does SspB itself help with linking protein synthesis and protein degradation? So is it possible that SspB could help promote binding of ClpX to polypeptides before full release to the ribosome? That's something people have wondered about. And initially this protein was classified as a stringent starvation protein. That's where the name comes from. So if we just take a quick look at its structure here, what do we see? So here is SspB. And then here we have structures from ribosomal proteins. And so in SspB, ClpX binds on this side. And effectively, here we look at the ribosomal proteins that bind RNA. And they have these RNA-binding sites there. So there's some similarities in terms of the alpha helix, in terms of the beta sheets here. And also, I'll just note, in terms of SspB and the ssrA tag-- so if we take this tag-- So this is our ssrA tag here. What's found is that ClpX recognition is on this end and SspB binding is on this end here. So in different points. So what this indicates here is that SspB and ClpX can bind simultaneously. OK. But this is small, so we can expect that there's some clash here for that. So where we're going to close is just looking at an overview as to how this machine works, and the model that then, starting on Friday, we'll look at experiments that were designed and performed to inform this model. So if we look at this in one type of cartoon, what are the stages? So we can think of three as depicted here, where there's some sort of initial recognition. So the ssrA tag of this condemned protein binds to the axial pore of ClpX. And this process does not require ATP hydrolysis. So here we see a folded substrate. This degron is another word for one of these tags, ssrA tag. So we see there's recognition here. Then what happens, ClpX unfolds this substrate. So somehow it has to grip and pull and apply a force that unfolds the polypeptide, and threads that unfolded polypeptide into the degradation chamber. So, you know, kind of this pulley system is shown here. This chopper-type thing is shown here. You can use your imagination in this unit for how to depict this machine. So we see that the polypeptide is being unfolded and threaded through ClpX into this chamber, where it gets chopped up by the serine protease active sites here. So for unfolding and translocation, ATP is needed. ClpX is hydrolyzing ATP to allow this to occur. In the degradation chamber, this degradation part is independent of ATP. Right? The serine protease doesn't need that here. So how can we kind of break this up further into a model that we can test? What I present here is the working model. And just note, the orientation is flipped here. So we have ClpX on the bottom and ClpP on top. So what happens here? We can look at this in terms of five steps. And we can begin here, with binding. So this ssrA-tagged protein needs to bind to ClpX. And that binding is associated with a dissociation constant, or Kd here. What do we see? After binding we have a second step, which is denaturation. So the polypeptide becomes unfolded. And that's defined by a rate constant for denaturation, as shown here. If we look next, we have translocation. So this polypeptide is moving through ClpX into the degradation chamber. And this is also associated with the rate constant-- so, rate constant for translocation. And both of these steps require the use of ATP. Once this polypeptide is in the chamber, we have step four, which is degradation. And again, we have k deg. This is fast. And then in this last step here, there's some release. So somehow these polypeptide fragments need to be released from the chamber. AUDIENCE: Is ClpP still a dimer at this point? PROFESSOR: Yes, yes. So often the cartoons are drawn just showing one of the heptamers. But think of it as a dimer, with these two back-to-back rings here for that. Right. So we have five steps here. Each one of these steps has a rate constant. And one question we want to ask with this is, what is the rate-determining step? And the quick answer where we'll end today, and as indicated in this overview, is that degradation is fast relative to denaturation and translocation. And there should be an intuitive aspect to that. We heard about last time how proteases give these tremendous rate accelerations. And if you have an unfolded peptide, those sites where cleavage will happen are going to be exposed there. So what we're going to ask is, is it possible to make experiments, design experience, where we can separate the denaturation process from the translocation process and analyze those-- and in the process of doing so, ask, what is the ATP utilization for each step? And what is the role for ATP in this process? And so on Friday we'll begin with discussing substrates, the design of substrates that have been used, to examine this model in more detail. Is there one question next? AUDIENCE: I was just wondering if the degradation step also removes translational modifications, or [INAUDIBLE] PROFESSOR: In the degradation step? That's going to depend. I mean, you can have different types of bonds with post-translational modifications, right? Right. So in the eukaryotic system, you have a post-translational modification to direct this condemned protein. And that machinery-- so they're ubiquitins, and you get this polyubiquitin tail. So you saw ubiquitin in recitation number one. And the eukaryotic proteasome has the ability to chop those ubiquitins ends off for recycling there, in that. So that's one example.
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MIT_508J_Biological_Chemistry_II_Spring_2016
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5_Protein_Synthesis_4.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ELIZABETH NOLAN: Today we should be completing the translation cycle. And the next topic that will come up is use of antibiotics as tools to study the ribosome and translation. So just a recap from last time, we went over the delivery of aminoacyl--tRNAs by EF-TU and looked at this model for understanding how that happens, OK? So recall, we discussed the initial binding of the ternary complex of EF-TU GTP and the aminoacyl-tRNA. And that's codon independent. When a codon/anticodon match occurs, we have a push in the forward direction. EF-TU's a GTPase. There's activation of the GTP center. So conformational changes, GTP hydrolysis. So we have EF-TU and the GTP bound form. There's conformational change, and ultimately accommodation of this tRNA in the A site. And that allows for peptide bond formation. So where we left off last time was discussing the conformational changes that occur in the decoding center and also in the GTP center of EF-TU. And just to highlight, I mentioned that there are conformational changes within the 16S rRNA, in particular three nucleotides that occur when it's a cognate codon/anticodon interaction. And these are just shown here. And effectively what we're looking at in these three panels are the 16S rRNA in the absence of the tRNA, in the presence of the tRNA, but the tRNA is removed from this image for simplicity, and then with the tRNA bound. So some of the easiest changes to see here are with A1492 and A1493. So if we look in the absence of tRNA, they're pointing down the bases. And here as a result of tRNA binding in the A site, we see that A1492 and 1493 are flipped, flipped up. OK. And if we look here, you can see how these are interacting with the bound tRNA. OK. So this conformational change helps to accelerate the forward steps. So that's in the decoding center. And then also just remember 70 angstroms away in the GTP center of EF-TU, there's conformational change of these hydrophobic residues that are thought to be a hydrophobic gate that allows histamine 84 to activate a water molecule for attack in GTP hydrolysis. So at this stage we're finally ready to have a peptide bond formed by the ribosome. And so we need to think about that mechanism and then what happens after. I'll say, so effectively what we have in the P site is the tRNA with some growing peptide chain. [WRITING ON BOARD] And then we have the aminoacyl-tRNA in the A site. [WRITING ON BOARD] And so what happens effectively, we have attack from here, release such that we end up with a P site with a deacylated tRNA. And in the A site we now have the peptidyl-tRNA that has grown by one amino acid monomer. [WRITING ON BOARD] Here, OK? OK. And so this is the N- terminal end of the protein or the polypeptide, and here's the C terminal end here. So thinking about this mechanism and having nucleophilic attack from this alpha amino group of the aminoacyl-tRNA in the A site, what do we need to think about? Is there anything surprising or unusual? AUDIENCE: Think about protonation state. ELIZABETH NOLAN: Yeah, right. Exactly. We need to think about the pKa. So typically do we think about an alpha amino group being protonated or deprotonated, that physiological pH. Yeah. Protonated we typically think about an H3+, not an H2 here. So what does that tell us? There has to be a general base somewhere that deprotonates this alpha amino group, such that we have this species that can attack, and then can imagine just formation and collapse of a tetrahedral intermediate here. So what is the mechanism of catalysis? OK. Our room's possessed. So what is the mechanism of catalysis here? What do we know? So we know from looking at the structure that the ribosome is a ribozyme. So no proteins in the catalytic center. What else do we know? There's no metal ions there and there's no covalent catalysis. So really what is a paradigm here? We have a paradigm of conformational change and effectively we have substrate positioning. You can imagine there's some protons shuttling in an electrostatic network that allows this to happen. And so as soon as this aminoacyl-tRNA enters the A site, we have formulation of this peptide bond. So what needs to happen next-- and once the screen gets fixed, we'll look at an actual depiction of these players in the PTC. What needs to happen after the peptide bond forms-- and we have now this peptidyl tRNA in the A site is that before the next round of elongation effectively we need to reset, and the mRNA and the tRNAs need to move relative to the ribosome. OK. So effectively we need to get this deacylated tRNA to the E site, and we need to get this peptidyl tRNA to the P site such that the A site is empty. Is it not going down? Pardon? OK, that's fine. So this process is called translocation here, and effectively we can just consider the three sites. We have the E site, the P site, and the A site. [WRITING ON BOARD] OK. And in this process another elongation factor, this time elongation factor G is involved. And the outcome is that we end up with the deacylated tRNA in the E site, the peptidyl tRNA in the P site. OK. And then the A site is empty such that the next aminoacyl-tRNA can come in. OK. So immediately after peptide bond formation, and this is the state after the process called translocation. So EFG is also a GTPase here. And effectively what happens is that EFG bound to GTP binds near the A site and GTP hydrolysis occurs. Bless you. OK. And as a result of GTP hydrolysis, there's conformational change. OK. And this results in translocation and then EFG is released. And in thinking about translocation we think about two steps. OK. And so the first step is something called formation of hybrid states, and then the second step is the actual movement, the mRNA and tRNA relative to the ribosome here. So we'll take a look at this. So here's just an overview of peptide bond formation backtracking a little bit, and then back to here. Thinking about confirmations and what this looks like, what we're seeing here in this depiction is the P site tRNA in green, we have the A site tRNA in this red color, and then we see the 23S rRNA shaded in light blue in the back. OK. So here's A76, here is an attached amino acid, and we see the nucleophile here, and attack there. OK. So substrate positioning within this active site. Here, as I talked about this translocation process, we're now at this stage as shown with a depiction of the ribosome. So this is where we're going using EFG in complex with GTP to allow the translocation process to occur. Here is another cartoon depiction. And so if we begin after peptide bond formation, we see the incoming EFG in complex with GTP. And if we look in this cartoon, it shows EFG binding near the A site. So it's not binding exactly in the A site, but nearby. OK. There's GTP hydrolysis. We now have EFG in the GTP bound form. OK. And now we see translocation, and that these tRNAs have moved. And after this step, EFG is released. And at this stage the ribosome is ready for its next round of the translation cycle. So let's take a look at what we know about the structure of EFG. So this is a really beautiful example of molecular mimicry, and EFG is shown here. And if we compare EFG to the complex of EF-TU with tRNA like the ternary complex shown here, what we see is they look very similar, and this domain IV of EFG resembles the tRNA quite well. So just looking at that we can begin to think maybe its domain IV that's coming in near that A site to have some interactions and cause this translocation event. Here is just a comparison of the ribosomes with either EF-TU and tRNA bound, which we've seen before, and here we see EFG in this red color. So quite similar, but are they the same? And the answer is no. So quite recently a crystal structure was obtained with EFG bound to the ribosome, and it was determined that in this structure EFG is bound in the post translational state. So what do we see? If we take a look here, we have a tRNA in the E site, a tRNA in the P site, and here in red we see EFG. And if you take a look, recall that ribosomal protein L12 that was involved in recruiting the ternary complex, we're seeing that there's some interaction here with EFG in that protein as well. If we look in this panel B-- so we look at a close up view near the A site, what's happening? Here's a P site tRNA, here is the mRNA, and here is EFG bound, and it's domain IV that's sticking its way in. OK. And so what are we seeing here? Basically these tRNAs have moved at this stage and we still have EFG bound here. So is EFG interacting with the A site codon of the mRNA based on this view? AUDIENCE: Not really. ELIZABETH NOLAN: Yeah. No. So not really. So it's not interacting in the same way as the tRNA. That's the take home here. They look the same, but the details are different here. OK. So what about these hybrid states? They have two different views, and the slides for looking at this-- effectively what the hybrid states are-- OK. These describe basically the orientation of the tRNAs. So effectively they can be P/E or A/P here. And the P/E state talks about having the anticodon end in P and the deacylated three prime end in E. OK. So here we're referring to this tRNA that ultimately needs to get ejected from the ribosome. And A/P we have the anticodon end in A and the peptidyl three prime end in P for the tRNA with the growing peptide chain. So effectively these hybrid states are describing the movement of the three prime ends of the tRNAs with respect to the 50S subunit. And that's shown in cartoon form on the slide here. OK. So effectively if we take a look, we have accommodation, so the aminoacyl-tRNA we see there's formation of a peptide bond. So there's some color coding here, and then look, rather than having these tRNAs straight up and down, we see that the three prime ends have shifted. So here we have the P anticodon end in P, three prime end in E, here AP with anticodon end in A, and peptidyl three prime end in P. OK. So first the three prime ends move, and then what do we see after the help of EFG? The five prime ends move and the A site is empty and able to take the next aminoacyl-tRNA. Here's just another view of the process for you to look at here. Yes? AUDIENCE: It's kind of from a few slides back, if that's OK. ELIZABETH NOLAN: That's OK. AUDIENCE: So on EFG where you have that anti-codon looking blue, there's no hydrogen bond interacting with the transcript. Is that the take away? ELIZABETH NOLAN: Can we say that based on what I'm showing you on this slide? AUDIENCE: I don't know. It just looks like it's around there, but I can't tell what the actual hydrogen bond interaction part. ELIZABETH NOLAN: So those details are outside of the scope. EFG will interact with that mRNA, the peptidyl tRNA, but it's interacting differently than a standard aminoacyl-tRNA. So it's not really interacting with the codon at this level of depiction. We're not seeing individual bonds or hydrogen bonds. So we can't make a conclusion about that based on this depiction here. AUDIENCE: What's the resolution of this structure? Is it high resolution or is it-- ELIZABETH NOLAN: Yeah. That's another issue here. I don't recall the resolution, but they're not great. So if you have a four angstrom resolution structure, for instance, is that type of information even available versus the resolution of maybe 1.5 or 1? Yeah. This resolution I don't recall, but that's a very good point to bring up. I don't think it's high enough to know that would be my guess here. Pardon? Oh, the resolution? So rewinding back to recitation last week. So crystal structures have a resolution. And so what does one angstrom resolution versus two versus four allow us to see? Oh. Oh. The question is, are there hydrogen bonding interactions between EFG, and say, the mRNA? Well, there's definitely going to be hydrogens because you're going to have C-H bonds or C-N bonds. But-- AUDIENCE: I can't see any of them, at least from this picture. And also, is this picture actually a picture or is it just like a [INAUDIBLE]? ELIZABETH NOLAN: This is from the crystal structure. AUDIENCE: But it's not itself a crystal structure. They take out all the hydrogens, wouldn't you, and you wouldn't see anything, right? ELIZABETH NOLAN: OK. So this is from the crystal structure and you can make choices as to what information you put in your depiction, whether or not you're going to show certain residues say, or just the backbone there. AUDIENCE: I'm not sure what we are expecting to see, but don't see. We will get the heteroatom distances. Yeah. And so the question is, how much error is there, and if there's one error, you can't tell where the analide regions are. AUDIENCE: So we're looking at distances like between potential hydrogen bonding sites as our measure-- ELIZABETH NOLAN: Or heteroatoms because you may not be able to see that hydrogen. But you can know something like, oh, if this heteroatom and that heteroatom are so many angstroms apart, is it likely that there's a hydrogen bonding interaction or not based on knowledge of bond distances here? So one thing I'll note and will come up as we're discussing antibiotics, I said nothing about how they've obtained the structure. And that's just something to keep in mind. And this also gets to the question of resolution, and what can you see? But they had tremendous difficulties getting this structure, and they had to use a mutant ribosome, and they strategically used an antibiotic to stall the ribosome here. So many, many attempts to get crystals that are even good enough to get some information here. OK. So back to these hybrid states and the formation of the hybrid states. Something important to know about-- and this is something I have a lot of time seeing in any cartoon that's presented is that the 30S subunit undergoes some conformational change called ratcheting. And effectively the ribosome can exist at this stage in either an unratcheted or ratcheted state and EFG selects from one over the other. So EFG will bind this ratcheted ribosome. And effectively what that terminology is describing is a small rotation of about 6 degrees of the 30S relative to the 50S in one direction. So the ribosome will be going between unratcheted and ratcheted. EFG can bind the ratcheted form. OK. And after that occurs, they'll be GTP hydrolysis on these translocation events here. So an awful lot is going on to get that one peptide bond formed and the ribosome ready to do it again. Where we're going to go at this stage is a brief discussion of the termination process in translation and the players that come up there. So effectively the elongation cycle is going to continue until a stop codon enters the A site. That's making the assumption some unforeseen circumstance hasn't happened to this ribosome. It hasn't stalled or prematurely stopped translation. So what happens when a stop codon enters the A site? OK. Again, we have translation factors. These translation factors are release factors that recognize the stop codon, and they have the responsibility of cleaving the polypeptide chain from the P site tRNA. And so there are two different classes of release factors. We have class 1, which are release factors 1 and 2. Release factor 1 and release factor 2 each recognize they're in stop codons. So, for instance, RF1 recognizes UAA and UAG. Whereas RF2 recognizes UAA and UGA here. There's a class 3 release factor RF3. This one is a GTPase and it has the job of accelerating dissociation of RF1 or RF2 after peptide release. So we'll look a little bit at structure and then one schematic for how this may all happen. So similar to EFG and the ternary complex of EF-TU GTP and the tRNA, we have another example of molecular mimicry with these release factors. And so initially when release factor 1 was crystallized, the structure shown here was obtained. So this was the protein crystallized in the absence of the ribosome, and it was a little difficult to reconcile this structure with function immediately. And then in later work RF1 was crystallized bound to the ribosome. And that structure is shown here. And so if we compare the left to the right or what's described as the closed to the open version of RF1, what we see is that there's a pretty substantial change in conformation when we're looking at RF1 on the ribosome. And if we use a little imagination we can think about RF1 resembling a tRNA. OK. We have this region here that's sticking out. And if we look at an overlay of RF1, so this structure of the ribosome bound structure in a tRNA, what do we see? So we have the tRNA, we have the anticodon end down here, we have the CCA end of the tRNA up here. And so what do we see? In terms of RF1, we have this PVT motif down here and we have this GGQ motif up here for that. And so this motif is important for hydrolysis of the peptidyl tRNA. And that's where it is. In terms of a schematic for termination as a way to thinking about this-- so here we have our ribosome that's then translating and now there's a stop codon in the A site. So here comes a release factor, either 1 or 2. It recognizes this stop and binds. So there's hydrolysis of this linkage-- and should think about that chemistry to what's happening. --peptide release. So what's shown in this depiction is that RF3 comes in and it was in GTP bound form. It binds in the region of the A site. There's some exchange. We have GTP coming in here and then some additional steps that involve GTP hydrolysis by RF3 involvement of the ribosome recycling factor. And we see that our friend EFG comes into play again here along with initiation factor 3. So some of these other translation factors seem to play a role in this termination cycle. And really, again, it's a question of looking at the data that's presented to you and interpreting that data and drawing some conclusions. So there's still a number of questions about this process and the ribosome recycling that remain. So if we look about this slide and where we've come in this discussion of translation effectively, all of the pieces are shown here for prokaryotes. So this is just a map to work your way through when studying the system. But we have initiation, we have elongation, and then this process of peptide release and ribosome recycling. OK. And so throughout this we're seeing the action of GTPases. So the power of GTP hydrolysis is needed. Conversion of chemical to mechanical energy. There's a lot of conformational change that's happening. The slides I've shown you don't do that justice but it's something to think about and keep in mind, and that this ribosome is amazingly dynamic. And so that is what's going to lead us into the next subtopic related to the ribosome, which is thinking about how have some of these observations been made? So how is it that we've obtained structural insights into the ribosome at different steps along this translation cycle? And just as for consideration, there's a little excerpt from a paper I like. So this was in 2010. So shortly after the Nobel Prize was awarded. And so there's a number of perspectives, retrospectives in the literature. And in this one called the Ribosome Comes Alive, Joachim Frank is talking about these pioneering work of the X-ray structure. And just in yellow here, he's stating, those who might have expected that the atomic resolution structure of this massive RNA protein complex would itself offer immediate insight into the mechanism of translation were thoroughly disappointed. And in fact the mechanism proposed from some of this early study ended up not being the correct mechanism here. There's a note about that on an earlier slide where the peptide bond formation step is shown. So what does he say? "I'd like to compare this situation to a visit to Earth by a martian who wants to understand how an automobile works." OK. So we can all think about flipping up the hood of our car and what do we see? "She looks under the hood of a parked car, perhaps even takes the engine apart, but still has no clue. It's clear she'll have much better luck if she's able to see that engine in motion." And so that's been a major goal in terms of thinking about the ribosome as well as other micro-molecular machines. How can you actually see these in motion and see the dynamics and conformational changes here? Really critical. So the question I pose is, is it possible to see the ribosome stopped at various points in translation cycle? And if so, how? So maybe we can't see the dynamics continually, but can we sort of park it at different steps? And the answer to that is yes. And basically a huge part of our understanding of this 70S ribosome does come from crystal structures, and researchers have been able to trap the ribosome at various points in the translation cycle using small molecules. And these small molecules are antibiotics. So where we're going to focus on for the rest of today and probably the beginning of Monday is thinking about the use as antibiotics. So small molecules that inhibit bacterial growth as tools for studying ribosome function. So a few questions related to that. First of all, what types of antibiotics target the ribosome? Where do they bind to the ribosome, and how can we use them experimentally? And also something just to think about, we have a crisis in the clinic in terms of a lack of new antibiotics and emerging antibiotic resistance. So how can fundamental understanding of the ribosome help in terms of therapeutic development? And this came up in a bit of a different context in seminar on Monday for anyone that was at Biological Chemistry Seminar. So we had Professor Matt Disney with us who was looking at small molecules to target RNA's. And one question that can come from that is, are there unknown molecules out there that might target the ribosome in different ways from the examples we currently have? I was super excited this morning to learn about a new book. So if any of you are interested in antibiotics, Professor Chris Walsh and Professor Tim Wencewicz at St. Louis have written a new book looking at antibiotics from a very chemocentric perspective here, and our friends on the cover. So I suspect be a wonderful read if you're curious. So let's take a look as a segue into thinking about these at the structure I just showed you a VFG bound to the ribosome. So we talked about how EFG is helping in the translocation process, and we saw the structure, and I told you in passing that this structure was very difficult for the researchers to obtain. And at the end of the day, they needed to use a mutant ribosome for reasons I won't go into. It's not relevant for this discussion. And also, a natural product that has antibacterial activity shown here. And so this small molecule binds EFG and it binds to EFG when EFG is bound to the ribosome. And moreover, it binds to EFG after GTP hydrolysis occurs. OK. So the result is that this natural product can be used to trap the ribosome in this post translocational state where EFG is still bound. So it's hydrolyzed GTP. There's been movement of the mRNA and tRNAs, but EFG cannot dissociate as a result of use of this small molecule. So you can begin to imagine how including this molecule or maybe other antibiotics that stop the ribosome at different steps can be used to obtain crystals and crystal structures here. And furthermore, they can also be used in a number of biochemical studies-- and we'll look at an example of that in the context of this lecture. --and also in recitations and problem sets. So where do antibiotics bind to the ribosome, and how many of them are out there that can bind the ribosome here? There's many options. So many antibiotics target the ribosome. And if we just look at a 30S subunit and a 50S subunit and take a handful of antibiotics that target the ribosome and see what we know about where they bind, we can make maps like these ones here and we can consider larger lists. You're not responsible for these details at all. Just the take home message is that there's many options and an extensive toolkit. Yeah? AUDIENCE: Are the eukaryotic and prokaryotic ribosomes similar enough that most of these antibiotics also affect the eukaryotic ribosomes? ELIZABETH NOLAN: Yeah. So that's a great question and something to think about. So that will depend on the molecule. There are many differences between the prokaryotic and eukaryotic ribosomes. Some will bind both. There is an example thiostrepton I believe that's quite specific, not for eukaryotic ribosomes. I mean, that's something also to think about. If they interact, are they interacting in the same way? And if they do inhibit the ribosome, is it by the same mechanism? And you can imagine implications related to therapeutic development in terms of that exact issue. Yeah? AUDIENCE: Do we have like a lot more ribosomes than prokaryotes? Is that also [INAUDIBLE] to have a lot more [INAUDIBLE] a lot more antibiotic [INAUDIBLE] that for us too? Do you know what I mean? ELIZABETH NOLAN: Yeah. I mean I think that's a little outside of the scope of our discussion because how do you get to counting ribosomes? Is that per cell or organism by organism and microbiome versus person? Is there another question? AUDIENCE: The ratio of the number of ribosomes we have to the number of proteins that we need to be producing, eukaryotic cells are more complex-- ELIZABETH NOLAN: Eukaryotic cells are definitely more complex right there. So what I say is overall case by case basis. So what about structures? Here are just some examples. Structures are highly variable and the ways in which these molecules can inhibit translation are highly variable. These are some examples that you may have come up with some of these in terms of laboratory work or maybe even been prescribed. So, for instance, chloroamphenicol This molecule here binds the 30S, prevents peptidyl transfer. Tetracycline binds the 50s and blocks accommodation. Gentamicin binds the 30S, causes premature termination. Erythromycin is a macrolide that for a long time has been thought to block exit of the polypeptide because it binds in the exit tunnel. But there's some new recent work suggesting a revision to that mechanism here. What are some general observations we can make? And keep in mind, there's always exceptions to the rule. So most of the antibiotics targeting the ribosome that we know about interact with the RNA, but, of course, some can interact with proteins. And we just saw an example of that with EFG. These antibiotics primarily target the decoding center and peptidyl transfer A center. Which makes sense if you're thinking about inhibiting translation. But, again, there's some exceptions. So thiostrepton interacts with the ribosomal protein that's not in that region. Magnesium might be necessary for antibiotic binding. So this is something to think about if using antibiotics in experiments. And just related to the earlier question, a given antibiotic may bind ribosomes of different species differently there. And so what are the consequences of that is something to think about. So here we're just looking at an overview of various antibiotics bound to the 50S. So this is taken from multiple different structures and the ribosome itself has been removed. But imagine that the A site tRNA is around here, here P site tRNA. What do we see? We have antibiotics called puromycins that are down by the A site. Here we have chloroamphenicol bound. Here we have the macrolides here. And just as an example of an antibiotic binding to the exit tunnel-- bless you. --here we're looking at the 50S. We have a P site tRNA, and here we have a nascent polypeptide coming through the exit tunnel, and here we have some examples from structural information about erythromycin, chloroamphenicol bound in this region here. So we're going to look at a puromycin as a case study for using antibiotics as a tool in a biochemical experiment. So the first thing that we need to think about is the chemical structure of puromycin and how that structure relates to its ability to inhibit translation. So these puromycins are molecules that cause chain termination. And we'll look at an example of a structure. [WRITING ON BOARD] So basically just want to use a little imagination when looking at this structure. What do we see? So what is this small molecule mimicking? Yeah. So what do we have up here? We have something that's adenosine like. Not exactly the same structure. But this may be similar to A76 of the three prime end of the tRNA. We have these methyl groups rather than an H2, but similar. What's going on down here? Pardon? Yes. It's similar to one amino acid or a peptide. So we have something here that's amino acid like. So if we're thinking about this as a mimic of the three prime end of the tRNA with the amino acid bound, what's fundamentally different here that's going to result in different chemistry happening? So how are the amino acids attached to the three prime end of the tRNA? What kind of linkage? Yeah. We have an ester in the normal circumstance. And what do we have here? Here we have an amide. So this is non-hydrolyzable. And what else do we have? Right here we have a nucleophile for the P site ester. So what can happen? Imagine that puromycin somehow can enter the A site. There's a nucleophile that will allow for chain transfer, such that the peptide that's on the P site tRNA gets transferred. But then what? OK. We're stuck because of this MI bond here. So effectively chain termination. OK. And so what's known is this molecule and its analogs can bind to the 50s A site. And that's something kind of incredible to think about. We talked about this machinery EF-TU that's needed to deliver the aminoacyl-tRNA. Puromycin can get there on its own. Which means maybe for an experiment, that is easier to do if you're going to use this in your experiment. Moreover, people have synthesized more complex versions. Just an example is shown here. C-pmn where we also have C75 of the tRNA mimicked here. And so this is just an overview of elongation, and then effectively chain termination happening after thinking about having a puromycin in the A site, a peptidyl tRNA, or some other molecule in the P site and the chemistry that occurs here. So we'll think about and close with one experiment that's been done using puromycin. And we won't have time to go through all of it in the last few minutes of today, but I'll just introduce the problem and we'll continue with the experiment next time. OK. And so what we're going to think about is a translation factor that hasn't come up yet in class. And this is elongation factor P. OK. And for a long time its function was unclear. [WRITING ON BOARD] OK. And so over the years this translation factor was implicated in a variety of cellular processes, but there wasn't any clear answer in terms of really what is its role? And so about two years ago there were two back to back papers-- one of these papers by Rodina and co-workers. So they're the authors of the paper being studied in recitation this week. --published work reporting on why EFP is important for translation. So prior to their work there were some preliminary studies indicating that somehow this elongation factor helps to modulate and accelerate peptide bond formation. But the questions are, when? So under what circumstances does EFP accelerate peptide bond formation? And then you can think kind of a follow up of that, how? And so we'll look at some experiments that were designed and performed using puromycin as a tool to address this question. And that's where we'll start on Monday.
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MIT_508J_Biological_Chemistry_II_Spring_2016
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12_Protein_Degradation_1.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ELIZABETH NOLAN: Where we'll spend the first part of today is finishing up where we left off last time with the experiments that were done to take a look at what types of polypeptides, the DNA, kDNA, J chaperone machinery interact with in E. coli. And once we finish up that, we'll transition into module 3, which is protein degradation. And most of today will just be some general background about proteases and protein degradation. And then on Wednesday, we'll begin to look at the macromolecular machines that are involved in those processes. OK. So last time we discussed DnaK, the chaperone, DnaJ, the co-chaperone. Right. So recall DnaJ went around and found some polypeptide in a non-native state and delivered it to DnaK, which can grip and hold on to the hydrophobic segments and somehow facilitate folding. And so where we left off were with experiments performed in a similar manner to what we saw for GroEL/GroES where pulse chase was done to label newly synthesized polypeptides. Right. So during the pulse period with this radio label methionine, newly synthesized polypeptides are labeled. And that allows us to see specifically what newly synthesized polypeptides DnaK, J are interacting with without the background of everything else in the cell. And why do we care about that? Imagine if we didn't somehow label to discriminate these newly synthesized polypeptides. Right. We can pull down many things in the precipitation, but we'd have no sense as to how long a given polypeptide existed in the cell. So maybe it was newly synthesized. Maybe it had been around a long time and something happened to it such that it wasn't in a native fold and DnaK interacted with it. OK. So that's the key point with these pulse chase and this labeling for a short time period. OK. So the researchers had an antibody to DnaK. They had to test its specificity as we discussed for GroEL/GroES. And then after immunoprecipitation, it's necessary to do the analysis. And so as I noted last time, in this particular study, the analysis were less sophisticated than what we saw for GroEL/GroES so they just used one dimensional SDF page, which we're all pretty familiar with. And they didn't extend it to mass spec. But with that said, there are a number of observations that are helpful that come from this study. So what we're going to do is examine their gels and see what conclusions we can come up with. OK. So this is their experiment number one. And what they did was look at the soluble crude cell extracts that were generated in this pulse chase experiment. OK. And so what do we see in terms of how the data is presented? Right. We have two lanes on the left, one and two, that are basically total cytoplasmic proteins. And then the lanes three through six on the right are four samples that were immunoprecipitated with this anti-DnaK antibody. OK. So something that was done in these experiments that was different than the GroEL/GroES work is that they use two different E. coli strains. So they used a wild type E. Coli strain. So that strain expresses DnaK. And they also used a mutant E. coli that is deficient in DnaK. So that's what this delta DnaK means. So they did some genetic manipulation and knocked out DnaK. OK. And as we learned in the introduction, these chaperone system is not essential. So what do we see if we work through this gel? And first, we just want to go over what the data show and then why that's important here. So if we compare lanes one and two, what do lanes one and two tell us? So these are the total cell lysates, soluble fraction, from either wild type E. Coli or delta DnaK. And why do we care to run these? So no immunoprecipitation. So I'll give you a start and then you all can contribute to the next ones. OK? So what I would say looking at these two lanes is first it looks like the total amount of protein and the distribution of these proteins is similar for both the wild type E. Coli and the delta DnaK knockout. OK. There's proteins that are distributed across a wide range of molecular weights, from below 14 kilodaltons to upwards of 100. OK. And why is this important to show? One, we want to see what the cell lysate looks like in the absence of immunoprecipitation. Right? And two, it's important to know whether or not knocking out DnaK has done anything to change those cells. And at least at the level here, it looks like in terms of the total cellular polypeptide pool, it's pretty much comparable in terms of total protein. So what happens now here where we have this immunoprecipitation? So we have four different lanes. We're going to focus on three, four, and five and ignore six. Basically, in three, we see immunoprecipitation from the wild type E. coli; in four, immunoprecipitation with wild type when the sample has been treated with SDS, so we need to think about why that was done and what that experiment shows; and then lane five in the delta DnaK knockout. So first of all, what does line three tell us? Kenny? AUDIENCE: Can you get enrichment of higher molecular weight proteins? And as the note says that 15 times more protein was loaded into this well, so I think that's just so you can see the signal. But I think that just shows to prove that the higher molecular weight proteins are more enriched in that lane. ELIZABETH NOLAN: Yeah. So are many proteins immunoprecipitated? AUDIENCE: Yes. ELIZABETH NOLAN: Yes. Right. We see many bands across a range of molecular weights. Right. And then as Kenny said, we're seeing much more intensity up here than down here. Right. So it looks like polypeptides in the range of about 20 to about 60 kilodaltons are enriched. So maybe there's some preference in that size range here. So that's a good observation. What do we see in lane four? What happened for this sample? So effectively, the crude cell lysate, those extracts were treated with SDS. AUDIENCE: They didn't bind anything? ELIZABETH NOLAN: Yeah. So nothing bound. Right. What do we see here? Just one band for DnaK. So why didn't DnaK bind anything? AUDIENCE: It's denatured by the SDS. ELIZABETH NOLAN: Yeah. Not the page... which is the sodium dodecyl sulfate. Right. It's a denaturant and it will denature things. They don't need to be in a gel. If you add that to a sample, you'll have denaturation, right? So when these samples were denatured, DnaK didn't bind. Do we think that the SDS denatured DnaK itself? Right. So we do see one band here. Is that surprising? AUDIENCE: No, because if the DnaK line is suggesting that that's the molecular weight of of DnaK, it's around there. And the antibody you're using to do immunoprecipitation... I mean, would likely probably sub the other bind. ELIZABETH NOLAN: Yeah. So the antibody was still able to bind. Right. What kind of gel is this? Well, it's SDS page, but how is it being monitored? AUDIENCE: It's a [INAUDIBLE] so it's a biodome radiogram. ELIZABETH NOLAN: Yeah. We're looking at radioactivity too so just to keep that in mind for the ban. Right. That's what's allowing us to see. What about lane five? AUDIENCE: It shows that in the delta DnaK line that nothing's pulled down by the immunoprecipitation. ELIZABETH NOLAN: Right. So no DnaK. Nothing's pulled down. This is a very helpful control, because imagine if you did see bands, that would indicate that there's some lack of selectivity with this immunoprecipitation step. So I think that's quite a nice experiment they added into this piece of work here. OK. So moving on to their next experiment, what happens during the chase if we look at different time points during the chase period? So this is similar again to what was done with the GroEL/GroES study. So what we're looking at is a one DSDS page. We have the molecular weight marker. All of these are with the immunoprecipitation. And we're looking at times from below one minute up to 10 minutes. So the question is what do we see in these data here. AUDIENCE: As time goes on, you see more and more concentrations of the smaller proteins. ELIZABETH NOLAN: Yes. So we're seeing fewer proteins as time passes. Right. Let's start with the first time point here. Does that look to be in pretty good agreement to what we saw on the prior slide? Right. We see that there's a number of polypeptides that DnaK is interacting with. Right. And they're over a range of molecular weights. Right. And then exactly as we just heard, as time progresses, what we see is, overall, there's fewer polypeptides. But it looks like there's fewer polypeptides of lower molecular weight here. So what does this suggest if you're going to interpret the data? AUDIENCE: Probably that lower molecular rate peptides are folded more quickly. ELIZABETH NOLAN: Yeah. So maybe the folding there is complete over this time or it's complete to a point that DnaK isn't needed anymore. Right here. What do you think about the DnaK band? AUDIENCE: Quite constant. AUDIENCE: Yeah. ELIZABETH NOLAN: Quite constant. So does that make sense? Yeah, why does that make sense? AUDIENCE: Because it's just the whole quantity is not changing. And it's not involving [INAUDIBLE].. Just like in the case of the less [INAUDIBLE] so when [INAUDIBLE]. ELIZABETH NOLAN: Yeah, right? So there was some newly-- what this indicates, right? There was some newly synthesized DnaK in those 15 seconds of the pulse. Right? And that has stuck around. And that's all precipitated to the same degree in each sample here. So what about this data here? And then, how helpful is this data? Right? So effectively, what was done is, radioactivity was measured by liquid scintillation counting. OK, so they measured the total radioactivity in each sample prior to separation. And then, they've converted that to some arbitrary scale of proteins bound to DnaK. Right? So we see that, over time, the total radioactivity decreases and effectively comes to some sort of plateau. So that's just some nice quantitation in terms of what we see here in the gel, right? It's quite easy to measure the total radioactivity in a sample. And you get a measure of that from liquid scintillation counting. And then, you can look at the gel, right? And they're in good agreement there. And as I said, this is completely arbitrary, what's on the y-axis. So do these experiments tell us much about the specifics of DnaK function? So we see polypeptides being bound. We see over time that fewer are bound. What's actually happening in the cell? AUDIENCE: I don't think we can conclude much from these. Only that we know that it interacts with the polypeptides for some amount of time. ELIZABETH NOLAN: Yeah. Right. I agree. So is it acting as a foldase? Is it acting as an holdase-- an un-foldase? That's not clear from the data presented in these experiments. And I'd say overall, there are studies that show different things, depending on the system there, for this. So that's where we're going to close with the chaperone systems. And where we're going to move into-- actually, one more comment, right, before closing on the chaperones. What happens if certain ones are deleted? So just to reiterate, not all of these systems are required for cell viability of E. coli. It's only GroEL/GroES. Right? So you might ask if trigger factor or DnaK or DnaJ is deleted, what happens in the cell to keep things functioning properly? And just one observation. If trigger factor is deleted, OK, there's no growth phenotype. Is that surprising? Right? That observation may depend on growth conditions. But say you're in some standard growth conditions. What's observed is that, in the absence of trigger factor, DnaK and J can basically compensate for that loss of function. OK? But then, if trigger factor and DnaK are deleted, at higher temperatures, that becomes lethal here. OK? The cells can't cope for that. But at lower temperature, GroEL/GroES can compensate for that loss of function. OK. So we're on to protein degradation. There's some incredible macromolecular machines involved in this unit here. And we'll move on to that one come Wednesday. Just if we think about where we're going with lifecycle of a protein, right, we've gone from synthesis to folding. We've learned that misfolding can occur. OK? And at some point these polypeptides, whether they're folded or unfolded, need to be degraded. So they have some lifetime in the cell. OK? And so we can think about proteases, so classical enzymes, like trypsin. And we can think about proteasomes, which are degradation chambers. And these players are really important because they have a role, big picture, in controlling the dynamics and lifetimes of all proteins and cells. So what are some of our questions for this module? Why are proteins degraded? We just said a little bit about that. How are proteins degraded? And what types of proteases exist? We'll briefly today touch upon the general catalytic mechanism because that's important to have this background for thinking about the degradation chambers. So what are the general mechanisms and what are the active site machineries? Protease inhibitors are really important at the lab bench, and they also have a big role in therapeutics. And so we'll talk about those a bit here. And then moving into protein degradation machines, we're going to look at ClpXP from E. coli as a case study. So we need to think about, what are the structures of these degradation machines, what are the mechanisms? How do they differ in prokaryotes and eukaryotes? So after spring break, Joanne will spend some time talking about the eukaryotic proteasome there. We won't talk about it as much immediately here, but we'll come back to that later. And how are proteins that are destined for degradation by a proteasome tagged to get to that destination? So here are our topics. An overview, which is where we'll focus today. And then, looking at ClpXP, and down the road, the 26S proteasome. So first, thinking about proteases. Some general points to get everyone up to speed. So because we all know proteases catalyze the hydrolysis of peptide bonds. So we can just think of some peptide and that peptide bond gets hydrolyzed to give us these products here. Why do we need a protease? The bottom line is just that spontaneous hydrolysis of peptide bonds is very slow, right? So we can leave a protein or a polypeptide on the bench top. And maybe it will unfold. Maybe it will precipitate. But it's not going to have the peptide bonds being broken unless something else has been done to it, right? So we can think about a half life on the order of seven years. And so proteases give tremendous rate accelerations on the order of 10 to the ninth. And we can just think about chemistry for a minute and what we might do as a chemist to hydrolyze a peptide bond. So hydrolysis is pH dependent. And so in chemistry we'll use acid or base to hydrolyze a peptide bond. And we can think about base catalyzed reactions, such as this one, where we have our OH-minus group attacking, or acid-catalyzed reactions, as this one here. OK? So effectively we can just think about pH dependence of hydrolysis. Just if we have rate and we have pH. Something on the order of this, right? Where we have enhancements at low and high pH and a relative minimum at neutral pH here for that. And so these types of chemistry is going to come up in the context of the protease enzymes, depending on the type, as we'll see in a few slides. So we can think about proteases as being irreversible biological switches, that these reactions are irreversible. And what does this mean from the standpoint of the cell? It means that the cell needs some way to handle and deal with these proteases, right, such that they don't cause unnecessary hydrolysis of polypeptides. That would be very deleterious to the cell, right, if a protease was running rampant and hydrolyzing proteins that it shouldn't here. So what are some strategies that the cell can use? One, cells are quite good at controlling protease activity, both in terms of space and time. And there's a variety of different strategies, depending on the locale and the protease. So regulation is really key here. And some examples are provided here. So one is that proteases will be stored as zymogens or inactivated precursors. And there'll have to be some event that activates this zymogen to give the active protease. Proteases can be stored in separate organelles here. So these might be zymogen granules or lysosomes. And sometimes they're stored with a protease inhibitor, as well. And another strategy, which is really the strategy we're going to focus on as we move forward in this module, is that degradation chambers are used, such that you have this huge macromolecular machine where all of the protease activity is in the inside. And what this means is that somehow a condemned protein that needs to be degraded by this machine needs to be tagged. And there needs to be some mechanism to get it in the inside of the chamber. So effectively, degradation will limit access of the active sites to the rest of the cellular environment. So that's what we see in ClpXP and this 26S proteasome. Just to note-- so just the other week in C&E News, there is a highlight of a pretty exciting paper. So I noted that proteases are of interest and important from therapeutic development. And here's a little excerpt about a molecule shown here that's found to hit the proteasome of malaria parasite. And so hopefully, by the end of this unit, if you go back and read this, you'll have some sense as to why is this a good inhibitor of the proteasome or a protease. And what's going on in terms of the proteasome machinery here. And how can we differentiate proteasomes from different organisms. Back to some of the strategies. Just an example is zymogen activation, and thinking a little bit from the perspective of the organism. So here, we can think about the gut. We're in the small intestine. So there's the epithelium, the cells, these are crypts. And here's the lumen, so the space where the food goes through, et cetera. What do we see? So inside the intestine, there's a protease named entarokinase. And it has a role of activating trypsinogen. So trypsinogen is a zymogen. It's produced by the pancreas. And the pancreas delivers trypsinogen and other things into the small intestine. And so once it reaches the small intestine where its activity is needed, it will be activated by the action of entarokinase to give trypsin. OK? And then what can happen? Trypsin can also activate trypsinogen, and it will also activate chymotrypsinogen to give chymotrypsin. Right? So the net result here is protease activity in the intestinal lumen, which is the extracellular space here. And so they travel from the pancreas in a form that's inactive and then become active in the intestinal lumen there. So as I said before, proteases are important. And if we think about this role in controlling dynamics and lifetimes of proteins and cells, what are some of those roles? And I guess I also point out this also-- they also can exist in the extracellular space. So if we think about homeostasis and how proteases can regulate homeostasis, just some examples. They can remove misfolded proteins or aggregated proteins. They can provide amino acids when needed, right? So after destruction of a polypeptide, you have small fragments or amino acid monomers. And they can modulate many cellular functions. So just some examples. And this is to show the broad range. We can think about blood clotting, the generation of hormones, just digestion and recycling of amino acids. So energy harvesting, the cell cycle, control of the cell cycle, and even cell deaths. So thinking about apoptosis here. And if we just select two of these cellular functions and how proteases play a role, what I have here is the maturation of insulin, a peptide hormone in the blood coagulation cascade. OK, so if we take a look, insulin is a really terrific molecule. And if you're looking from some trivia not shown here, it also binds zinc and forms an interesting oligomer. So if you're interested in metals, that's a good one. But what do we see? We see that insulin is synthesized as a prepropeptide. And so in blue, we have a signal sequence. And then we have these chains here. And look, there's a bunch of cysteines, right? So there's action of a protease. And what do we see? The signal sequence is cleaved and at some point in this process, there's formation of disulfide bonds, right, in some regiospecific manner. So this is pro insulin. And then what happens? There's another protease cleavage event that gives us the mature form of insulin. This grape chain here is removed. OK? So this is an example of a hormone being stored as an inactive precursor. And actually, there's many peptides that are stored as inactive precursors. And then some protease has to come and cleave a pro region. So in my group, we're interested in a family of antibacterial peptides called defensins that are in the intestine. And they have a pro region. And it's trypsin or another protease that comes along and has a cleavage event to release the active peptide. So not only limited to insulin here. If we look at the blood coagulation cascade, we can imagine that we don't want blood to coagulate on whim, right? That'd be a huge problem. So proteases are required to allow coagulation to occur. And what we can see here is that prothrombin is converted to thrombin by a protease. And thrombin is a serine protease. And we'll hear more about serine proteases in a little bit. That converts fibrinogen to fibrin. And as a result, coagulation occurs. And that's important for wounds. And there's many, many other examples. So if we think about types of proteases and mechanisms of catalysis, what I just would like you all to be aware of is that there's two general mechanisms. And we can think about four different mechanistic varieties within that. And so we can divide these up by proteases that are involved in covalent catalysis. So there's formation of a covalent acylenzyme intermediate. And this is what we'll see for serine proteases, cysteine proteases, and threonine proteases. So examples here are quite relevant to this module as ClpXP is a serine protease. And as you'll see later on, the eukaryotic proteasome is an end terminal threonine protease. The second general type are proteases that accelerate the direct attack of water on the substrate. OK, so this is non-covalent catalysis. And the types here are aspartyl proteases and zinc proteases. So there was a question a few lectures ago if there's metal-dependent proteases. And the answer is yes, zinc proteases. And we can also think about these from the standpoint of the acid and base catalyzed chemistry we saw before. So just for some trivia. If we think about the human proteome-- 533 proteases. And this is a count here. So on the order of 200 serine proteases, 140 cysteine, around 190 metalloproteases, and 21 aspartyl proteases. So we have many of these enzymes to act at different places and points. If we take a look at the active site machinery, what do we see? So here we have the serine proteases. They have a catalytic triad comprised of aspartate, a histidine and a serine here. Cysteine proteases-- we see a cysteine and a histidine. And so these are the ones involved in covalent catalysis. Here we have the non-covalent catalysis. So the aspartic acid or aspartyl protease. We have two asp residues. And here we have an example of a zinc protease, where we see a single zinc ion coordinated by two histidines, and in this case, a glutamate and a bound water. OK? So if we think about just the covalent versus non-covalent catalysis here. So when I get further along. So imagine we just have some dipeptide. What we find in these enzymes is that they have what's called an oxyanion hole here. And we can think about the enzyme allowing attack as such. So that or some nucleophile here. So what do we get? We get a covalent acylenzyme intermediate. OK, we have the oxyanion hole. And these are the serine and the cysteine proteases here. OK? And we'll go through in more detail the mechanism in a minute. If we think about non-covalent catalysis, and again, we have our dipeptide. We can just think about for a minute one of the metalloproteases, right? So in these cases, the protease is accelerating the direct attack by water. So I imagine we have some metal here that has water bound, right? What happens? Imagine we can de-proteinate the water molecule. And then there can be attack. OK, so why does the metalloprotease allow this to occur? So what's happening when the water binds to the metal that will facilitate this? So we can think about the pKa of a water molecule, right? And what happens when a water molecule is bound to a metal, right? Say zinc. So how do we think about a metal? AUDIENCE: A Lewis acid. ELIZABETH NOLAN: Yeah. Right? We have a general Lewis acid here. Here. So what's going to be effect of the pKa of the bound water relative to unbound water? AUDIENCE: It'll be more acidic. ELIZABETH NOLAN: Right. We're going to lower the pKa of the bound water, which is going to help generate the nucleophile. Right? So that's how it's facilitating the direct attack here. OK, so what we're going to do is look at the serine protease example in a bit more detail. AUDIENCE: Are the end termini de-pertinated? Or is it-- ELIZABETH NOLAN: I am just-- what would it be at physiological pH? AUDIENCE: NH3? ELIZABETH NOLAN: The embryote NH3 plus. Right. So these are just showing a simple dipeptide that we have NH3 plus and O minus in terms of the acid ends there. OK. So just thinking about here, this covalent catalysis. So here's each protease type, the active site, and the nucleophile. So in the case of the serine proteases, the nucleophile is the serine side chain. And I'm showing this because ClpXP-- ClpP protease-- uses serine protease chemistry here. So what we observe in this overview is a generally accepted mechanism. And we see formation and collapse of this covalent acylenzyme intermediate. So if we take a look here, we have a bound polypeptide. This is the oxyanion hole provided by these two NH. Here we see the aspartate, the histidine, and the serine. Right? So what do we see happening here? First, there's formation of a tetrahedral intermediate. So there's an attack. OK? And here we have loss of the RNH2. And here what do we see? We see, basically, the histidine working on this water molecule. We have collapse of this acylenzyme intermediate, another tetrahedral intermediate, and release of the acid product here. OK? So in thinking about this, we think about the histidine as being a general acid general base involved in general acid-base catalysis, a proton carrier. We see this oxyanion hole providing stabilization here and here. Right? We have this negative charge. And something that you need to think about are the pKas. If we think about just pKas of amino acids and how this chemistry is happening. Right? So what is a little bit mysterious here, based on our knowledge of pKas of the catalytic triad? And they give some approximate values just for serine, histidine, and aspartate here. AUDIENCE: Well, each proton abstraction is being done by something that should, theoretically, have a lower pKa. So you have an aspartate abstracting a proton from a histidine, which is abstracting a proton from serine. So there has to be a lot of perturbation of the system for that to happen. ELIZABETH NOLAN: Yeah. Right. There needs to be a lot of perturbation to pKas for this to work, right? How easy is it to de-proteinate the serine by a typical histidine? Is that going to happen based on pKa? AUDIENCE: No. ELIZABETH NOLAN: Right? So there's something about this active site and the environment that's going to give perturbation of these values. If we just move beyond this cartoon form for a minute, and just look at the catalytic triad from chymotrypsin from a crystal structure. This is the orientation of the serine histidine and aspartate. And something to keep in mind is that different serine proteases, or different proteases in general, have different substrate specificity's, which means they prefer to cut before or after a given type of amino acid, depending on the side chain. And this is just a cartoon depiction indicating that, here's the peptide, here's some side chain, and there's some recognition site here. So there's a degree of substrate discrimination. For instance, trypsin likes to cut after arginine and lysine. But it will cut at other places, as well. Right? Chymotrypsin likes aromatic hydrophobic residues. Elastase likes small and uncharged residues here for that. So you may have seen diagrams or cartoons of specificity pockets for thinking about substrate discrimination amongst these proteases. And I guess what I would just say is, it's not so simple as those types of cartoons. If we look at the structures of serine proteases, just to compare, what we see is that, for trypsin, elastase, and chymotrypsin, they have similar overall structure. So this is an overlay of the three enzymes. And the catalytic triad is shown in red. OK? So despite this similar overall structure, they have distinct substrate preferences. And it's just something to be aware of. So you're not responsible for the origins of this substrate discrimination. And here's just a view showing more about the secondary structure of trypsin shown here. And in this case, there's an inhibitor bound. And I'll just note, in terms of the substrate preference, and these are the three types of activity we'll end up seeing within the eukaryotic proteasome. So if we just have some polypeptide. OK? OK, and so imagine we're thinking about hydrolysis here. OK, so the C terminal end of this amino acid with our one side chain. We think about the enzyme and the identity of R1. For trypsin, it prefers to cut after arginine or lysine. OK, so a positive charge. For chymotrypsin, phenylalanine, tyrosine, and also other ones like valine, leucine, and isoleucine. OK, so aromatic plus hydrophobic. And then elastase here. And we find that elastase prefers to cut after small. And if I'm boxing it, these are the ones we think is most preferred, small and uncharged residues. OK? So some discrimination based on side chain identity. So where we'll close the general background is just with a note on proteases and disease and protease inhibition. So if we consider various human diseases and proteases, there's many, many links. And many proteases are implicated in a variety of diseases and pathologies. And so this is a table just to give you a sense of the breadth. What we see in terms of the class is that all of the classes are represented here. And that we see diseases ranging from cardiovascular problems, to cancer, et cetera, cystic fibrosis, inflammation here. OK? So as a result, there is quite a bit of interest in terms of the possibility of protease inhibitors as therapeutics. And beyond that, they're also widely used in the lab. So how do these inhibitors work? Just as a general rule of thumb to think about, generally they react to form a covalent bond with the catalytic nucleophile. So for instance, for the serine proteases, they'll form some covalent bond with the active site serine residue. And we can classify these inhibitors as being either reversible inhibitors or irreversible inhibitors. So as those names indicate, if it's a reversible inhibitor, that covalent linkage between the protease or the proteasome and the inhibitor can be broken down. And types of reversible inhibitors, for instance, use aldehydes as the reactive group. So in contrast, the irreversible inhibitors form a covalent linkage that is not readily broken down with the catalytic nucleophile. And so irreversible inhibitors include vinylsulfones. And if you go back and look at that little excerpt from C&E News about this molecule that's inhibiting the proteasome of malaria, you'll see that it has a vinylsulfone on its terminus here. Epoxides are also employed. OK, and generally, if we have inhibitors that block the function of a protease or a proteasome, they're going to interfere with many critical cellular functions right here. And just in terms of cancer, just some observations. So it's been found that proliferating cells are sensitive to proteasome inhibitors. And there's some proteasome inhibitors that can selectively induce apoptosis in proliferating cells. And so cancer cells are proliferating, and there's interest in the use of these as anti-cancer drugs. So what I've included in these slides are some examples of inhibitors of each class, and then the mechanisms. Here are just three molecules. So we have either reversible or irreversible inhibitors, right? And what is there to note looking at these molecules? They're all polypeptide-like. Right? So there's amino acids or moieties that, with a little imagination, we can think about as being somewhat similar. And then we see these reactive groups on the terminus. So the aldehyde, for instance, the vinylsulfone. And so if you look at the structure of this molecule being used to inhibit the malaria proteasome, there's some clear similarities to these here. In terms of mechanisms, we can think about these reversible inhibitors. So for instance, the chemistry with the peptide aldehyde. Here we're seeing the nucleophile of the eukaryotic proteasome, which is really interesting because it's an end terminal threonine. That's why we're seeing it drawn as such here. So we can have formation and collapse of this species here. Or in the case of the irreversible inhibitors, we have the vinylsulfone and the chemistry that happens here for that. And so if you're interested in these, I encourage you to look at the mechanisms a bit more. And we'll see a little bit more on inhibitors being used experimentally as we go through the rest of this module. So where we'll start on Wednesday is looking at the structure of E. coli ClpXP, which is a degradation machine used to degrade certain condemned proteins there. OK?
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MIT_508J_Biological_Chemistry_II_Spring_2016
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8_Protein_Folding_1.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ELIZABETH NOLAN: So we're going to spend the first few minutes finishing up our discussions of the evolved orthogonal ribosome called Ribo-X and experiments that were done to characterize it and determine whether it did a better job at allowing the tRNA to suppress the amber stop code. And then we'll be transitioning into module 2. So we'll be starting protein folding today and be spending the next couple of days on that area and thinking about some of the macromolecular machines and other proteins involved. So just as a recap, where we left off last time with thinking about incorporating unnatural amino acids using a new ribosome. And so we discussed how an orthogonal ribosome was designed that binds to orthogonal mRNA. And then the idea was, can this orthogonal ribosome be improved, the immunogenicity and selection, to get a new orthogonal ribosome, where suppression of the amber stop codon by the tRNA is favored over termination by release factor 1? So we talked about this issue with really factor 1 causing truncated protein phenotypes. And so where we left off was in a series of three types of experiments to look at how well this Ribo-X works. So we looked at protein yield to ask, is it translating polypeptide, as well as the starting orthogonal ribosome. And then the next experiment we were looking at, and where we left off with, was experiment 2, which was the question of amino acid incorporation. And so what was done, just in recap, is that a GST maltose-binding protein, MBP, fusion protein was designed. And recall that this protein had a protease cleavage site and that the GST portion contained cysteine and MBP does not contain cysteine. So the idea was to use radio labeled cysteine as a probe for misincorporation of cysteine into maltose-binding protein. So in terms of the actual experiment, what was done, imagine that we express this fusion. In the presence of the radio labelled cysteine, we use a protease. And in this case, it was thrombin. This protease gives us two fragments, GST plus MBP. And these two proteins have different size, so we can separate. So just doing SDS page. And ask, where did the proteins migrate on the gel, and then where do we see radioactivity? And so the data are shown here that were reported by the authors. So what are we looking at? We're looking at, on the left, the Coomassie stain for total protein. And then on the right, we're looking at radioactivity here. So what do we have here, in terms of the lanes? They've done some labeling to help us. So maltose-binding protein is running up here. GST is running down here. And we have four different conditions. What do we see? We have, in lane 1, Ribo-X is what's used. In lane 2, we have a system where it's the initial orthogonal ribosome that was not involved. So that's our point of comparison. In lane 3, we have the native ribosome only. And in the 4, lane 4, a control for no gene. So what do we see? AUDIENCE: All the ribosomes look like they're doing the same thing, except for when there's no gene. ELIZABETH NOLAN: Right, no gene to translate. Yeah. So why do we come to that conclusion, they're all doing the same thing? What we see on the left here, in the Coomassie stain, is that in lanes 1 through 3, we see a band for MVP, and they're all similar, in terms of intensity. And likewise, for GST, we see a band, all similar intensity. So it looks like, in all cases, a similar amount of protein is being synthesized, and that this protease cleavage worked in a comparable manner. AUDIENCE: Do you ever have different RNAs, depending on whether or not the orthogonal ribosome versus the native ribosome, and do you worry about how that might influence this experiment? ELIZABETH NOLAN: Yeah, right. The native ribosome is not going to bind to the orthogonal mRNA, so you need to give a plasma that has a ribosome binding site compatible with the native ribosome. And yes, it's possible that native ribosome won't work as well. Or reverse. I'd say reverse is more likely, that one of these mutants won't work as well. But that doesn't appear to be the case here. So what does the gel on the right show? So where do we see radioactivity? AUDIENCE: Mostly at GST, but then it looks like there's a lot of background. ELIZABETH NOLAN: Yeah. Do we expect background? Where might the background come from? Right, so we see a strong band here, and this is at the same places where we see GST. So that's a good indication. We know that there's cysteine and GST, so we should see a band here. So what about background and what about MBP? AUDIENCE: You're going to have some misappropriation in MBP. It's very much falsely compared with GST. But a little band in MBP could be some amino acid misappropriation. ELIZABETH NOLAN: OK, is that band MBP? Do we know that definitively in this gel? AUDIENCE: Not necessarily because the Coomassie band's a little wider, so it can be a slightly smaller molecular weight then. ELIZABETH NOLAN: Right. There's many species in these gels. It isn't just two bands, one for GST and one for MBP. So where did these other bands come from? Could it be from the initial protein purification and there's some contaminants? Could it be a result of the thrombin cleavage, and maybe thrombin cut at some places other than just this cleavage site? So that's preferred, but maybe it can cut some other places. So there's other species in the gel. And cysteine comes up in other proteins, so it makes sense that there'd be some background to say this protein wasn't 99%, 100% pure, or maybe from cleavage there. In terms of this little band, is that MBP or not? A little hard to say. Where they placed the arrow indicates maybe not, right? But what could you do to find that out? AUDIENCE: Western. ELIZABETH NOLAN: Yeah. So maybe a Western Blot. If there's an antibody, you could run something less concentrated. You could do mass spec. There's options if you wanted to track that down. But the bottom line is if we compare the radioactivity here to what's seen here, there's much more with GST, which is what we would expect. And so through some further analysis, they conclude the error frequency is less than 1 in 10 to the 3 for Ribo-X, at least for cysteine. So what's the limitation of this experiment, from the standpoint of misincorporation, which is what I asked you to think about in closing last time? AUDIENCE: What you're testing for is cysteine in this incorporation. ELIZABETH NOLAN: Only testing for cysteine in this incorporation. So we don't really know about other amino acids. So this is good news, but there's other information we're not getting from this experiment there. So is it possible that other amino acids are misincorporated? It would take some additional experiments to get at that. So the last experiment we're going to look at is asking the question, does Ribo-X actually do a better job than the progenitor or o-ribosome? So what we want to look is at the efficiency of suppression of the amber stop codon. So last time, we talked about how one limitation of the Schultz method is that there's a truncated phenotype because RF one enters the A state, rather than the tRNA. So in order to look at this, they continued to use this type of fusion. But instead of having a protease cleavage site here, they stuck to the amber stop codon in between GST and MBP. So they made two different plasmids, so GST, a plasma encoding, GST, the stop, and malE. So this is the gene named for MBP. So if this gets translated, the question is, do we get the GST-MBP fusion or GST? Because if the tRNA does its job, polypeptide synthesis will continue and we'll get the fusion protein. If release factor 1 ends up here at this position, we get termination and only GST. And they also made an additional construct. So in addition to asking, what happens if there is a codon for one unnatural amino acid, what happens if there's two? So recall, with the Schultz method, what we saw last time is that when attempts were made to incorporate two unnatural amino acids in one polypeptide, the efficiency went down to below 1% there. So what if two of these are here? Again, the question is, when these get translated, do we got GST-MBP or GST? And as we can see from the gel, we can differentiate pieces by size. OK. So in addition to comparing these, what was done is in both sets of experiments, there was a comparison of the Schultz method we discussed first, and Ribo-X here. And just in terms of size, so GST-MBP is about 70 kilodaltons and GST is about 20 kilodaltons, 26. So easy to separate on the gel. So what are the data? And what was the unnatural amino acid employed? So they ended up using an unnatural amino acid called BPA, which is a benzophenone, so a crosslinker. Here. And so I'm going to write up some details that came out from the gel on the board because there's a lot of things to navigate in that gel. So effectively, what are the comparisons we want to make? So in lane 3, the method we have is the Schultz method. There is one codon for incorporating one unnatural amino acid. And what they see is they get about 24% efficiency of full length fusion protein. In lane 5, we have the Shultz method with two here, and we get about 1% efficiency In lane 7, analyzing the orthogonal ribosome Ribo-X, and one. And what's seen is 64% efficiency, so quite an improvement there. And then in lane 9, what we have is Ribo-X with two and an efficiency of 22% here. OK so here, we're looking at the wild type ribosome, and here, we're looking at the orthogonal ribosome, orthogonal mRNA with the two mutations we saw before. So these values come up from quantification of the data here. So you can convince yourself by comparing the bands for GST, resulting from truncated phenotype translation termination and the bands for the fusion protein GST-MBP. So showing that there was successful suppression of the amber stop codon here. So what are the major conclusions? The major conclusion is that, at least with this system, what we see is that Ribo-X has minimized this truncated peptide phenotype compared to the wild type ribosome, and that it's been possible to, basically, diverge the decoding properties of the orthogonal ribosome from the indigenous cellular machinery. AUDIENCE: I'm trying to understand you right. So the percent here, that's the percent of the total expressed protein? Oh, so the other 99% would have been the GST only? ELIZABETH NOLAN: Yeah. So here, for instance, if we take a look at lane-- let's compare lanes 3 and lane 5. So here's lane 3, and what do we see? So in here, we have incorporation of one natural amino acid. And we see that there is a band for the fusion protein and there's a band-- let me make sure I'm in the right lane, lane 3-- and a band for GST itself. And the intensity of this band is greater than the intensity of that band, and you can imagine doing quantitation, whereas if we look at lane 5, where we're trying to incorporate two by this method, we see a band for GST. And what do we see up here? Very little. If we look at lane 7, we're seeing 64%. Lane 7 here, we see that we have this band for the GST-MBP fusion and a weaker band for GST alone there. So percent efficiency, percent of the total. So there's other things happening in this field. So the Schultz method and these orthogonal ribosomes are two examples. One thing that came up after this work with Ribo-X was to design ribosomes that can use quadrupling codons, rather than triplets. So a lot of creativity and things to look up if you're curious. But with that, we're going to close the translation module. We will not leave the ribosome. It will keep popping up throughout modules 2 and 3. But we're going to move into what happens to a polypeptide as it leaves the ribosome. So how does it get its native fold? And so what happens to nascent polypeptides emerging from the ribosome, and how do polypeptides fold? And so there's reading posted for module 2 on Stellar and listed here, one required paper, which is a really wonderful review that came out about two years ago. So let's think about folding. And as a point to thinking about that, let's think about our ribosome. And there's some emerging polypeptide chain, so the nascent polypeptide. So what happens to this polypeptide? And the first thing to keep in mind is something we need to think about, is where is this polypeptide destined to go? Is this a polypeptide that will be in the cytoplasm? Is this a polypeptide that will become a membrane protein, or part of the secretory system? And so we can think about cytoplasmic protein versus membrane proteins, or a new karyote secretory here. And so we're going to focus this module in terms of thinking about what's happening in the cytoplasm. We might touch upon this if there's time, but I think there won't be. So the cytoplasmic proteins are folded by chaperones that they can come into contact with as they're emerging from the ribosome, or also after the polypeptide is released. AUDIENCE: Do extracellular matrix proteins fall into either of these categories? ELIZABETH NOLAN: I actually don't know. Joanne, where do extracellular matrix proteins fall? JOANNE: Well, they get made inside the cell. ELIZABETH NOLAN: They're made inside the cell and then they have get shuttled. JOANNE: So you should go talk to Matt Shoulders because if you look at collagen, that's exactly [laughter] ELIZABETH NOLAN: OK, so these interact with a player called signal recognition particle, which allows for targeting to the membrane or endoplasmic reticulum in eukaryotes, and then folding can happen here. So we're going to be focused in the cytoplasm, just realize there's other machineries involved for membrane proteins. So here's just another view of our ribosome. We saw this early on in the ribosome unit. And we want to think about this exit tunnel and the emerging polypeptide chain. So it's the 50S subunit. And as we discussed before, this exit tunnel is long and it's also quite narrow and it's lined by both ribosomes RNA and proteins. And I know a few of you asked about the hydrophobic residues of proteins that line this tunnel after lecture 2. And the thing to keep in mind is that it's not all hydrophobic. There's also RNA there. There will also be other residues. And something just to think about, like can water molecules get in there, as well? So we see for instance, there's two proteins, L4 and L22, that line part of the tunnel. We have protein L23 at the exit. But there's also a lot of RNA there, so don't forget that. So a question, just to address early on, does protein folding occur in the exit tunnel? And I'd say this has been a bit of a controversial question over the years and there've been camps arguing both possibilities, yes or no. I think the thing to keep in mind is that the dimensions are limited. And although we can imagine some confirmation of flexibility and dynamics in this exit tunnel, it can't undergo some tremendous change to, say, accommodate something like ubiquitin that you saw early on. That doesn't just make sense. So is it possible for some alpha-helical fold to occur in this exit tunnel? Presumably. There is some work that indicates there's folding zones in the exit tunnel, so maybe some folding happens there. But really, the main conclusion is that most folding occurs outside of the ribosome and after the polypeptide emerges from the 50S here. So if we're thinking about most folding of polypeptides as occurring in the cytoplasm for cytoplasmic proteins, what we need to think about is that environment. And we learned in the introductory lectures, or had a reminder, that the cellular environment is very crowded. So we have this issue of macromolecular crowding. And in thinking about that, we need to ask the question, how does this emerging polypeptide fold to its native form in this type of environment? What type of machinery is there to help protect it? How is misfolding avoided and intermolecular molecular interactions that are non-productive avoided? So where are we going to go in this module? We're going to look at protein folding from both the in vitro test tube perspectives and also from in the cell. And so in thinking about protein folding in vitro, we'll discuss some of the seminal study, so Anfinsen's hypothesis and folding of ribonucelic A, Levinthal's paradox, which brings us to thinking about energy landscapes, and also touch upon some of the experimental methods that are employed. And then in terms of machines, we'll think about, largely, post-translational protein folding in the cytoplasm, so GroEL, GroES, DnaK and J. We'll also talk about a protein called trigger factor that associates with the ribosome, and those nascent polypeptide chains. OK. So these machineries fold soluble proteins, not membrane proteins. And I'd also like to point out, and again, we may or may not get to these systems, depending on time, but in addition to these chaperones and macromolecular machines involved in folding, there are classical enzymes that are really important. And these include enzymes that, say, isomerize prolene, also thyle oxidases and isomerases there. So what are our questions for this module? So why and how are proteins folded? And in terms of how, in the lab versus in the cell, classical enzymes and micromolecular machines. What happens when proteins are misfolded? How does protein folding relate to disease? What methods are employed to study these phenomenon? And for the case studies, we'll look at, in terms of cytoplasmic players, we want to understand, really, what are the structural properties of these different chaperones and their partners? How do their structures relate to function? How do they help peptides attain the native fold? And how good is our understanding of these systems? We'll see in the case of DnaK/J, it's actually pretty difficult to know what they're actually doing here. And really, what is the experimental basis for our understanding? If we just take an overview of folding and misfolding-- and this is diagram for a eukaryotic cell and from many, many different types of studies-- what do we see? So we see that some sort of biomolecule called chaperone keeps coming up again and again. So these are proteins that assist with folding, or unfolding, disaggregation. There's many possibilities for the trajectory of a protein here. So here, we see a nascent polypeptide emerging from the ribosome. And imagine that some folding intermediate is released. So this is not fully at the native fold, but it's somewhere along that pathway. What might happen? Right here, what we see is some chaperones allow this intermediate to form a native protein. But look, there can also be unfolding, and this could work its way back. This native protein could unfold to a misfolded state. We can think about remodeling, and maybe there's chaperones involved and taking this misfolded state back to an intermediate that's on a productive pathway. What happens here? Maybe there's some trouble, and rather than reaching its native fold, this intermediate ends up aggregating. It forms some sort of protein aggregate, and maybe that can form oligomers, or some sort of amyloid fibril, like what we hear about with Alzheimer's disease. Here, we see there's chaperones that can be involved in having disaggregate activity, and they can help in breaking down these aggregates and getting back to some productive place here. OK. So there's inherent complexity here and many players and relationships between protein misfolding and disease, just to be aware of. So we typically think about the protein fold providing function and protein misfolding can result in improper function. And there's many different types of improper function. It could be loss of function. It could be gain of function. It could be formation of some sort of aggregate that's deleterious to the cell for one reason or another. And if we just take a look, in terms of human diseases that are associated with protein misfolding, what do we see? So there's examples out there, like Alzheimer's, Parkinson's, familial ALS, and mad cow. So Alzheimer's disease is associated with formation of Abeta plaques in the brain. In Parkinson's there's a peptide called alpha-synnuclein that aggregates in familial ALS, also called Lou Gehrig's disease. There are single point mutations in an enzyme called superoxide dismutase that results in misfolding and some negative consequences there. And then misfolding of the prion protein. So a lot of these, in terms of neurological disorders. So in addition to fundamental studies, there's significant interest in understanding protein misfolding from the standpoint of disease and prevention. And I'll just note, sometimes questions about natively unfolded proteins come up and those are outside of the scope of our discussions today. But be aware, there are proteins that are natively unfolded. You saw a little bit of that with some of the ribosome proteins that had those unfolded extensions going into the interior. So in terms of thinking about protein folding in the test tube, where we're going to begin is with Anfinsen's hypothesis and his seminal experiment on protein folding. So Anfinsen is responsible for the thermodynamic hypothesis of protein folding. And he performed seminal experiments on a protein, an enzyme, called ribonuclease A. And so what Anfinsen hypothesized is that, in terms of a protein shape or fold, it's the primary sequence, so the sequence of amino acids, that dictates this final shape in aqueous solution. So whatever that primary sequence is, it dictates, basically, the array of possibilities and the thermodynamically most favorable result. So what was the experiment Anfinsen did to probe this? What he did is look at denaturation and refolding of ribonuclease A. So this enzyme cleaves RNA single stranded. It's 124 amino acids in length. And in the native form, it contains four disulfide bonds. And since there's four disulfide bonds, there's eight cysteines in the primary sequence. So two cysteine side chains can come together to form a disulfide. And so if we think about eight cysteines forming four disulfide bonds, there's many possibilities, in terms of how those cysteines are matched and the linkages. So different regioisomers, over 100 possible combinations of these eight cysteines to get four disulfides. And only one regioisomer, so one of these combinations, is the native form. So one out of over 100. So these native disulfide linkages that are formed indigenously are required for activity. So what was Anfinsen's experiment? The experiment he did was to take native ribonuclease A, and I'm going to just sketch that. So imagine we have the four disulfides. So first, step 1, he reduced it. So he added a reducing agent to reduce these disulfides. We'll talk a little bit more what that might be in a minute. And so the end result is, rather than having these disulfides, we have eight free cysteines. So free meaning not in a disulfide, indicated by SH. OK, so this is reduced. And so over the course of this, Anfinsen developed some assays to monitor for activity of this enzyme. And what was found is that there is a loss of activity. Next step, add a denaturant. So a denaturant is some chemical, like urea or guanidinium, that is going to disrupt the fold of the protein. And in this case, he used urea. So as this is sketched, this is still folded, but the disulfides are gone. OK, so what's the result here? We get some unfolded polypeptide with the cysteine somewhere. So this is denatured, so we have no disulfides, no native fold, and inactive. It can't cleave the single-stranded RNA. OK, so we've succeeded in destroying activity and destroying the fold of this protein. What did he do next? So the next step in this experiment was to ask, OK, if we start with this unfolded polypeptide that's completely denatured and there's no disulfides, can it return to this native form by removal of the denaturant, and then allowing it to oxidize? So imagine here, we work backwards. And step 3, remove the denaturant. So how that might be done, dialysis is a way to dialyze away the denaturant. And then what happens if we allow this to oxidize? So for instance, air oxidation. So what he found is that in this order of steps, so the denaturant's removed and then the protein is allowed to oxidize, that greater than 90% of the enzymatic activity was restored. So you have this denatured polypeptide and dilute aqueous solution, and work your way back and you can restore this activity. You have a question? AUDIENCE: The intermediate stuff, where it's reduced but not yet denatured, how do you confirm that the native fold is still the same or similar? And what were the results, in terms of activity, for that intermediate? ELIZABETH NOLAN: Yeah. So how could we confirm if the fold is perturbed? What might be a method to do that? AUDIENCE: Circular dichroism. ELIZABETH NOLAN: Yeah, circular dichroism. So that's one possibility. Did he have that available? That's another question, but that will give you a readout on alpha helix C or beta sheet. That's one possibility. You can imagine other possibilities. Maybe it would run differently on some form of column there, as a possibility. There was a loss of activity here. Was it 100% or less than 100%? I'm not sure about that detail. Joanne? JOANNE: I don't know. Are you sure they didn't denature before they reduced? ELIZABETH NOLAN: I think he's done it other ways. JOANNE: I mean, the protein with a lot of disulfides in it, they may not be accessible to reductant. ELIZABETH NOLAN: I mean, often, you add them together to get here, right? And he did a lot of experiments, as well, with additives. And then definitely, in this direction, my understanding is he performed this both ways. So effectively, if the denaturant's removed first and then it oxidizes, versus oxidizing it and then removing the denaturant, and when it was that later scenario, that the disulfides were allowed to form first, the end result was a sample that had negligible activity, less than 1%. And so from that you can imagine why because if this was allowed to oxidize, it's not pre-folded to allow the correct disulfides to form here. AUDIENCE: So with those two steps, the polypeptide and the karyote enzyme just folded back up into its original state? ELIZABETH NOLAN: Yeah, so isn't that incredible? AUDIENCE: Yeah. How long did it take? ELIZABETH NOLAN: For this case, I don't know. And depending on the polypeptide, it can vary from a short period of time. We have examples in my lab, where maybe in 30 minutes, you can see the properly folded form today, or even faster than that, depending. Like, seconds to days, depending on the protein and the size. Yeah, but that's what's really incredible about this experiment, just beyond the details of the ordering. And what happened is the fact that he could take this 124 residue polypeptide that needs to have four specific disulfides, and just in dilute aqueous solution-- without any help, minus here-- it could come to its native fold. So this was support of this hypothesis, that the primary sequence of a polypeptide that can dictate shape. And if these polypeptides are allowed to fold under dilute conditions, where intermolecular molecular interactions aren't a problem, they can achieve the thermodynamically most favorable result. And he did plenty of additional experiments, too, in terms of putting additives in and asking, how do these perturb the results? So what did he actually have to say from his experiment? In his words, "the results suggest that the native molecule is the most stable configuration, thermodynamically speaking, and the major force in the correct pairing of sulfhydryl groups and disulfide linkages is the concerted interaction of psi chain functional groups distributed along the primary sequence." So this primary sequence dictates the array of possibilities. So in thinking about that, that brings us to the paradox of Levinthal here. So he was thinking about this problem of protein folding and just thought, well, imagine we have one polypeptide with 100 amino acids. So smaller than ribonuclease A. What if each amino acid had only two possible confirmations? What does that mean, in terms of possibilities? We have 2 to the 100th. So if that polypeptide were to sample every possible confirmation during folding, taking just a picosecond per transition, the time required to fold the protein would be what? And based on his back of the envelope work here, it would be ridiculous, longer than the time of the universe. And that tells us that just can't be, in terms of how we think about this here. So each amino acid can't adopt its shape independently. That's just not working on a biological timescale. So how do we think about this? We can use energy landscapes here. So thinking about tumbling through hills and valleys. And so basically, we can depict protein folding, and this is an example, say, in a test tube, where there's some ensemble of starting unfolded, or partially folded, structures, and these are of higher energy. And there'll be some sort of stochastic search, and basically, these forms will give us ensemble of partially folded structures and, ultimately, converge to a native structure here that's of lower energy than that. So we have an ensemble of many denatured proteins that needs to make its way to the native form. And just looking ahead a bit to our discussions of chaperones, these proteins that assist in folding, this is another view of an energy landscape, but it's taking chaperones into account. So we have energy here. And this depiction is from the assigned reading. It basically divides things up in terms of productive intramolecular contacts versus intermolecular contacts that lead to situations like oligomers and aggregates and fibrils. So up here at high energy, we have unfolded, or partially folded, species. And what we see here-- bless you-- is that these chaperones are helping to allow these partially folded states to reach a native state by helping getting over these barriers. And the chaperones do not want to have the proteins going in this direction here to species that are potentially deleterious and result from intermolecular contact, so oligomers, fibrils, and aggregates here. What are some methods, in terms of experimental methods, for folding? There are many that can be employed. So you just need to take studies by a case by case basis. Commonly used fluorescence, whether that be native emission from a protein. So if you imagine, you have, say, a tryptophan. Emission can vary, depending on where it is in a protein. Methods like FRET. We just heard about circular dichroism, which tells us about secondary structure, NMR, FTIR, stopped-flow, and there's a large field in computation in theory, looking at protein folding as well. What are some methods to denature a protein? So here, we saw urea used. There's many others, whether that be heat or pH. And denatured protein means unfolded protein, in the context of the lectures in this course. So often, studies in vitro start from using an unfolded protein sample, and then you look at how folding progresses. What are some lessons from in vitro folding studies, just to keep in mind? 1, every protein's different. And even proteins that seem similar are very different. So maybe they have the same secondary or tertiary structure, some small peptide, but when you try to fold them, they may require different conditions. There are multi-dimensional energy landscapes, like what we saw on the prior slides. You can often see intermediates along the folding process. And in dilute aqueous solution, as Anfinsen hypothesized, primary sequence dictates fold. Just a note to anyone doing experimental work, why do we like to use the ice bucket in the cold room when working with proteins and enzymes? Many native proteins are only marginally stable under physiological conditions. So we can think about a delta G of denaturation per amino acid. So what this means is use your ice bucket in the cold room when working with your samples. Just closing, thinking about protein folding in vitro versus in vivo. Do studies in vitro really enhance our understanding of what's happening in the cell? Just some observations to keep in mind. So on the benchtop, folding can occur over a tremendous timescale. From nanoseconds to hours, I have here. It can be days, depending on your peptide and conditions here. The studies are generally performed in dilute buffer and in the absence of any additional protein. So you have some pure polypeptide that you want to fold or study and that's what you work with. And it's found that small proteins will often fold without assistance here. So they don't need helpers. In the cell, how do we think about the rate of folding? So from one point of view, the rate of folding is limited by the rate of polypeptide biosynthesis and how quickly that polypeptide is emerging from the ribosome, if you're thinking about a nascent chain. And we can think about the concentration of peptide coming off the ribosome, which is often quoted as low micromolar. And as I mentioned earlier, we really need to keep in mind that this cellular environment is very crowded with many different biomolecules and players. And as a consequence of this crowding, there's many proteins that help in folding, especially the chaperones. We'll see that a number of these chaperones protect the polypeptide that needs to be folded from this environment here. So a take home is that just spontaneous protein folding in the cell is error prone, if it were to happen, and that inter and intramolecular interactions are a big issue. And so these chaperones are available to help overcome these issues here. So where we'll begin on Monday is looking at trigger factor, GroEL and GroES and DnaK/J as an overview. And then we'll work our way through these different systems for protein folding in the cytoplasm. [SIDE CONVERSATIONS]
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. ELIZABETH NOLAN: We're going to move on with GroEL/GroES and a few more comments about where we closed yesterday and then talk about experiments that were done to determine what polypeptides are folded by this machinery. So I'm just curious. Has anyone stuck trigger factor or GroEL into PubMed to see how many hits you get? Yeah, so Rebecca's question yesterday, or on Monday, about trigger factor and active versus passive folding motivated me to take a look. So just to give you some scope, if you put trigger factor in PubMed, as of last night, there's 11,810 hits there. GroEL is closer to 2,000 to 3,000-- in that range. If you put trigger factor active folding, you end up with 34 hits. Most of those are about using trigger factor in protein overexpression. So if you also express trigger factor, does that help? And it looked like there was one paper in those 34 that suggests an active folding role for one of the domains. But that is just looking at an abstract. And so, the point there is there are many, many studies that consider these chaperones and a huge literature to search. So what we're able to cover here is really just the tip of the iceberg for that. There's also a new review out on GroEL/GroES, which is not required reading, but we're posting it on Stellar. So it just came out last month, and I really enjoyed reading this review. I thought they did a very good job of talking about current questions that are unanswered yet in terms of models and presenting different models for how this folding chamber works-- so passive versus active, for instance. And they also give a summary of the substrate scope-- so the experiment we'll talk about today. So where we left off last time, we went over the structure of this folding chamber and here's just another depiction of the overview. So effectively, we have to back-to-back heptamer rings as shown here. Some polypeptide in its non-native state can bind. It initially binds up at the top by these apical domains, and there are some hydrophobic interactions. OK, ATP also binds, and we have all seven ATPs found within one ring, the ring that has the polypeptide. OK, we see the lid come on, and then this polypeptide has some time, a residency time, in this chamber to fold. And then after the residency time, which is generally quoted on the order of 6 to 10 seconds, the lid comes off, and it gets ejected. And during that time, the ATPs are hydrolyzed. So somehow, this ATP hydrolysis gives conformational changes that drive this cycle. OK, and then we see, again, we flip to having function in the other ring. So one point to make involved cooperativity, so I hope you've all seen cooperativity before, probably in the context of hemoglobin. We have examples here of positive cooperativity and negative cooperativity. So within one heptamer ring, ATP binds to all seven subunits. So that's positive cooperativity. And then we can think about negative cooperativity between the two rings, where we only have ATPs bound to one ring. So the other heptamer ring will not have ATP bound here. So what is happening inside this chamber? The polypeptide enters the chamber, and it's given this protected environment to fold. And we saw that when the GroES lid comes in that the hydrophilic nature, hydrophobic nature of the interior changes. And it becomes more hydrophilic. So I just want to point out-- and this also builds upon Rebecca's question from last time-- is this passive folding in the chamber so effectively in Anfinsen's cage, where the primary sequence dictates the trajectory? Or does the actual chamber itself play a role? So that would be active folding. And effectively, is there forced unfolding or refolding by GroEL itself? So perhaps the apical domains can force unfolding before polypeptide is released into the chamber. And the cartoon that was just up indicated that to some degree. Maybe the cavity walls are involved. And what I would say is that the pendulum on this has swayed quite a bit over the years in terms of whether or not GroEL is a passive folding cage or actively involved in folding. And some of the debates in the literature have resulted from experimental set-up that may bias results to indicate one thing or the other. And that's something the community is striving to work out these days. And I'll talk about that a bit more on the next slide. But I'll just note-- these questions are still there, and the recent review I just noted discusses these questions. There was a study just a few years ago that was performed with very dilute polypeptide substrate-- so below one nanomolar. And what they conclude from this study is that GroEL is involved in active folding of a maltose-binding protein mutant. One question I'll just spring up with this is, maltose-binding protein is a nice model polypeptide, but what happens for a native GroEL substrate? And is there utility in studying those? So why have I emphasized this dilute protein sample point here? So what happened in some early work, in terms of studies that were done to try to differentiate active or passive folding, is that there were some complexities in in vitro studies. So, here, I just have a cartoon of folding in the chamber. And if we think about only one polypeptide within the GroEL chamber, it's folding in isolation. So there's no possibility for it to form an aggregate or a ligamer with other polypeptides. It's all alone here. So this folding in the chamber avoids the complications of the folding landscape we talked about in the introductory lecture to this module. So what happens in aqueous solution, right? There's the possibility that, depending on your conditions, maybe there's some sort of aggregate that forms. And if this aggregate forms, what does that mean in terms of what you see? And so, in earlier work, there were some in vitro kinetic studies that indicated GroEL accelerates folding relative to folding in dilute aqueous solution. But some of these comparisons weren't appropriate, because as it turns out, oligomerization might compete with what you're watching for. And so, if there's some oligomerization happening, it might indicate that the rate is slower than you think. So there's ways to monitor for this. And it's just a point in terms of what control studies do you need to do to make sure your experimental setup is appropriate there. I think it'll be exciting to see what's to come in future years about this question and what kinds of biophysical techniques are applied, including single-molecule studies here. So where we're going to go, moving on, is to think about what actually are the substrates for GroEL. So what polypeptides get folded in this chamber? And how do we begin to address that question from the standpoint of what's happening in the cell? OK, so first, we're just going to consider some observations. And then we're going to go into the experiments here. So here are some observations. So the first one is that polypeptides, up to 60 kilodaltons, can fold in this chamber. So that's quite big-- 60 kilodaltons. Some proteins or polypeptides need to enter the GroEL multiple times to be folded. So that means the chaperone has the ability to bind and release and re-bind the polypeptide here. So when studies are done in vitro, what's found is that almost all polypeptides interact with GroEL. So you just saw even an example of that in terms of this non-native maltose-binding protein. So many polypeptides will interact. And this really contrasts what's observed in the cell, where, in vivo GroEL is involved in only folding about 10% of E. coli proteins here. OK, so what observations three and four suggest is that GroEL has some preference for particular endogenous polypeptides. And what we want to answer is, what are these polypeptides, and what are their properties here? OK, so Hartl's group did some nice studies to look at this, what needs to be done. First of all, there needs to be a way to isolate the polypeptides that are interacting with GroEL in the cell. And then, once these polypeptides are isolated, they need to be analyzed in order to learn about their identity and properties. OK, so we're going to look at experiments that were done to address this. And they involve pulse-chase labeling of newly synthesized proteins, amino precipitation, and analysis here. So in terms of addressing what are these substrates, we're going to begin with pulse-chase labeling. OK, so basically, the goal of this experiment and why we're starting here is we want to determine which proteins interact with GroEL. And, in addition to which proteins, we want to determine how long they interact. OK, so what is the experiment? These experiments are going to be done with like E. coli cells. So we want to know what's happening in the cell. So imagine we have an E. coli. And so these bacteria are grown in some culture medium. And the trick here is that they're going to be grown in medium that's depleted in methionine. So incubate, or grow, in medium with no methionine. OK, so effectively, we're depleting them of that amino acid. OK, so then after some period of growth, what are we going to do? We're going to spike the culture with radiolabeled methionine. And this is the pulse. So we're going to add 35S methionine. And we're then going to incubate for 15 seconds. OK, and so that's the pulse with a radiolabeled amino acid. Then what are we going to do? And after we go through the steps, we'll go through why. After this stage, we're going to add excess unlabeled methionine. And we're going to then continue this culture for 10 minutes. OK, this is the chase here. And during this chase period, basically, samples will be taken at varying time points. OK, and then, at some point, we're just going to stop this. OK, so just say, stop culture and experiment. So what's happening in each of these steps? And why are we doing this? So what we want to do is think about newly translated polypeptides. OK, so we have a living E. coli. It has ribosomes. And these ribosomes are going to be synthesizing polypeptides over the course of this experiment. So during the pulse period, all proteins, or all polypeptides, synthesized are radiolabeled. Right, because the methionine has been depleted from the culture medium. And so effectively, the methionine that these organisms are seeing are the S35-labeled methionine. And all polypeptides have an informal methionine from the initiator tRNA and what other methionines are in the sequence. So, if we think about doing this for 15 seconds, and we think about the translation rate, which I gave as 6 to 20 amino acids per second when we were discussing the ribosome, we want to think about how long are these polypeptides going to be? So we have a translation rate of 6 to 20 amino acids per second. OK, so, if we think about 15 seconds of a pulse, we're getting polypeptides on the order of 90 to 300 amino acids synthesized during that time. So newly synthesized polypeptides in these 15 seconds are radiolabeled. What happens next? OK, we have this chase period where we flood the system with unlabeled methionine here. Why are we doing this? So certainly, there are some polypeptides that are longer than 300 amino acids that still need time to be synthesized. And if there's new peptides being synthesized that start in this stage, we won't see them, because this unlabeled methionine is in vast access over the radiolabeled methionine that was added early. So here, we have, the synthesis of larger polypeptides can be completed. And we have, no longer producing radiolabeled new peptides. OK, so this allows us to only see the peptides that were radiolabeled during this pulse period here. So what are we going to do in terms of the sampling at various time points? So let's say we want to sample at one minute, five minutes, ten minutes. What do we need to do? So can we just aliquot some of these E. coli and put them on our bench? We could, but that's not going to be very helpful to us, because what we want to do is stop the translation machinery and all of the cellular machinery here. AUDIENCE: You need some kind of clench? ELIZABETH NOLAN: Yeah, we need a clench. And not only do we need a clench, we're dealing with a living organism too, right? So we need to break open the E. coli in whatever this condition is to stop the reaction. OK, so we're going to take aliquots at varying time points. And basically, we care about time, so you have to immediately lyse, or break open, the cells. And this was done in the presence of EDTA. So what is EDTA? AUDIENCE: Ethylenediaminetetraacetic acid. ELIZABETH NOLAN: Yeah, ethylenediaminetetraacetic acid. So it's the chelator. And why might this lysis be done in the presence of this metal chelator? AUDIENCE: [INAUDIBLE] processes like-- [INAUDIBLE] magnesium, which would help [INAUDIBLE] AUDIENCE: Are the proteases that are not binding? ELIZABETH NOLAN: There certainly are zinc proteases. So that that's one class of protease. So EDTA will chelate many, many different metals. The main point here is we want to stop stop translation, shut down processes here. OK, so we have these samples. What do we need to do next? We need to detect these newly synthesized proteins that interact with GroEL. And we want to do this at each time point. So how are we going to do this? We have a very complex mixture that has all of the cellular components. So the next step in this will be immunoprecipitation. And so, what will happen in immunoprecipitation in these experiments is that the researchers had an antibody that binds to GroEL. And this antibody was put on a bead and used to fish out GroEL from this complex mixture. And we need to talk about these antibodies a little more. But just in starting, I imagine there's a bead. And we think about antibodies as being Y-shaped biomolecules. So here, we have a GroEL. And imagine that, in this mixture, we have GroEL that has some polypeptide bound. That's one of its endogenous substrates. So, if these are mixed together, then the antibody binds GroEL with the polypeptide attached. OK, here, we can imagine "capture" of this species here and using the bead to separate, say, by centrifugation. So let's think about this a little bit and a little background to have everyone up to speed. If you need to learn more about antibodies, please see a basic biology textbook for further details. But these are Y-shaped molecules that are produced by a type of immune cell called B cells. And they're used by the immune system to detect foreign biomolecules and help to neutralize them. And so, in these, the tip of the Y contains the paratope that ideally binds specifically to a particular epitope-- in this case, GroEL here. And so, we often think about a lock-and-key model with antibody and think about the antibody binding its target with precision here. So for these experiments that were done, just realize the researchers had to come up with an antibody to GroEL. How is that done? They may have immunized, say, a rabbit or given a rabbit GroEL and allowed that rabbit to produce antibodies. And then they isolate the antibodies here. So something we want you to take home from this course is, yes, the antibodies should bind the target with precision. But there's huge problems in terms of use of antibodies in research. This is just the start of an article that was published last year around this time. And it's focused on pharma and clinical trials. But this is much more broad. And often, antibodies aren't as specific as indicated by the label on the container from the supplier here. And it's pretty dismal what they quote in this terms of how difficult it is to reproduce data here. So if you're going to use an antibody, you always need to test it to see whether it is selective or not for the species of interest that you want to detect there and have that information on hand so you don't misinterpret your data here for that. So what are the steps for this immunoprecipitation? Basically, as shown on the board, beads will be functionalized with the antibody and then just added to the cell lysate. And the antibody can recognize GroEL. And the goal and hope are that whatever polypeptides are associated with GroEL are pulled down together. So that's something a bit incredible here that these polypeptides remain bound to GroEL during the steps of this process. You can imagine, if there's a low-affinity binder, it could be lost. So the sample can be centrifuged. And then, you can isolate these beads here. So, in cartoon form, a complex cell lysate in your microcentrifuge tube. You can add the antibody, centrifuge. And see, down here, we've pelleted the beads with GroEL attached. And then some sort of workup needs to be done to dissociate the protein, or polypeptide, substrates here. And then they can be analyzed. AUDIENCE: How long do they do that for? Do you know how many-- ELIZABETH NOLAN: How long do they centrifuge for? AUDIENCE: No, for the immunoprecipitation. Is it 30 minutes? Is it-- ELIZABETH NOLAN: I don't know how long the incubation time is. Need to go back to the experimental, but that's getting right back to this question as to how do they stay bound. AUDIENCE: How do they stay bound? ELIZABETH NOLAN: Yeah. So, see the point here. If you have a high-affinity complex, that's one thing. If you have low-affinity association between GroEL and the polypeptide, you can imagine it might get lost during this workup. And how much do we know about those affinities there? AUDIENCE: You said that they would just give rabbits GroEL, and hopefully antibodies would just happen. But if a rabbit's immune system encountered GroEL, would it actually see it as an antigen that it had to develop antibodies against? ELIZABETH NOLAN: So, yeah. So here's the point-- would it? So, if it's E. coli GroEL, would the rabbit recognize this, yes or no? And if no, then what can you do to provoke an antibody response? And so, what can be done is, say, you could take a GroEL subunit and attach that to something immunogenic. So there are carrier proteins that will mount an immune response. So one of the subunits of cholera toxin is an example that can be used. And then the idea is you're mounting an immune response against that carrier protein. But you'll also get antibodies to whatever is attached. So that's another strategy for doing it if direct injection doesn't work. And too, not going off on a big tangent, but there are some decisions that need to be made. So would they use the full-length GroEL? Or maybe they would just use a polypeptide region, like some shorter polypeptide that's a portion of GroEL. So there's a lot of possibilities there in terms of what you use to generate the antibody for that there. And it's something that a lot of companies do these days. You can send them your protein or your polypeptide fragment. And they'll conjugate it to one of these carriers and treat the rabbits or whatever animal and then isolate those antibodies. And then they need to be characterized there for that. OK, so how are these samples going to be analyzed? That's the next step. So, for the analysis, effectively, we're going to have some mixture. And, at the onset, we don't really know how complicated this mixture will be. I told you initially that about 10% of E. coli polypeptides are thought to be substrates for GroEL, which is quite a large number if we think about the total number of proteins in E. coli. And the other point is we have this radiolabel, which we're going to use for detection there. OK, so, for analysis-- OK, there's two things. We need to separate these various polypeptides in each sample. And then we need to determine what their identities are here. So-- that were bound to GroEL from one another. OK, and then, we need to determine identities. And once we know the identities, we can think about their properties. And this needs to be done in every sample that was collected along this time course, which is also going to give some temporal information. So what are the methods that have been used? So, in order to separate the proteins in this complex sample, the method is a 2-D gel-- so 2-D gel electrophoresis. OK, and in terms of determining the identities, what's done, once these polypeptides are separated, is to do a protease digest and then mass spectrometry. Has anyone here ever run a 2-D gel or seen the equipment? One person. Has anyone heard of 2-D gels? Fair number. OK, so, we'll go over this briefly in terms of 2-D gel. So, in terms of 2-D gel electrophoresis, we talk about running these gels in two dimensions. And, in each dimension, we separate based on a different property. So, in the first dimension, the separation is based on charge. And effectively, we can talk about the pI of a protein. So the pI is the isoelectric point. And it's the pH where the net charge on the protein is zero. And so, the type of gel we use here is called isoelectric focusing, or IEF. And effectively, what's done is that the gel electrophoresis is done through a continuous and stable pH gradient. And, in this gel, the protein will migrate to a position where the pH corresponds to the pI. Then the anode is low pH and the cathode high pH. So that's quite different than SDS, where, in an SDS-PAGE gel, we're coating the protein with negative charge. So then, the second dimension is something most of us are familiar with, is SDS-PAGE. And so, what happens in SDS-PAGE? We have separation based on size here-- on molecular weight. So has anyone not run an SDS-PAGE gel? And this is totally fine. I never ran one till I was a postdoc. So it's not something to be ashamed about if you haven't. OK, so everyone has. So what's the ratio of SDS molecules to amino acids? So if you take your protein sample and you put it in your loading buffer and run your SDS-PAGE, what is the ratio of binding? What is SDS? AUDIENCE: Sodium dodecyl sulfate. ELIZABETH NOLAN: And what does it do? What happens to your protein in SDS? AUDIENCE: Denatures it. ELIZABETH NOLAN: OK, what else? So it's a denaturant. So it denatures the protein. So why does SDS-PAGE let you separate based on molecular weight, more or less? AUDIENCE: It coats the protein, more or less, uniformly with negative charge. ELIZABETH NOLAN: Yeah. AUDIENCE: Do we know the exact ratio of binding? ELIZABETH NOLAN: Yeah, so what's the ratio of binding that can be done in terms of grams of SDS per grams of protein or number of SDS molecules per amino acid. What is it? And there'll be some error, but there's approximates. But it's something to think about, right? You're putting your sample into this. So it's about 1.4 grams of SDS per gram of protein. That's the ratio there. And as said, the idea is that SDS is giving the protein a large net negative charge. So it's going to override whatever the intrinsic charge is of the protein. And so, it gives all proteins a similar mass-to-charge ratio here. With that said, sometimes, there are proteins that migrate in the gel in a manner that's not reflective of their molecular weight. That's just something to keep an eye out on. So within the slides that will be posted on Stellar, there'll be some background information about both of these methods-- the IEF gel and SDS-PAGE, which I encourage you to take a look there. OK, so back to the 2-D gel-- how is this actually going to be run? So it's one gel. First, it needs to run the IEF gel. And you need a special apparatus for. This it's called a cylinder, or tube, gel-- so not flat like what you're all accustomed to for SDS-PAGE. Then, this gel needs to be equilibrated in the SDS-PAGE buffer. And then, you run the SDS-PAGE separation. And, in this step, just to note, the gel is rotated 90 degrees. OK, so what you get-- you get a gel where we have molecular weight here. We have pI here. And if it's a cell lysate, there's going to be many, many spots. These should all be spots unless you did a poor job running the gel. So this 2-D gel is being used, because it's going to provide better separation than a standard 1-D gel. Imagine trying to separate peptides out of some cell lysate using just a 1-D gel. Even after this immunoprecipitation, we'll see that these samples are very complicated here for that. So what we need is some way to detect the spots that indicate different polypeptides. So what are methods? Maybe Coomassie stain for total protein. We can use the radiolabel-- autoradiography, for instance, which is what's done here. We're looking at the S35 radiolabel-- or maybe Western blot here. So how are we going to get from this gel to knowing the identity of each of these spots? AUDIENCE: You have to identify your spot, excise it, extract the protein from the gel, adjust it, and then run NS and line it up with known protein for evidence. ELIZABETH NOLAN: Exactly. So what will be done is that each spot of interest will be cut out of the gel. So you need a way to mark them. You'll see they're numbered in the data that we'll look at. The protein needs to be extracted out of the gel. Then the protein will be incubated with a protease that will give some number of fragments. Trypsin was used in this work. And then that digest can be analyzed by mass spec. And so, for each sample, you get all of the m over z values for the different polypeptides that resulted from the digest. And then, effectively, you can compare that to some database of E. coli protein sequences. So further details are provided throughout here. So what are the major questions? And what are we going to look for answers for in the data here? So first, how many proteins interact with GroEL? We can imagine getting an answer to this by counting the number of spots. What are the identities and structural features and properties of the proteins that interact with GroEL? We're going to get that from the mass spec analysis and then literature studies. And then another question we can get at is asking, how long do proteins interact with GroEL? Because we're calling the pulse-chase samples were taken at various time points over that 10-minute period. So, at two minutes, do we see the same polypeptides associated as we see at 10 minutes? Or if we monitor one given polypeptide, when does it show up and potentially disappear from the gels? So all of these samples can be addressed with these methods. And where we'll begin on Friday is going through the data in some detail. But just as a prelude to that in the last minute, here's the data from the paper for these gels. So this is looking at the 2-D gels for, on the top, total soluble cytoplasmic proteins at zero minutes and then total cytoplasmic proteins at 10 minutes. So this is without the immunoprecipitation. And then, at the bottom here, what we're looking at are the polypeptides that we're isolated from the immunoprecipitation with the anti-GroEL antibody at zero minutes and 10 minutes. And so, before we meet next time, what I encourage you to do is take a close look at these gels and see what information can you pull out just from a qualitative look. So simple questions, like, we see a lot of proteins here. And please don't go and try and count all the spots. I'll give you the numbers next time. How do these gels here from the immunoprecipitation differ from these up top? And it's not just the total number of proteins. There's some additional subtleties in these data. OK, so next time we'll begin examining these data, looking at what polypeptides were pulled down. And then we'll move into looking at the chaperone DnaK, DnaKJ system there.
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MIT_508J_Biological_Chemistry_II_Spring_2016
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28_Metal_Ion_Homeostasis_4.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JOANNE STUBBE: So we've been talking about iron metabolism in general in the first lecture. And in the second lecture we started to focus on iron metabolism in humans, and the third set of lectures is going to be iron metabolism and bacteria with a focus on hemes. And the two things you want to talk about in the lecture today are, how does iron get taken up into cells in humans, with a focus on receptor mediated endocytosis, and then we're going to start talking about hopefully iron regulation-- how you sense iron, ion regulation at the translational level. By sort of a unique mechanism, at least at the time of its discovery. So in the last lecture, we introduced you to some key features about iron chemistry in general that we're going to use throughout this lecture and next lecture. So you need to go back and review your notes if you don't remember that. Or hopefully you've had it somewhere before, and it's a review from you from freshman chemistry or inorganic chemistry. And so iron metabolism-- what do we know? We know the average human being has 3 to 4 grams of iron. We talked about this at the end of the last class, of how is the iron distributed. We all went through that most of our iron is in our red blood cells in the form of hemoglobin. But it's also-- so in the form of hemoglobin, it also can be stored in proteins called ferritins, which we're not going to spend much time on, but I will introduce you to today. And then many of you may know that red blood cells die every 120 days. And we'll see that the iron is really continually recycled, and we'll talk a little bit about the mechanism of how that's regulated. So instead of excreting it, what happens is you recycle. The iron unit's recycled by macrophages in the spleen. And so the other place you see a fair amount of iron is in the macrophages. And the third place you see a fair amount of iron is in the tissues, because myoglobin, again, has to deliver oxygen to the respiratory chain. So what I want to do now, and I'm going to go back and forth between the PowerPoint and notes. And so some things I'm going to write down some things not. Hopefully you have these cartoons in front of you so you can write down some of the things that I will say here, and say it again and say it again. So this is sort of the big picture that I took from some review. And most of these big pictures have some issues with them. But I think it still gives you the big picture. So here's a duodenum, where we can take up iron from the diet. And we'll talk about this in more detail, but a key player in allowing the iron from the diet to go into our system is going to be FPN-- that's going to be ferroportin, I'm going to describe this again. But you're going to see FPN over and over again. It allows iron to be transferred in the plus 2 state, and that's going to be important. And so what we see, that if you look at iron from the diet, there's not that much. [AUDIO OUT] somebody's guess as to how much there is. A few milligrams. And the question is, where does it go in the bloodstream? And it goes to a protein that we're going to talk about that's a carrier for iron in the plus 3 state. So we're going to see plus 2, plus 3 into conversions over and over again. And sort of what the strategy that has evolved to be able to deal with these different oxidation states is. We'll see that this little protein, TF, is transferrin, and we're going to look at transferrin for a-- very briefly, but it binds iron 3 and bicarbonate, and then delivers this to tissues, and also delivers it to marrow. And marrow, which is-- accounts for approximately, by mass, 4% of the body weight, makes all of our red and white blood cells. So that's going to be important. And so the marrow makes the erythrocyte, the heme for the erythrocytes makes the erythrocytes, and the erythrocytes are the red blood cells that have all the hemoglobin. So out of the 4 grams, you have 2 and 1/2 grams of hemoglobin. And then these red blood cells die every 120 days, and instead of just discarding everything, they're recycled. And they're recycled by the macrophages in the spleen. And somehow you want to take the iron from these red blood cells and reuse it. And so there's a series of reactions that happen. Ultimately you get iron 2, and the iron 2-- here's again our iron 2 transporter, ferroportin, is going to take the iron that's recovered and put it back into transferrin, where, again, it can be distributed, depending on the sensing of iron. Now, the major player in the sensing and storage of iron is the liver. So the liver, we're going to see there's a protein there not indicated on the slide called ferritin, and ferritin binds 4,500 molecules of iron. And this is also-- the liver is the organ that generates, biosynthesizes the key regulator of iron homeostasis, which is a peptide hormone that we're not going to spend a lot of time on, but I'm going to show you what it does. So that's called hepcidin. And what we'll see is hepcidin in some way controls the levels of ferroportin. So we also see that we lose some iron daily, but the iron losses are small. So we have a lot of iron units, but the iron is continually recycled, and the question is, how does that happen? So I just want to look at one place where, in the duodenum, where we're going to take up iron. So what I'm going to do is-- this is a cartoon of what I just showed you in more detail. But I'm going to focus on iron absorption from the diet. And I want to make a couple points about this, which are general. And so what we'll see is we have enterocytes, so this is an enterocyte. And you have an apical brush border membrane. And then you have a second membrane which is going to get us into the bloodstream. So this is called a basolateral membrane. So we get iron from our diets mostly in the plus 3 state. But to do anything with iron, probably because of the ligand exchange issues we talked about last time, the rate constants for exchange are much slower with iron 3 than iron 2. So from the diet, we have iron 3. And iron 3 needs to be reduced to iron 2. And that can be done-- we'll see this is going to happen over and over again. And this can be done by a ferric reductase. And what we will see is in this membrane, we're going to have an iron 2 transporter. So in addition to the ferroportin I just briefly introduced you to, and will introduce you to again, we have an iron 2 transporter, that's called DMT 1. Again, the acronyms are horrible. But it's a divalent predominantly iron 2 metal transporter. And we're going to see, when we think about regulation of iron homeostasis, this is going to be a key player. Because it takes iron from the diet into our cells. And in this membrane of the enterocyte, what we will see is that we have-- and this is what you saw in the previous slide-- you have ferroportin-- so I'm only going to write this down once. But this is going to take the iron 2 and then transfer it into, ultimately, the carrier in the bloodstream, which is going to be transferrin. So here we have iron 2, but for it to get picked up by transferrin, it gets oxidized to iron 3. So what you're going to see over and over again is going back and forth between iron 2 and iron 3. And so this gets oxidized to iron 3. And these proteins-- there's a copper iron oxidase. And if you look at the handouts, you'll see that this is also called-- again, I don't expect you remember the names. What I think is key here is that you need to transfer this to the plus 3 oxidation state. So now what happens in the plus 3 oxidation state-- so let's go over to the next board here-- we have a protein called transferrin, and we'll look at this a little bit. And transferrin is going to bind iron in the plus 3 state, but it also requires bicarbonate. So in the blood, is that unusual that you would require bicarbonate? Or why might you require bicarbonate? What do you know about blood cells and hemoglobin? So we have iron 3 that's regenerated enzymatically, through some kind of oxidation reduction equipment. And we're going to see this, again, over and over again. And they each have different names, so that's confusing as well. But you're cycling between 2 and 3. And then transferrin, we have a structure of this picks up the iron in the plus 3 state, and also picks up bicarbonate. So where do you think that bicarbonate comes from in blood cells? AUDIENCE: CO2. JOANNE STUBBE: Yeah, so it comes from CO2. Why? Because a major function of red blood cells is to transfer CO2 from the tissues back to the lungs. So CO2 is not there, at pH 7, it gets rapidly hydrated to form bicarbonate and protons. And so this is unusual. I think this is one of the few systems where you have-- we'll see bicarbonate as a ligand. So in addition to these enterocytes, which again are involved in iron uptake, we also have macrophages in the spleen. And so this, again, is due to the diet. And this is due to basically recycling-- iron recycling. And so what you have is macrophages in the spleen, and you have in the macrophages dead red blood cells, which I'll abbreviate RBC. And so the idea is we want to get the iron out of the red blood cells somehow to reuse it. So that's the goal. And so somehow in a complicated process, we get iron 2 out. And then iron 2-- here we have our friend ferroportin, that I just showed you in the previous slide, is going to take and put into the extracellular mirror in the plasma the iron 2. So what happens to the iron 2? We just saw over here, the iron 2 gets oxidized to iron 3. The same thing is going to happen over here. So we have iron 2 that needs to get oxidized to iron 3. And again, let's just call it a copper iron oxidase. I'm not going to go through the details. And then what happens to the iron 3? So the iron 3 then gets picked up by the transferrin. And then depending on what the needs are the cell, the transferrin can deliver. If you have a lot of iron, it could deliver it back to the liver. We'll see that's the storage place for the iron. So the iron 3 transferrin needs to get taken up, just like we saw with cholesterol. Or if we need iron in some other tissues, we'll see that there are receptors for iron 3 transferrin that can, again, take iron into the cells to meet the needs of the cell for iron requirement. Now, the one thing I wanted to tell you in the first slide, which I had forgot, was that in addition to all of these requirements for iron, and the predominant form being hemoglobin and myoglobin, what we see is that iron is found in only 4% of the metabolic enzymes. So iron is found in many proteins that catalyze all kinds of reactions, like we talked about last time. But that's a small percentage of the total amount of iron. So this sort of diagram is pointing out a few things that sort of is indicative of iron mediated metabolism in many cases. And so what I briefly want to do is sort of summarize the functions of these different proteins. So this is phenomenological. And if you're going to have-- if you were given an exam on this, I'll give you the names of all of these things. Because I think the names are actually confusing. So number one, we have DMT1. And again, when ion is trans-- it's a transporter of iron 2. And so that's an important thing to remember. But even though it's transferred into the cell and it moves around inside the cell as iron 2, likely because, again, the ligand exchange, though iron starts here and it needs to move here, and it needs to move here, and this is the way nature-- because of the exchangeability of the ligands-- has decided to move iron, and also oftentimes copper 1 around, instead of in the oxidized state that this transporter deals with iron 2. The second key thing is ferroportin, and this goes-- brings, again, iron 2 to the extracellular milieu. And so this is bringing it inside the cell. This is bringing it extracellularly the outside the cell. And this leads to the next thing that we see over and over again-- while iron 2 is brought outside the cell, it then gets oxidized to do anything with it. So then we have general ways of iron 2 being oxidized to iron 3. And this could be a copper iron oxidase. But again, there are multiple-- there are multiple names for these [INAUDIBLE]---- we'll see in a few minutes, steep is one. I mean, they have five different iron oxidases. And iron 3 is going to be the key for transferring this to ferritin, which is the way that iron is transferred, just like the LDL particles are the way cholesterol is transferred around the cell. Ferritin-- transferrin is the way the iron is transferred around the cell. So so this iron 3-- so iron 3 is picked up by transferrin. And again, this is iron 3. I'll show you-- we have structures of all these proteins. This is, again, iron 3 bicarbonate. And then the question is, how does this transferrin get into cells? So this is the major carrier. Iron. And it's carried in the plus 3 oxidation state. Maybe, and we'll see that the KD for binding-- what do you think the KD for binding to a transferrin might be? Do you think it's weak? Do you think it's strong? And what would you, if you were designing something that was carrying around iron to all the tissues, what would you design? Something weak or strong? Say it was weak, what would happen? Yeah, it comes unbound. And then if it gets reduced, into the realm where you have iron 2 and then you have reactive oxygen species. And so nature has developed, I would say, you've seen with siderophores you can get things 10 to the minus 35 for dissociation constants to 10 to the minus 50. The KD for this is 10 to the minus 23 molar for iron binding to transferrin. And so the next thing that happens is that the iron binding to transference goes to the transferrin receptor. And so transferrin then binds to the transferrin receptor, just like the LDL particle binds to the LDL receptor. So this is the transferrin receptor. And so what you're going to see is that, in contrast with iron transported across-- in the case of the enterocyte, or in the case of ferroportin, where it's iron 2, this is all transferred in the iron 3 state. So this-- again, this is important to see the differences in the oxidation states that are used to control uptake into the cell. And this occurs by-- we'll briefly look at this, but it's very similar to what you saw with the LDL receptor. The receptor mediated endocytosis. So we're going to look at a cartoon of this. So there's one other player that I want to introduce you to. And this player becomes really critical because we don't have ways-- we don't produce a lot of excess iron and then export it. All the iron is recycled. So what controls that iron recycling? So the key regulator is a peptide hormone which I introduced you to in the previous slide, called hepcidin. And we know quite a bit, actually, about the structure of this peptide hormone. And I'll tell you what its proposed function is. We're not going to spend a lot of time discussing this. But it is made in the liver. So it's bio synthesized in the liver. And it's basically-- its function is, it's a major site of regulation, and it controls iron from the diet, and iron cycling through extracellular factors, like the transferrin-- like transferrin. So how does it do this? So here we have a little peptide hormone. It's made in the liver. And how can a control iron recycling? And so the one guy that we see now is ferroportin, ferroportin. And so its major function-- it has a lot of functions, and it's complicated, and people are still studying this. But one of the major functions is to control the amount of ferroportin. So if you look at the way it's described, the hepcidin combined extracellularly to the ferroportin. So I'll draw a little cartoon of that. And then targets it for degradation by the proteosome inside the cell. So that's the key feature of hepcidin that you need to remember. So we're going to see, if you look at-- if you look at a lot of the cartoons I've given you, you have your ferroportin, watch transfers iron from the inside extracellularly. I forgot my colored chalk today. I was on drugs or something. But people were bothering me up until five minutes. I didn't have time to think before this lecture. So I'm sorry I'm a little discombobbled here. But this is hepcidin-- hep-cid-in. And so it binds to the extracellular side. And what does that does when it binds? It causes-- somehow things change, and it causes it to be degraded inside the cell by the proteosome. So. This interaction, extracellular, causes ferroportin to be degraded inside the cell by our friend the proteosome. So does everybody sort of understand what the model is? So this is the key regulator. And you've seen ferroportin-- we only looked at two cell types. We looked at the enterocyte, and we looked at the macrophages in the spleen, both of which have ferroportins, but ferroportins that are in a number of additional cell types. And when we look at regulation, one of the key regulators of everything is going to be that we need to control are the levels of ferroportin. Because that allows all the iron to somehow be recycled. It's a key player controlled by hepcidin that allows the iron to be recycled to the different tissues. So we have a number of proteins that I'm going to very briefly introduce you to, in addition to these guys. And so we're getting into more acronyms cities. But the additional proteins that we need to think about-- so involved in iron homeostasis. Our number one, the ferritin, which in the introductory slide-- and let me just show you. So what I'm going to do, these are the list of proteins that I'm going to go through one by one and tell you a little bit. This is sort of an amazing protein. It has 24 protein subunits. It has two kinds of protein subunits. You don't need to remember this. But what is this function? It's a key-- and this is found in all organisms-- it's involved in iron storage. And why is this important? It's important because it keeps iron soluble so that it's not precipitating sort of as rust. There are, in yeast, if you look at some of yeast homeostasis, when things start going awry you can you can look at it in an electron microscope, you see iron all over the inside of the mitochondria, just these big black blobs where the iron has precipitated and mineralized. So we need to keep iron soluble, and we need to keep iron non-toxic. So what do I mean by non-toxic? In the last lecture, I told you that iron 2 can easily be oxidized to iron 3 by oxygen. We're going to talk about that in module 7 a little bit. And that can result in all kinds of damage inside the cell if it's not controlled. So this protein is sort of amazing. You can bind 4,500 irons, most of them are in the iron 3 state. But when you start out, it binds iron 2. So iron 2, again, inside the cell is what gets transferred around in general. So iron 2 binds, and then each ferritin has an oxidase activity that I'm not going to go into in detail that can oxidize it to iron 3, which puts it into this mineral structure that you see in these 4,500 atoms of iron. OK, you don't see it there, all you see is the protein there. So this gets oxidized to iron 3, and this is how it's stored in mineral form. So now the question is, say you needed iron. So we have a lot of iron, we want to keep it sequestered so we don't have to worry about reactive-- it doing chemistry that's aberrant. We want to keep it soluble. So we have iron stored in the plus 3 state in some kind of mineral form. How would you, if you wanted to use iron, now what would you do? Do you think you can get it out of the iron 3 mineral? No. What do you have to do to it to make the ligands more labile? All you need to reduce it. So to use it, you now-- and people are still arguing about what the reductants are-- so you need to reduce iron 2 plus 2 so you can use it. So that's ferritin. Does anybody have any questions about ferritin? It's got a complex structure, we have lots of structures of it. You can have-- every ferritin is sort of different, it has different ways of dealing with these issues of how you mineralize, and how you remove it. But this is a major storage protein in all organisms of ferritins. It's sort of an amazing structure. So what we were talking about before is that we get iron 3 transferrin. What does iron 3 transferrin look like? So we take iron from the diet, or we're recycling iron from red blood cells. We need to get it to the plus 3 state, where it gets picked up by transferrin. That's what we need to do. And so if you look at this-- So we've picked up iron 3 in transferrin. And the structures of transferrin are known. So now we need to look at transferrin-- whoops. And if you look at the structure, it is composed-- the protein is composed of two domains, each of which can bind iron 3 bicarbonate. So it has two little lobes over here. You can see this lobe and this lobe, the N terminal and the C terminal lobe. And they each bind-- if you look at this carefully, there is the iron, there is the bicarbonate. It has two tyrosines, a histidine, and an aspartate as ligands. And it's in an octahedral environment. So again, why bicarbonate? And people thought for a long time the bicarbonate was related potentially to how do you deliver this iron 3 out of the transferrin into something that's useful, namely the enzymes that are going to use it to catalyze transformations. And what is the bi-- is there a role for bicarbonate in that process? So what's unusual about the transferrin, again, I get-- the KD is tight. What's most unusual is it's got bicarbonate, it's got two tyrosines, and it's got a histidine, and it's got an aspartate, and it's an octahedral environment. And how do you think-- what do you think the proteination state of the tyrosines are? Everybody know what tyrosine is? Do you think it's proteinated? Non-proteinated? This brings up another sort of general principle we talked about last time. If you have water attached to a metal, what can it do to the pKa of the water? It decreases it so that you lose the proton under physiological conditions. What's the pKa of tyronsine? It's on the order of 10, 10 1/2. And in fact, this is bound-- it's the phenylate. So both of these are in the phenylate form. So both of these are phenylate. And again, if you want to think more about this, both Liz and Lippert have taught a course, are teaching a course now, in bio inorganic chemistry, where you really sort of talk about the details of these kinds of interactions, which are key to the way everything functions. So we have transferrin, and the unusual part is the binding of bicarbonate, and then, again, let me just re-emphasize it's in the plus 3 state, and you have fairly tight binding. And what we're going to see is, it's going to bind just like the LDL particles bind to the LDL receptors, it's going to bind to the transferrin receptor. So we now have a transferrin receptor. So this is the receptor. And we know we have structures, actually, of the receptors. It's a 90 kilodalton dimer. So and its transmembrane. So you have-- so this is the transferrin receptor. I'm going to show you a cartoon of this in a minute. 90 killodalton dimer, and so this is extracellular. This is intracellular. And this is the membrane. So let me just show you that cartoon over here. So extracellular, intracellular. And if you remember back to the LDL receptor, how did we trigger receptor mediated endocytosis? We had a zip code. Here we also have a zip code. And the zip code is YTRF. So there's also, on the intracellular side, a zip code for triggering transferrin uptake. So those are the players that we need to think about. So the transferrin, in the transferrin receptor, have parallels with LDL. LDL receptor-- of course every one of these things is different. But this was one of the other systems that had been characterized quite extensively, the first one being the LDL receptor. And so the model is shown here. This model hasn't really been-- this model's not completely correct. I'll tell you where things need to be changed a little bit. But really people haven't studied this model in a long time, even though there's a lot we don't understand. So here's the surface. Here's transferrrin, these little things here. Here's the transferrin receptor purple. So the transferrin binds to the transferrin receptor. To get uptake into the cell, you need to have clustering. So that's not shown here, because this cartoon was drawn before we realized that you had a cluster-- the transferrin receptors. When you transfer, when you cluster, and you bind transferrin, again, just like we saw with the LDL receptor, in some way, you have machinery that attracts the clathrin, and then it's going to pinch off the clathrin coated vesicle. And they skip here the clathrin coated vesicle. So that should be in between-- this is clathrin. And then what happens, just like in the LDL receptor, you remove the clathrin from the external part of your little vesicle. So that's what's indicated here. So what do we have? We have the transferrin receptor, and transferrin, and this is-- the internal pH of this system is about 5-5. So if you think about this, how would you-- how would you remove the iron from the transferrin? Why might bicarbonate be there? So I just told you that bicarbonate in iron are bound to the transferrin. Can you think of a mechanism by which that could happen? X inside the cell, at lower pH? We don't know the answer to this. It's still open to debate. So-- but what happens to the bicarbonate at low pH? Think about hemoglobin. Think about 5.07 and hemoglobin. We spend so much time talking about bicarbonate as a key player inside red blood cells. What happens to bicarbonate in the presence of acid? Yeah, so it forms carbonic acid. What happened to the carbonic acid? To CO2 in the water. Yeah. So this is something we banged into you over and over again in 5.07. There's an equilibrium that happens over and over again inside cells. So maybe that's a way to deliver the iron. I don't know. So we somehow lose iron. But the iron is in the plus 3 state. To get it into the cytosol, which is where we're going to use it, to deliver it to all of the proteins, what do we need to do? Hopefully you now remember this. We need to reduce it. So steep is a reductase, a ferric reductase, that converts this into iron 2. Where did we see this guy before? DMT1. We've see that before as a key player in uptake into enterocytes. So you see these same players over and over again. You see this shift from iron 2 to iron 3 over and over again, actually, in yeast, where I know a lot about iron metabolism as well as in human systems. Now-- so we've got iron 2 out of the transferrin, transferrin receptor. And then the iron 2 goes into the cytosol. And then we've got to figure out how to use it in a way so that we don't have oxidative stress and deliver it to the proteins to biosynthesize all our co-factors. So then the question is, remember in the LDL receptor, it got recycled. So what happens here is distinct from what happens in the LDL receptor. In that now the transferrin and the transferrin receptor are both recycled. So that's distinct from what we briefly talked about in the case of cholesterol. So we have two ways of taking iron into the cell one-- is through these di-- iron 2 transporters, the DMT molecules, and the second way is through iron transfer-- iron transferrin which circulates in the blood and delivers it to all the tissues. So these are the major mechanisms of iron delivery, and recycling within the cell controlled by hepcidin, this peptide hormone. So what I want to do now is look at how this iron is sensed. How do we control everything? And iron sensing-- So iron sensing, there are going to be two players. And so we're going to look at iron sensing. And I'm going to introduce you to the two players, and then I'm going to show you the general logic of how you control all these proteins we've talked about-- ferritin, DMT1, transferrin receptor-- all of these things are going to be controlled by the mechanism we're going to talk about now, which is regulation at the translational level. So this is iron sensing by translational control. So who are the two players? They're written up there. But we have iron responsive element, and we're going to see that's a little piece of RNA. So-- and I'll show you what it looks like. So this is RNA, a little piece of RNA, stem loop piece of RNA, that has defined characteristics. I'm going to show you what it is. And then we have iron responsive protein 1, or iron responsive binding protein 1. They're called both of these things, I don't remember what was in the articles you had to read. They're sort of used interchangeably. And there are two of these, so there's a 1 and there's a 2. And they're structurally homologous to each other, and I'll tell you a little bit about each one of these. So we also have a one and a two. So those are the two guys. These are proteins. So these are proteins, that's why the name binding protein. So it turns out that iron responsive binding proteins are homologous to aconitase-- where you seen aconitase before? Yeah, so in the TCA cycle. It catalyzes the conversion of citrate to isocitrate. So-- and where is the TCA-- TCA cycle located? In the mitochondria. So this is a TCA cycle enzyme found in the mitochondria. But what we'll see is, we're working on RNA, we're going to regulate somehow. We're going to use interaction between this protein and a piece of RNA to control the translational process, where is that located in the cytosol? So these proteins are located in the cytosol. So if you think about what happens with aconitase, let me just write that down for you. So we have citrate. And I asked the question, do you think it's interesting that citrate is involved in this overall process that I'm going to be describing? What do we know about citrate, besides the fact that it's an intermediate in the TCA cycle? So this is citrate. It undergoes a dehydration reaction. So we're going to lose water to form aconitate, cis-aconitate-- and then it becomes rehydrated. So that's the reaction you learned about a long time ago in the Krebs cycle or the TCA cycle. Why is it interesting that citrate is involved? I don't know why it's really involved. But do you think it's interesting? What is citrate, if you look at the structure of it? Yeah. AUDIENCE: Combined iron. JOANNE STUBBE: Yeah, combined iron. And in fact, there are iron siderophores that use citrate. I don't think this is an accident. And thinking about, again, how nature uses primary metabolites over and over again in ways other than what you see in primary metabolism. So what's unusual about this protein is the following. And this is the key to the way the sensing is going to work for the iron responsive binding proteins. So if you look at-- if you go back and you look-- if you go back and you look at the Krebs cycle, or you go back and you think about this, this is something that probably confused you all. You have an iron 3, a 4 iron 4 sulfur cluster. Remember I talked a little bit about this, trying to show you that this was going to be highlighted later on? and what we have in this 4 iron 4 sulfur cluster-- you have a cysteine attached to three of the irons. We have one iron that's unique, OK that doesn't have the cysteine that you see in normal 4 iron 4 sulfur clusters. So this is the unique iron. So if you look at that over here-- so here's the cartoon of this. So here you have cysteine, cysteine, cysteine in the 4 iron 4 sulfur cluster. Here's citrate. And that iron-- so most of you probably learned in respiration, iron sulfur clusters are involved in electron transfer. They do one electron chemistry. They undergo oxidation reduction, which we briefly discussed in the last lecture. But what's it doing here? What it's doing here is binding the citrate. So here's citrate. This is the hydroxyl that we're going to eliminate to lose water to form cis-aconitate. So this is the first example. But this was discovered by Helmut Beinert at Wisconsin many years ago, where the iron sulfur classes were doing something other than redox chemistry. This is just the tip of the iceberg. Remember, I talked to you about radical SAM proteins, 100,000 proteins doing interesting chemistry. This is the first example of this. And these really are seminal experiments to figure out how this all worked. So the unusual thing is that most iron sulfur clusters look like this, and they all have 16 on each of the iron, and they do redox chemistry, but now we're finding that a lot of iron sulfur clusters have unique iron they can end up doing interesting chemistry as well, namely binding S-adenosyl methionine. So if you go back and you think about what happens, this is helping dehydration. So you're going to dehydrate. But now you have to reorganize the thing. This is one where they talk about the Ferris-- spinning around the Ferris wheel if you look at an introductory TCA cycle thing, how this reorganizes. I don't think this is a very good picture. But it needs to reorganize because you're going to rehydrate another carbon, using the same iron. So if you sit here and you stare at this, what you see is this carboxylate. Now, here was the initial carboxylate bound, this one wasn't bound. Now, this one ends up being bound. And now you're adding water back across this double bond. So the purpose of this system is simply to catalyze the dehydration reaction. So what the heck are we doing with an iron responsive binding protein being a cytosolic aconitase equivalent? And so what I'm going to come back and tell you one Friday is, this is going to be the key switch for iron sensing. Whether the iron is in the apostate, with no metal, or whether it moves to the 4 iron 4 sulfur cluster state. And we'll talk a little bit then about how those two states, and the presence of RNA, can control which of all these proteins I've thrown at you today actually get translated. OK.
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MIT_508J_Biological_Chemistry_II_Spring_2016
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R11_Mass_Spectrometry.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JOANNE STUBBE: This recitation on mass spec is supposedly associated with reactive oxygen species. So [INAUDIBLE],, which happens all the time in this course, because we can't describe all the techniques as we go along. So what I'm going to do is just give you a two second overview of what you need to think about to put the paper you read into the big picture. I don't think the paper-- the paper also explains it. And this week we're going to focus on the mass spec paper, which is mostly sort of trying to figure out the technology, and then the next week is focused on the biology. And so the major unsolved problem-- so everybody and his brother is using mass spectrometry as a tool nowadays. There has been a revolution in mass spectrometry. [INAUDIBLE] The instrumentation is cheaper. The mass spectrometric methods have just really taken off. And people didn't even know who mass spectrometrists were, but they're starting to win major prizes, because it's revolutionized what we can do. I was talking to somebody yesterday, and they just got a mass on a protein that's 3.3 million. How do you get a protein into the gas phase that's 3.3 million? Right. Doesn't that sort of blow your mind? Anyhow, it's been a revolution. And we're going to be looking at-- this module seven, which is on reactive oxygen species, and we've been talking about the question of homeostasis. And so one of the things with these reactive oxygen species is they are used by us to kill bacteria, viruses, or parasites. But now, in the last five years or so, everybody's focusing on the fact that here are these reactive oxygen species that play a key role in signaling, which is everywhere, and the signaling process we're going to be looking at next time and is alluded to in this particular paper is epidermal growth factor receptor and epidermal growth factor. There are hundreds of these proteins that have receptors that are involved in growth and epidermal growth factor receptor [INAUDIBLE] of successful cancer therapeutics. So it's interesting what happens up here, what happens down here, how do you control all of that, and people are studying this. So we've already seen cystine is unique. And if you have a reactive oxygen species, and we'll see that the reactive oxygen species we'll be looking at, when we're going to be looking at a number is actually superoxide. So that's one electron reduced oxygen, which in the presence of protons can rapidly disproportionally give[?] oxygen gas to hydrogen peroxide. And hydrogen peroxide can react with cystines to form sulfenic acids, which is the subject of the paper you had to read. And so the question is how prevalent is this, and the question is, is this important and interesting in terms of regulation inside the cell? And so the key issue is-- even cystines aren't all that stable, you know, if you have proteins with cystines, and you let it sit around for a long time, you could form-- and the protein's concentrated, you could form disulfides. It's not a straightforward reaction, but you can form disulfides. The question is if you had hydrogen peroxide inside the cell, which you do, can you form sulfenic acids, and do they have a consequence biologically? OK, and that's the question we're going to address next time. And so the issue is this is unstable. So if you want to develop a method to look for this species, and you start cracking open cells, and you start working it up, what happens is this falls apart and reacts and gets destroyed. And an example of this is the area of DNA therapeutics and DNA drug interactions, therapeutics that interact with DNA. For decades, you see lesions on your DNA. How do you determine what the lesions are? Mass spec has been a major method to look at that. Almost all the lesions in the early days were complete artifacts of the analytical chemistry to work them up. They had to get them into some form that you could stabilize the lesion and then analyze it. And what was happening because they weren't careful enough and quantitative enough, they changed it to something else. And so the data in the early years was all completely misinterpreted. So the issue in this paper is that other people had developed this, and Kate Caroll has taken this on. Can we have a way of derivativizing this [? mentally ?] inside the cell, because if you disrupt-- if you disrupt this by cracking open the cells and trying to purify things, it undergoes further reaction, and what this can undergo further reaction to is SO2 minus, sulfonic acids or SO3 minus. Sulfinic acids and sulfonic acids. OK. And it turns out this reaction is also reversible with hydrogen peroxide a lot of people are looking at that at this stage is irreversible. Anyhow. So the question is can you develop methods to look at all of these things. And in fact Tannenbaum, who was in the chemistry department, but also [INAUDIBLE],, he is looking at nitrosation of SHs, again forming a reactive species, and he's developed new methods sort of like Carroll has to try to specifically look at these modifications. And in the end, what you want to do, and this is the key, you might be able to detect this-- the question is, is this interesting? So you have to have a way to connect this back to the biology inside the cell. And that's what the second paper is focused on. So what we're doing today is simply looking at the technology that's been developed to try to get a handle, how do you look at sulfenylation, you're not really focusing on the biology of the consequences. And so what we're using is mass spec. And we're using a method of mass-- how many of you have done mass spec? So if you know something and I say something wrong, you should speak up, because I'm not a mass spec expert. And actually, I've got a whole bunch of information from, say, the Broad, and I thought it was not very good. So we need a way of trying to figure out that you're going to see-- there's hundreds of variations on the theme. I'm going to give you a very simplified overview of what things you need to think about. And so if I say something that you don't agree with, tell me. OK, so when looking at mass spec-- this didn't exist when I was your age-- using soft ionization methods, and what does that mean? It means that you don't want your molecules to crack. So the issue is that what mass spec is about-- so really looking at mass spec, and the key issue of what you wind up looking at is mass to charge. OK. So m over z. OK, so the problem is how do we get something charged enough so that the mass is small enough so that you can see it, taking a look at the mass analyzer, which is going to be part of all mass spectrometers. OK. So there are two different ways you could change the mass to charge. You could dump an electron in. And if you dump an electron in, that produces radical species, which can then fragment. We want to avoid that. That's not soft ionization methods. But how can we control this? The way we can control this is dumping in protons. So what we do is we can control it by adding protons or by subtracting protons. And we'll see that the different methods we're going to be looking at, we'll see there are two main methods that most of you have probably heard about your classes. One is electrode spray ionization, so ESI. And I think, if you're in Brad's lab, they have a lot of these. Yesterday's class had people that had used these, but really didn't know much about what's inside the machine. So this is the kind of thing I think your generation, if you're going to use this as a tool, need to roll up your sleeves and understand a lot more about what's going on, and MALDI, maser MALDI. Matrix Assisted Laser Desorption-- it will become clear why it's called that in a minute. So these are the two methods. And what we do is we can protonate, so that we can move this into the analyzer range, where we can actually read it. So what we'll see is the analyzer-- I'm going to show you sort of what the three parts of a mass spectrometer are-- can only read 1,000 to 2,000 daltons. OK, so if you look at your protein, much, much bigger. So you're going to have to stick a lot of charges on there to be able to see anything. So that's the whole thing, and the question is, how do you do it by one method or by using the other method? OK, so all mass specs have sort of the same components. And you can go to websites. The Broad does have a website, and what the Broad will tell you is what all these spectrometers are, but I don't think they do a particularly good job telling you what's useful for what, and why it's useful, which is, I think, what you need to use if you're only going to use it fleetingly and then move out. So you have a source. So you have an inlet. How do you get your sample from the liquid phase or the solid phase into the gas phase? OK, so that's going to be that. And so what is the distinct ionization method? And we will see that there are many ways that you can ionize, and we're just going to briefly look at in a cartoon overview of how this happens. And then so once you ionize it, it needs to move from the source. So you need to have ion movement into the analyzer. So this is the mass analyzer. And this becomes important. And we will see in a second that there are many methods to do the mass analysis, mass to charge analysis, and then after you do this, you have a detector. And then, furthermore-- and I think this is a big part of it now, if you're doing wholesale anything, you have to have a really sophisticated method of data analysis. And so that's the other thing that I get frustrated about all the time. So you see people-- I mean, people do experiments where they've spent-- last year, somebody spent three months trying to get all the proteins out of a cell, 10,000 proteins out of the cell by mass spectrometry right. Now, because the technology is changing, they can do it in four days. But what do you do with all this information? And how do you use this information in a constructive way, and how do you know if it's correct or not? So those are the kinds of things. I think if you're going to use this-- I think everybody is going to be using this technology. You need to educate yourself about how to look at this. OK, so that's what the issue is. And so we have a source, an analyzer, and a detector. OK, so this is just a cartoon of that, which describes this in more detail. And I think he put this on the web. I think he put the PowerPoint on the web. I was doing this at the last minute yesterday. So it's different from the handout I gave you that's written out. This is a PowerPoint. OK, so you can go back and look at this, but one of the other things I wanted to say is that sometimes when you analyze your mass, you want to analyze it further, and that was true-- many of you might not have caught it, but that was true in the analysis that was carried out in this paper. Did anybody recognize that you had to analyze this using more than one mass spec? Did you look at the data carefully enough? So also you probably didn't read the supplementary information, which also is critical to think about. I mean, if you want to look at the methods, you need to get in there and roll up your sleeves and look at them. So we're going to see that the methods that people often use is they don't look at the whole protein, but they degrade it down into pieces. So then you can find here a whole bunch of pieces, OK. But that doesn't tell you anything. The mass does tell you something. It might tell you whether it's sulfenylated or hopefully, you can distinguish between any other modification, but it doesn't tell you the location of the sulfenylation. And so you can do a second method. So you could have some other gas. There are many ways to do this that you bring this in to now take a peptide. So you pick one mass charge. You throw in something that's going to degrade it by fragmentation, and then I'll show you in a minute we understand what kind of-- using certain methods, we understand the fragmentation patterns, which actually allow you to sequence the amino acids. And the reason I'm bringing that in is when I first got to MIT, Klaus Biemann was in the lab, and I did many experiments with him. And these are the first experiments that were done to sequence peptides by mass spec as opposed to doing Edman sequencing, which the mass spec was actually better, and there are pluses and minuses, but I noticed from looking at the literature, people were still using the same method that he developed. So this is just a cartoon. And it just shows you that there are many ionization methods. We're focusing on these two, FAB, fast atom bombardment. We didn't have any of these when I was your age. Fast atom bombardment was something I used a lot because I've worked on DNA drug interactions, and it allows you to look at nucleic acids. And a lot of these other methods don't. I mean, we're focused on proteomics in this particular paper, and then mass analyzer. So you have time of flight. I think Brad's lab has MALDI time of flight. So what does that mean? You've got a long tube in here, and what happens is you have mass to charge, and they're different sizes. And so the smaller ones fly faster. They don't want to keep away from the walls, but the smaller ones fly faster than the bigger ones. So that helps you differentiate between all the ions you're actually looking at. I guess somebody just told me you guys just got a new quadrupole ion trap. Anyhow, if you want to look at this, I have notes on all these things. But I think this is something you'd have to study in detail. And so while I have pictures of them all and how you can differentiate one from the other, I think it doesn't really mean that much to me, because I don't know enough about the physics of how they were designed. I mean, this really has revolutionized what you can do. OK. So that's the components of all mass spectrometry. What I want to do is very just briefly look at the ESI and then look at the MALDI and then show you what the issues are in general, and then we'll focus right in on the paper, and the recitation I did on Thursday, we didn't quite get through all of it. We got through most of it, but then we'll continue next week and also attach this to the biology, which is the second paper, the nature chemical biology paper also written by the Carroll group. ESI. So that's the one we want to look at next. And so that's up there. This is a cartoon of how this works. So what do you do, and how do you do this? So the first thing is you have your protein of interest, which I'll call the analyte, because we want to charge. Lots of times you put it under more acidic conditions, pH 6 or something, 6 1/2, depends on the protein. So you get more charge states. And if you're trying to look at something big, you need a lot of charges on there to get it into this mass range of 1 to 2,000 to be able to see it using this method, and apparently what you do here-- can you see this? Have you done this? Can you see this capillary? Can you look at what's going on? AUDIENCE: I don't think so. JOANNE STUBBE: OK. So I was just wondering, because I haven't ever. So it's all closed off. It's in a box, and so you can't-- there's not like a thing where you can watch what's going on? AUDIENCE: Not that I've seen. JOANNE STUBBE: OK, because I think it's sort of amazing. How do you get this huge protein and solution into the gas phase? Right. I mean, that, to me, is like mind boggling, OK? I mean, these guys were geniuses. And you know, there's been a number of Nobel Prizes for this, but I wouldn't have a clue how to do something like that. So what you do is apparently, you put it down a capillary and then you spray it out, and then you have to-- so what you get at the end of this, this plume of spray, apparently you've got a lot of the analytes and a lot of solvent molecules, and then the goal is during this process, to get into the analyzer is to get rid of-- to separate all the analytes mixed together into a single analyte and remove all the solvent. OK, so that's the goal. And apparently, according to the people that were here yesterday, this is taken from, I think, sort of one of the papers that was first out. This is the way they did it in the old days. I don't know if they still do it this way, but the goal is really, to get a single analyte with no solvent on it. OK, and so the question is, how do you do this, and the chamber they had was at atmospheric pressure, and then they had a potential and pressure gradient, which allowed it to get into the mass, before the mass analyzer. So you start here with the initial spray, and then as you go farther, you remove some water molecules. You finally get to the place where you've removed enough water molecules that all these positively charged species come together, they're incredibly unhappy. And then they fragment apart. I mean, that's the way they describe it. It sounds reasonable. So you get smaller and smaller till eventually, you get to a place where you have an analyte that you can look at specifically and the water has been removed, and that's what you look at. OK. So again, we need to be in the range of 1 to 2,000. So that's the way these things work. Although, I think, again, how you get to looking at the single ions I think in different mass spectrometers. And so what the issues are, I think, are shown here, and this is the beauty of this methodology. So if you have a protein of 10,000 molecular weight, you couldn't see it, because the mass analyzer is limited. So you have to go all the way down to eight charges on it to be able to see it. And then, you divide that by that, and you get-- what do you have? You have to do some corrections, but you get something that's this size. OK, but you can see it now because of all the charges on it, but the beauty is if you add more charges, you get another peak. And you get another peak. And it all has the same information, and it just differs by the number of charge. So you have all this information. You can use that-- all these informations together to give you a very accurate mass on this system. So this method by analyzing all the data, and now the computers do this, I guess, routinely can give you a very accurate mass. So if you look at this printout, it doesn't look like that. This is what it looks like. And what do you think's going on there? So we look at mass charge, and we're in the range of 1,000 to 2,000 daltons. And then what is this all-- what is all of these peaks associated with? Anybody got a clue? AUDIENCE: Isotopes. JOANNE STUBBE: Yeah, so isotopes. So where are we seeing isotopes before? So these are mostly stable isotopes. We spent recitation two and three looking at radio isotopes. OK. I would say, you know, radioactivity is pretty important. Stable isotopes are extremely important to mass spectrometry. So if you get into this, you're going to be able-- you'll see that being able to label things with different kinds of stable isotopes is key to really deconvoluting the complexity when you're looking at a whole proteome and thousands of peptides. We're getting down-- it becomes very complicated, and you have to be able to compute what you expect based on the normal natural abundance isotopic distribution. So that's the key thing. So we look at the normal isotopic distribution. And if you look at that, I think in the next one, I show you an example of that. So what are the isotopes-- you probably can't read this here, but if you pull out your computer, you'll see this. So we have C12, C13. OK, we have hundreds of amino acids with carbons. So you have C12 and C13. C12 is 99%. C13 is 1%. That's an actual abundance. OK, so every one of these has different natural abundance. We know what they are. In fact, if you're an organic chemist, you can measure isotope of x using a mass spectrometer, if you have something that's really accurate, which we do. I've measured a lot of C13 isotope effects, using a mass spectrometer, based on differences in natural abundance and changes. Yeah. AUDIENCE: [INAUDIBLE] JOANNE STUBBE: The what? AUDIENCE: The natural abundance of deuterium? JOANNE STUBBE: Yeah, I think it's up here. So it's up here. I think it's-- let's see, 3%. Yeah, protons deuterium 3%. AUDIENCE: Would you expect a huge distribution from that? JOANNE STUBBE: You see isotope effects on everything. You see-- if you do mass spec, I mean, this is something I think that's not appreciated, and you have a linker with deuteriums in it, and even if you chromatograph it, you change the chromatographic properties based on the deuterium, and so you might think it's migrating here, and it doesn't. It has an isotope effect on how it migrates. So yeah, you need to pay attention to all of this stuff. OK, and it seems like a small amount, but the beauty is that it is a small amount, but it's incredibly informative, and we have very powerful computers that can allow us to do the analysis. So we do have protons. You see deuterium used. You saw deuterium used in this paper you read today. They did CD3 and CH3's. OK, you can also see the tritium. OK, that's much smaller. I don't know what the ratio is, but you can look at it. But you also-- this one is also incredibly important and is widely used in proteomics-- N14 and 15, and people do isotopic labeling. So they might see N15 labeled lysine or arginine or deuterated lysine or arginine. And why do you think they would deuterate the lysine or the arginine or N15 label it? What do we know about lysine and arginine in terms of thinking about proteins and analysis of proteins? What do you think about lysine and arginine? You've seen it several times over the course of this semester, and you probably saw it in 5.07. AUDIENCE: [INAUDIBLE] JOANNE STUBBE: What? AUDIENCE: The protons will exchange? JOANNE STUBBE: Well, now as you put it-- no. So that that's true if it was on a hydrogen and a nitrogen, it would exchange, but they put the deuteriums in on carbon, so they're not exchanging. OK, so why that would happen in any amino acid, why lysine and arginine? And the reason is that almost all-- and this was also done in this paper, you don't work on the huge protein. You cleave it to pieces. And you cleave it into pieces, and where you cleave is with trypsin, which is the major-- you've seen this used now over and over again. That's a major thing you use because it cleaves next to basic amino acids. So these become really important in labeling experiments, if you read much mass spec data, or if you look at Alice Ting's work, everything is N15 and deuterium labeled, and lysine and arginine to try to make sure they have coverage of the whole proteome, which is what her lab actually looks at. OK. So we have isotopic labels, and we can take advantage of these, and we can calculate what the distribution should look like, OK, of the isotopes should be, depending on what the-- we know what the sequence is. We know what the abundance is. And so you can calculate the whole mass spec. So let's see. So there's going to be a number of things that we want to do, and what we're going to be describing today and the next time is a "workflow." These are the words that people use all the time, and "platform." And what we're trying to do in the case of the Carroll papers is simply look at whether the protein is modified or not. But as with most post-translational modifications, do you think this is going to be 100% modified? No. In fact, it's only partially modified. That adds to the complexity of understanding whether the biology is interesting or not, so what you have then is something that's modified and something that's more non-modified. So then the question is, how do you tell how much is modified and how much is non-modified? If this enhances the rate only a factor of two, and this is 99.8%, of this, are you ever going to be able to see an effect of this modification? That's the question that you have to focus on, and everybody and his brother is doing experiments like this. We will see in a second, hundreds of post translational modifications, and the question is what are they doing in terms of thinking about the biology of the system. OK, so what's the platform? What's the platform we're going to use? So there are two ways you can look at this. So we have a protein that has been modified. You're going to-- if you had a huge protein, and you only had a single OH on it, even if it was 100%, and the protein was, say, 300,000 molecular weight, you might not be able to see it. You need to do a calculation to see whether you could see it or not. If you have a small protein of molecular weight 30,000, or whatever-- I think the 22,000 or 23,000 like glutathione peroxidase, used in this paper, you could see it. So you could look at the protein directly. But how else could you do this? You would enrich. If you were doing this in the whole cell, you would want to separate this away from everything else. OK, so to do that, you want to be able to have a way to stabilize this, OK, and that's what this paper is all about, and then not only to stabilize it, but to separate the stabilized form out. So where does this happen? And in this particular cartoon, where do you see post translational modifications? Probably the most popular one is phosphorylation. So we have signaling cascades in kinases. And in fact, if you look at the epidermal growth factor receptor, it's a tyrosine kinase, and it gets phosphorylated and is regulated. And this sulfenylation is supposed to be on top of the phosphorylation. So you have multiple post-translational modifications that can affect activity. So Forest White, for example, in BE, works on kinase signaling cascades. And so he's developed a method, as have others, to be able to pull phosphorylated proteins out of a crude gemisch. OK. So, you know, if you look at this, here he's got iron bound to a phosphate and bound to some bead. So the iron's bound to some chelate around the bead, just like your nickel affinity column, which then binds to the protein. But this raises the issue that I was discussing in class, which I spent a lot of time on over and over again, but you need to think about, do you think these bonds are tight, how tight do you think those bonds are? What do you need to think about for this kind of analysis to work? It's the same thing with nickel affinity column that you talked about when you were looking at purification of proteins. AUDIENCE: It has to be stable enough. JOANNE STUBBE: It has to be stable enough. That's the key. So you have to undergo ligand exchange. It's got to-- if you didn't have-- when you start, you don't have phosphorylated form of your protein around. You have nothing. You have water there. OK, so the waters have to undergo exchange, so the phosphate can then bind, but it's an equilibrium, and so up and down the column is coming off and on. Yeah. AUDIENCE: [INAUDIBLE] JOANNE STUBBE: It could. I mean, so it's a question of what out competes what. It's a question of relative Kds. So what you have to do is study all of this to figure out how to optimize this, how did they arrive at this? Probably somebody did a lot of studies. OK. This is a new method. I don't know how new it is, but it's a method I don't know that much about, again, of pulling phosphates out. So that's one way. So you have-- so you usually have an affinity purification. And if we look at the Carroll paper, what she does in the next paper is she's going to figure out a way-- she's derivatized, she's made a dimedone derivative, which stabilizes the sulfenic acid, and then she attaches something to it that's going to allow us to affinity purify that. We'll come back and talk about that later. So what are they using over here? They're using-- this is-- if you look at histones that get acetylated or methylated, they have an antibody that's specific for the acetylated lysine, so they use antibodies to pull something out. So that's a method-- the second way of pulling things out are using antibodies. That's quite frequently used. And what did they use in this paper? Did it detect the modified sulfenic acid? Does anybody remember? Did you read the paper carefully enough? AUDIENCE: Like, a anti-dimedone antibody? JOANNE STUBBE: Yeah, so they use an antidimedone antibody. OK, so that becomes really critical that you know that your antibodies are actually working effectively. So we have antibodies, and then, another thing that people are interested in this department, the Imperiali lab, is sugars. We have sugars everywhere. OK, we don't really understand the function of these sugars. We understand some of them, but it's amazingly complex. And what we have are proteins called lectins, and any of you heard Laura Kiessling talk, maybe undergraduates wouldn't have done this, but she discovered a new lectin and discovered the basis, the structure the sugar that binds to this lectin. And so you can selectively move that type of sugar. Again, it's an equilibrium. So they're coming off and on, but it binds, hopefully, enough so that the other stuff washes through, and you enrich in the protein of interest. So these are sort of some of the tricks that are actually used. We're going to see, in the case of the Carroll paper, next time we use click chemistry to make something with a biotin on it, because biotin you all know can bind to streptavidin, which has pluses, and it has minuses, but it allows you to pull things out more easily, because the interaction is so tight. So you could do this-- the workflow could be on the intact protein, or it could be on peptides. OK. And so the bottom half of this graph shows what happens after you treat this with trypsin. So with trypsin, and you're always cleaving next to lysine or arginine. So the C terminus of your protein is always a lysine or an arginine. And you can find that more easily if you deuterate or N15 label it. That's what people routinely do in the [? Broad. ?] And then you have, I think this is the most amazing thing, so you have a protein. And then you have an HPLC column. Have any of you done HPLC? And so do you think-- you could have a protein of 300,000 molecular weight, and look at the separation of your peptides. But if you look at any one of these things, do you think it's pure? So it's not pure. So every one of these peaks, if it's 300,000 molecular weight, you can calculate-- the reason people use trypsin is-- does anybody know why use trypsin, besides that cleaves at lysines in its specifics? Why do people use trypsin as a thing to cleave a big thing down into a little thing? AUDIENCE: What's the rationale for cleaving it? [INAUDIBLE] JOANNE STUBBE: So the rationale for cleaving it is just to make it smaller and easier to analyze. That's the rationale for cleaving it. So a peptide, a small peptide. But the question is, how big is the small peptide that's easy to analyze? And so that's the rationale. It gives you a distribution of peptides that's pretty good, that are all accessible to mass spec methods. So I don't know what the distribution is, but you know, people have done that calculation. And so almost always the peptides fly, whereas if you use other things, and you have something much bigger, it might not get ionized in the appropriate way or in a quantitative way, and you completely miss it. So the trypsin has been most successful. But each one of these little peaks is not one peak. You'll see when you put it into the mass analyzer, and if you read this paper carefully, you will see they got multiple mass charge species, which then they associated with specific peptides, OK. They know the sequence of their protein. And then they always use tosyl phenyl chloro ketone. Why do they use that? Anybody have an any idea? So in the experiments where they're doing the trypsin cleavage, they put in tosyl phenyl chloro ketone. Anybody know why? OK. No good. This is something that-- so tosyl phenyl chloro ketone is an alpha halo ketone. So it's activated for nucleophilic attack, and what you do is you have an acylated N terminus and an aromatic, and that's specific for chymotrypsin, like proteases. And so what this does is that covalently modifies the active site of chymotrypsin, and kills chymotrypsin. If you choose the wrong time to cleave with trypsin, you don't start getting cleavage next to hydrophobics, which then makes the analysis of the peptides much more complex. So the analysis of the peptide, a lot of people have done a lot of peptide chemistry, and I was telling this story before. I always go off on tangents. But Stein and Moore won the Nobel Prize. Maybe this is what you do when you get old, but Stein and Moore won the Nobel Prize, you know, in the 1950s, the 1950s, for separating amino acids. Do you know that they had a three story column of Dowex that was composed of anion exchange Dowex and cations? It was all polystyrene backbones of anion and cation polystyrenes, to be able to separate the amino acids. OK. And when you do that, of course, it gets stuck on the resin. Your recovery's out of the bottom of this chromatography. You need tons of stuff to put on the column in the first place. And this is what's happened. I mean, you have a little tiny HPLC column that has huge number of theoretical place that allows you amazing separations. I mean, again, the technology is sort of mind boggling, what you can do now. OK, so what you're doing here is then you're just asking the question, if you have a post-translational modification, x, you can either look at the entire protein. And so you could probably tell it was modified, but telling the location of the modified location, you can't, or you can treat it with trypsin. And then you get, again, with trypsin, you have little pieces. And one of these little pieces will have an x on it. And then you can define it. And then if you want to do sophisticated analysis, you can hit it-- use a second mass spectrometer, and actually sequence this. OK. So I think the next one just briefly goes to MALDI. And MALDI-- so Matrix Assistant Laser Disorption-- have any of you ever done that? OK. So where do you do that? Do you do that in [INAUDIBLE] lab? AUDIENCE: No, in the undergrad lab. JOANNE STUBBE: Oh, OK, because this is Brad's new thing. OK. OK, so you're looking at peptides. OK, so what do you use as the matrix? AUDIENCE: We used some aromatic acid. I don't remember. JOANNE STUBBE: OK. So you probably used sinapinic acid. AUDIENCE: [INAUDIBLE] JOANNE STUBBE: OK. So this is so-- you're using a different one still from this one-- this is-- I don't know. I got this idea somewhere. I don't know. So when I've done this-- I did do this maybe 10 years ago-- I've looked at a lot of peptides. We went through five or six of them before we found one that really worked well. So I don't know how state of the art has become, you know what it is. But the other one in the book that I got this from was, again, an acid. And so what is the idea? So the first thing you have to do is you have to ionize. So the way you do that is you mix your matrix and solution with your protein of interest, your analyte, then you evaporate it. So you have a solid on a little plate. And then you use a laser beam at 337 nanometers. And the light is absorbed by whatever the matrix is and causes you to have a plume of material. This is, again, amazing to me that the protein goes into the gas phase. And then, you have to go through this, go into the analyzer. Did you do time of flight? OK, so you have time of flight. So you guys know what it is then. And in the end, you do detection. So, again, the protocol is the same, but the method is different, and this is widely used and easy really easy to use nowadays. So the issue then is this is what you face when you're looking at a whole proteome. So you just can't calculate the mass of all the proteins from the gene sequences. Why? Because almost every single amino acid in your proteins are modified. So that adds complexity to all of this. So de-convoluting the mass spec becomes more complicated. So this just shows you, you don't need to look at this, but if you look at cystine, you could form disulfides. You can attach a prenyl group, an isoprene group on it. You can attatch palmitic acid on it. You can sulfenylate it. You can nitrosate it. So you have many, many modifications of the amino acids that are chemically reactive and involved on catalysis, and then not only involving catalysis, they are involved in regulation. So that then adds to the complexity of trying to deconvolute what the mass spec, I think, is actually telling you. And then, sorry, it went backwards. And so then what that does is tells you-- whoops. I'm just completely discombobulated here. OK, so what that does is that, again, you're just adding different masses on to all of these amino acids. The problem is that you have modified, and you have unmodified. And the question is what's the distribution? OK, and so if you have a very non-abundant protein, and most of it's unmodified, it's going to be much harder to find. So these are just things you need to think about, and your technology to look needs to be extremely well worked out, so that when you look and you don't find something, you know what the lower limits of detection are. So here we are at our system. Now we're into the Carroll paper, and so what we're looking at is sulfenic acids, degenerated by hydrogen peroxide. We'll see-- do you think that's a fast reaction, hydrogen peroxide with a cystine? Anybody have any intuition? I think these reactive oxygen species you're going to find are not so intuitive about the chemical reactivity. I'll give you a table with what we think we know in general. But I think it's not so intuitive. If you look at the rate constants for reaction of a hydrogen peroxide with a cystine it's 1 per molar per second, really slow. OK. So then the question you have to ask yourself, so this was something that was debated in the literature for 15 years. Is this so slow that this could never happen inside the cell? Because I just gave you a second order rate concept. So we have two molecules interacting at the concentration, this could be high. This is really low. You can calculate the rate constant for the actual reaction. It's really, really slow. OK, so we'll see that there are some proteins, peroxiredoxins that are in humans, are there in quite high levels that can increase this rate to 10 to the fourth per molar per second. So there's a huge rate increase but you need to think about all this kinetic stuff to really understand if this modification can happen inside the cell. Otherwise, well, if it can't happen, why are you wasting your time looking for it? Which is what a lot of people are doing scientifically. OK, so let me see what the next-- OK. So now we're into making a reagent that can specifically modify this, or specifically modify this. OK. So the reagent that they chose-- she didn't invent this reagent-- was dimedone. And this reagent specifically interacts with sulfenic acids. It doesn't react with the free cystine. So you've got to study all of this. And if you're going to use this as a reagent inside the cell, you want it to be fast. You don't want to take 30 hours to do the reaction. You want it to be over fast, and you want it to happen at pH 7. So how do you think this reaction works? Where's the most reactive part of this molecule? AUDIENCE: Those two protons? JOANNE STUBBE: So two protons. So this these have low pKas, so you can easily form the enolate. Depends on the details, the experimental details. And now you have this, and what you end up with is this molecule. And so the question is, does this go in 5%? You need something that goes in quantitative yield at pH 7, rapidly. OK, we're going to come back and talk about what the issues are, because the issues are even harder if you want this region to work inside the cell. OK, we're doing this on glutathione peroxidase, which is what he's using as a model to see if all of this stuff works. OK. So what you really want to do if you're thinking about regulation in the end, is you want to know how much is in each form, and you know, if you read hundreds of papers published on methods trying to figure this all out, but what she did in this case, was she developed a second reagent with an iodo group. OK. And as you can see, what is the product of the reaction? The product of the reaction is the same as the product of this reagent. But this reagent does not react with sulfenic acid. OK, so you get no reaction. So how does this reaction work? What do you think? The what? AUDIENCE: SN2. JOANNE STUBBE: So it could work by an SN2, but the way probably works is it attacks the iodine. So you form-- this is probably the mechanism from what's been done in the literature. So you attack this, and you form this, which then gets attacked by the enolate. So it doesn't really matter what the mechanism is, but the key thing is for this to react-- if you're interested in a mechanism, which I am, it does matter what it is. So the key thing is now you have the same reagents. So how could you ever use it attached? How could you ever use it to distinguish sulfenylation from a cystine. So what did they do in this paper? AUDIENCE: [INAUDIBLE] JOANNE STUBBE: Yeah, so they put the deuterated form on this. So what they did then was in this paper, so you got to keep these straight, if they see deuteriums present, so they made this deuterium label, and this protonated so now you have a mass difference of 6. OK. And in the system, they're using glutathione peroxidase, which has three cystines in it. And one of the cystines is more reactive than the other two, but for proof of concept they mutated two of these cystines into serine initially, so you only had a single reactive cystine, but then they went back and studied the whole protein. OK, so let me just introduce you to this, and then we'll come back and talk about this next time. Let me just do one more thing. OK, so here is the difference in mass between these two species. So this is what you're looking at. If they start out with deuterium labeled dimedone, the peak that they observe is going to be associated with sulfenylation, and if they start out with the protonated material, the peak they observe is going to be associated with the [INAUDIBLE] group. OK, so that's the idea. And then what they did was they simply took their protein, and they have, in this case, 50 micromolar of their protein, and then they increase the concentration of hydrogen peroxide. They don't really talk very much about how they design the timing, but they use, you know, two equivalents. So they use variable amounts of hydrogen peroxide. And what you can see is the maximum amount. So now what you're using, we talked about this before, but we're using anti dimedone antibodies for the detection. And here, they're starting with no hydrogen peroxide. So you don't see any dimedone derivative, and then you increase the concentration. But you get to the highest concentration here that they looked at. So it's 100 micromolar versus 50 micromolar in the protein they used. But what did this immediately tell you? Did any of you look at this data very carefully? What is this? This guy here is associated with a [INAUDIBLE] group that is only reacted with iododimedone, so if you got 100% yield, what does that tell you? This tells you the maximum amount of material you're going to observe. So if you look at this peak, and you look at that peak, you can't do this by eyeball. You need to do this quantitatively. The phosphor images or methods that allow us to do this quantitatively. What do you see? AUDIENCE: It's not at the max. JOANNE STUBBE: Yeah, it's not at the max. And so what we'll do next time-- so these are sort of controls, and the question is how effective is this reagent, and if you start hanging stuff off of your dimedone over here, are you going to change the rate of modification? Can it get into the active site where this SOH actually is, these are the kinds of things we're going to talk about next time when we look a little bit more at the details of the reaction with this, and you should look at the reaction with gap dehydrogenase, which is another control enzyme they ended up looking at it, because what they do is address what the issues are that you're going to encounter when you get into something real that you care about. And that's much more complicated. OK so that's it.
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MIT_508J_Biological_Chemistry_II_Spring_2016
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21_Cholesterol_Biosynthesis_3_Cholesterol_Homeostasis_1.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JOANNE STUBBE: So what we were doing last time is we were still focused the first two lectures were trying to understand the biosynthetic pathway for cholesterol bio-- it's good, thanks-- for cholesterol biosynthesis. And we almost got to where we wanted to go, but we didn't quite get there. So what we've been focusing on is a new way of forming carbon-carbon bonds using C5 units, isopentenyl pyrophosphate and dimethylallyl pyrophosphate. And to do that, we had an initiation process where these molecules were generated from acetyl CoA. And then the last lecture we were focused on how we did the elongation process where we took a bunch of these IPP units, strung them together to make farnesyl pyrophosphate, which is C15, and I showed you that C15 could be linear or cyclized. And we went through the general rules of what you're going to see with all turpine chemistry, which is quite diverse, given that there are estimated to be 70,000 natural products in the terpenome. So we had gotten to production of farnesyl pryophosphate and now the next step-- remember, cholesterol, if you look at its structure-- this is a precursor to its structure-- is a C30. And so the next step is quite an interesting enzymatic reaction which we're not going to talk about in any detail, but those of you who are interested can go look it up. But how do you take two C15s and form a C30 so you lose your pyrophosphates? And you can see when you generate this, now you have a linear c30, which, of course, is a complete hydrocarbon and is insoluble. So this now sort of defines that you need to be in the membrane to be able to do any further chemistry. So those of you who are interested in mechanisms of how things work, that's really sort of a fascinating system it's really pretty well worked out at this stage. But today what I mean to do is focus on the next step is, how do we get from C30, which is this linear squalene hydrocarbon, into lanosterol, which is then the precursor to steroids but also the precursor to cholesterol, which is what we're focusing on in this particular module. So what we're going to be looking at is how we went from two FPPs-- we're still doing chain elongation-- to a C30. And then the question is, how do you get from C30, which is linear, to a linear epoxide. And I'm not going to draw the whole structure out, but we're still linear. And then the next step is the step I want to talk about. So this is when lanosterol synthase. So that's where we're going in the next few minutes to get to our final product. So if you look at this reaction, remember, we're going to do a cyclization. And what do you need to do to do cyclizations? What was the general rule that I gave you last time? Does anybody remember? If you want to cyclize something, we talked about it. We looked at a number of examples. What did we do in those examples? Anybody remember? So here's a second example. I gave you two rules. If you go back and you look at your notes, we protonated the olefin and that triggered off the cyclization. And here, perhaps you could have protonated the olefin to trigger off the cyclization, but in the end, cholesterol has a hydroxyl group in the C3 position. So the next step in the pathway, which also will involve, ultimately, protonation and ring cyclization, so those are the two rules I gave you during the last lecture, to get to this epoxide, we have to do some chemistry. Does anybody know what cofactors you would use to do this reaction? Anybody got any ideas from introductory biochemistry? You have a vitamin bottle. What vitamin would be involved in doing this transformation or could be involved with doing this kind of a transformation? It's an oxidation. Requires oxygen gas. So what are the possibilities? AUDIENCE: NAD. JOANNE STUBBE: So NAD. Does NAD-- this is a good teaching point. Does NAD react with oxygen? Who suggested NAD? Why doesn't it make you react with oxygen? That's one of the things you learn in any introductory course. NAD does not react with oxygen. Why? What is the chemistry of NAD/NADH? Whoa. Maybe I should be teaching 5.07. So NAD/NADH, we just went through this with conversion of acetyl CoA moiety of mevalonic acid to the alcohol. It involves hydride transfer. And if you tried to do this chemistry instead of two electrons at a time, one electron at a time, and you looked at the reduction potentials, it would be way uphill, thermodynamically. So NAD/NADH never does one electron in chemistry. So that's not going to be a possibility. Yeah? AUDIENCE: You could use something that's like iron? JOANNE STUBBE: So that would be one thing. And we're going to see iron-- heme irons play a key role in all of this process. This turns out to be a flavoprotein. That's the other redox active cofactor. So this is a flavin monooxygenase. You don't need to remember this. We understand the details. I'm not going to talk about the detailed mechanism, but flavin cofactors are extremely well understood. The chemistry of them is extremely well understood now. So we've gotten to our oxidosqualene and now we've finally gotten to this really cool step. So how do we go from this step-- so this is this molecule here. And what I'm emphasizing again is we're going from a linear step into the cyclic product. So remember, triggering off cyclization, there were two rules-- protonation, protonation of an olefin. In this case, you have some kind of protonation of the epoxide. Epoxides are not very good leaving groups. You need to protonate it. And that is then going to trigger off this cascade of reactions to allow you to generate a molecule with four rings. And this all occurs with a single enzymatic step. And so the way you can visualize this happening-- and again, you don't need to copy this down. It's all-- if you look at your handouts ahead of time, there's some things that are written down that would take you 10 minutes to copy and then you probably get it written down incorrectly because you're looking like this is. The hard things that are hard to write down are all given to you in your handouts. You can write it down if you want, that's fine. So what we want to do is we want a ring open, so we need to protonate the epoxide, and that generates what? A carbocation. And then now what happens? We generate another carbocation. And now what happens? We generate another carbocation. And now what happens? We generate another carbocation and we end up with a carbocation at this position. So I'm going to draw the structure of this. So we have ring opened, and let me also emphasize that the key to this process occurring to give us lanosterol is the conformation of the linear molecule. So what do we see here? What does this look like? In terms of cyclohexanes, what does this look like? If you have cyclohexyl rings, what kinds of conformations do they have? AUDIENCE: Chair? JOANNE STUBBE: Chair and chair and boat. So the key here is that you have a chair conformation here. You have a chair conformation here, but here you have a boat conformation. And one of the general rules I told you last time about terpene chemistry in general was, what do the enzymes do in the active site to transform something that's linear into something that's cyclic? They need to fold the molecule into the right conformation. And that can, in part, be done, the fact is the active site is very hydrophobic. We talked about that. And you can also have aromatics that could potentially-- I'm not drawing out all these intermediates, but could potentially facilitate not only the conformation but stabilization somewhat of the intermediates that you observe along the reaction pathway. So here's another example of the importance of shape to defining the chemistry that's actually going to happen. And in contrast to the enzymes we talked about last time, which were type I. You probably don't remember that. But this is, again, a different super family involved that you observe, and it's observed quite frequently. So these are type II. So if you look up the structures, and in the article you had to read by Christiansen, the second type of structure. There are two general types of structure. This is the second type of structure involved in making interesting terpene molecules. So what I'm doing now is showing you how we've cyclized this to leave us with a carbocation. And remember, if you have just a stick as opposed to a stick with a hydrogen, that's a methyl group. So here at the ring juncture, we have a hydrogen. We have a trans ring juncture. And again, if we have a stick with nothing on it, it's a methyl group. And we're into a chair conformation again, and then we need to attach the last ring so we have three six-membered rings and a five-membered ring. And in the end, what have we generated? We've generated a carbocation. So I've written this as a single step. Nobody has seen the intermediates. You could write it is multiple steps. I mean, the fact is it would be-- it's pretty hard to trap any of these carbocations, and people have spent a lot of time trapping them. So what you see, I think, is quite amazing, but we aren't finished yet because we have a carbocation and we need to get rid of that. And what you need to do and this is-- you will have one of these problems on the problem set that will be due next week. You'll be given something simple, not as complicated as cholesterol. But what you need to think about is where do all these methyl groups end up in. What's the stereochemistry of the reaction? So then this geometry becomes critical if you're thinking-- you need to think about the stereo electronic control of hydride and methyl anion equivalent migrations. So what you have in this particular reaction is you're going to have-- and I like this example because, again, I gave you a set of rules that you can see that are associated, typically, with carbocation reactions in general, and this one does all of them. So one of the rules was that you have hydrogen migrate with a pair of electrons, so that's a hydride. Again, you have a second hydrogen migrate with a pair of electrons. So I'm not drawing out all the intermediates. Now what we have is a methyl group migrate with its electrons. We now have a second methyl group migrate with its electrons. And in the end, we're left with a cation here, and the last step in many of these reactions is loss of a proton. So here we would have loss of a proton. And if you look at the chemistry and you look at the final product, which I'm not going to draw out, you end up with this molecule. So this is a flat rendition of what I've actually drawn on the board. So this is an example of all of the chemistry I talked about as being general in all of these 70,000 terpenes. You'll find most of them don't do all of the chemistry. This one does all the different kind of chemistries associated with carbocation type chemistries that hopefully some of you have learned about in introductory organic chemistry classes. So again, to me, this is the most amazing reaction I think I've ever seen. I told you already that I heard about this in 1969 when they'd just figured out that this could happen enzymatically. And this became the basis, for those of you, if there are any synthetic people here, people doing cascade reactions. Kim Jameson's lab does this, but back in those days, they were using this approach, trying to define the folding to do all these steps, just like nature had figured out how to do this. And if you look at the number of asymmetric centers, you end up with seven asymmetric centers and no other products that people could detect. So this is quite an amazing feat. So this is the model that I just drew on the board. And so we still aren't quite there yet because if you look at this structure and you look at the final structure of cholesterol, you have a methyl group here, here, and you're going to have a methyl group here. So we have 1, 2, 3 methyl groups. And if you look at the final product, cholesterol, they're all gone. So you need 19 more steps to get to cholesterol. This is not a simple biosynthetic pathway. So to get from cholesterol-- so this is lanosterol. So we've gotten to the precursor to steroids and cholesterol. And when we start talking about regulation, you'll see that lanosterol is, again, a central player because it can partition between different kinds of natural products that we aren't going to be talking about, other kinds of natural products we aren't going to be talking about. But to get to cholesterol, which I'm abbreviating from now on as Ch, it's 19 steps. So let's go over here. My goal is not to teach you about the chemistry of all this. I'm not sure how easily you can see it. Hopefully, you have the handouts with you, but we have this methyl group, this methyl group, and this methyl group that need to be removed over here. So that methyl group is gone and these two methyl groups are gone. So how do we do that? And so all of this reaction-- so we have loss of three methyl groups. And all of these reactions are catalyzed by one kind of enzyme, which is a cytochrome P450 monooxygenase. So we're going to see that all the reactions are catalyzed by a cytochrome P450 monooxygenase, not a flavin monooxygenase. And if you look at the chemistry, flavins are not anywhere near as strong are oxidants as heme-dependent oxidation. So if you have something really hard to oxidize, you're never going to use a flavin. You're going to use a heme. And what do we know about all these enzymes? I'm not going to talk about this in detail, but you have an iron 3 heme. And for those of you who don't remember what heme is, we're going to be talking about this in more detail in the section on reactive oxygen species. It's a protoporphyrin IX. That's what you see in hemoglobin. It's the exact same co-factor you see in hemoglobin, but what's distinct about this is that instead of having a histadine ligand, it has a thiolate ligand. And that's key to why P450s can catalyze these inactivated hydroxylations-- can catalyze hydroxylations of unactivated bonds where this hemoglobin reversibly binds oxygen. So these P450s use this heme system in an oxygen system. And what did they do? And so what I do is refer you over here to-- let's look simply at 7 through 10 and we're removing this methyl group. So we're removing this little methyl group in the A ring. The first ring is the A ring. Sorry. So stereo specific, and so I'm not drawing the rest of the structure. And our goal is if we go through 9 through 10 and then 11 through 13, we want to get rid of both of these methyl groups. And it's thought that one enzyme, but they don't know, can catalyze multiple oxidations. And why don't they know? Where do you think all was chemistry happens? You have cholesterol. What do we know about the structure of cholesterol? It's a grease ball. So where do you think the chemistry happens? AUDIENCE: In the membranes. JOANNE STUBBE: In the membranes, yeah. And so that's been-- P450s, you go to meetings, thousands of people still go to P450 meetings on the major targets of all kinds of therapeutics, and they're almost all membrane-associated, which has been problematic in terms of isolation and characterization. And here, despite a lot of effort, people really still don't know the sequence of events or have isolated and purified the enzymes. They're all in the ER, which is what we're going to come back to, and there are a membrane-associated. So what happens in these reactions is you take a methyl group and then you oxidize it with one P450. So we somehow use oxygen-iron chemistry to do a hydroxylation reaction. Have you seen that before, in the first part of the semester? Anybody remember seeing it? Maybe you didn't see it. I missed a couple of lectures. Do you remember seeing hydroxylation reactions anywhere? Liz, do you talk-- was that in any of the natural products? AUDIENCE: Sometimes [INAUDIBLE] P450s [INAUDIBLE].. JOANNE STUBBE: But what you'll see-- I think this would be, like, a decorating module that you saw in the non-ribosomal peptide synthetases. But here these things, as in the non-ribosomal peptide synthetases, are absolutely specific. And so you have one hydroxylation, you have a second hydroxylation, you have a third hydroxylation, which is chemically distinct. And then the question is, how do you get rid of this altogether? Because our goal is to remove the methyl. That's what our goal is. So we've gone hydroxymethyl, the aldehyde, the acid. So now you have an acid next to the alcohol. How do you get rid of that? Has anybody-- what kind of chemistry could you do to allow you to lose the CO2? AUDIENCE: [INAUDIBLE]. JOANNE STUBBE: You need to speak louder. Don't be-- I mean, just tell me what you think. AUDIENCE: Decarboxylation. JOANNE STUBBE: The what? Decarboxylation. But can you decarboxylate-- so you're right. We want to decarboxylate. AUDIENCE: [INAUDIBLE]. JOANNE STUBBE: What do you have to do to decarboxylate? AUDIENCE: You reduce the alcohol [INAUDIBLE].. JOANNE STUBBE: Reduce the alcohol? AUDIENCE: [INAUDIBLE]. JOANNE STUBBE: What? What are you going to do? These are the kinds of-- you'll see these reactions happen over and over again in biochemical pathways. AUDIENCE: Oxidize [INAUDIBLE]. JOANNE STUBBE: Right. You want to oxidize it. So what happens, if you look at this pathway over here, in this step, you use-- it should be NADP, so you use NADP. And what does that do? I'm not going to write this out, but it oxidizes this to a ketone. And now what do you have? You have a beta-ketoacid. And beta-ketoacids rapidly undergo decarboxylation reactions. So this is a strategy that nature uses over and over again in many biosynthetic pathways. And the thing that's interesting, if you look at that pathway in detail-- and again, you're not responsible for that-- but then it does the same thing on the next methyl. So in the end, you end up with a carbon with two hydrogens here. But it's not straightforward, but this kind sequence of events you actually see a lot in metabolic pathways. So I don't want to really say much more about this. In 19 steps, you need to remove three methyl groups. All the enzymes are ER bound, making it difficult to study the individual enzymatic reactions. And we would like to know the order, but we don't know it at this stage. What we know is what we see at the end. So finally, I wanted to get here at the end of lecture 2. We've gotten here a little later. We've started with acetyl CoA. We've made the major building blocks for all terpenes, IPP and dimethyl APP. And we've gotten to form this very complicated molecule. Everything starts with acetyl CoA and you can-- this was classic work by Konrad Bloch, who won the Nobel Prize for this work, who then by doing label chasing, which you learned about, hopefully, in introductory chemistry, helped them to figure out this complex biosynthetic pathway, which isn't so easy because things are membrane bound and very lipophilic. So we've gotten to cholesterol. So this module is on cholesterol and we've been able to biosynthesize it through an amazing sequence of reactions that have been studied over the decades. But we can also get cholesterol-- we want to ask the question, first of all, why are we interested in cholesterol? I think you've already seen hints of that with the statins inhibiting HMG-CoA reductase. We have issues when cholesterol levels are too high or too low. We need to control the levels of cholesterol. And the second way we can get cholesterol besides making it is we get it from our diet. So if we get it from the diet, the molecule we'll see is not very soluble. How is it distributed into the tissues? And then if you've distributed a lot of cholesterol from your diet, you certainly don't want to keep making cholesterol. So the question is, how do you control those two events? What are the general mechanisms of regulation of the levels of cholesterol? And we're going to at the end look at some of the classic experiments that Brown and Goldstein did to understand how cholesterol, which from the diet can get into the bloodstream, can get transferred into cells. And so that's where we're going. And again, the reading is a reading I've already given you before. So why do we care about this? We have a 30-step synthesis. We're getting it from the diet. We have key issues in homeostasis, which is what our focus is going to be. So why do we care about cholesterol? We care about cholesterol because it's associated with human health and coronary artery disease. Probably many of people who have had heart attacks. And so elevated cholesterol levels have been known for some time to be associated with plaques, artherosclerotic plaques, which can lead to heart attacks and strokes. So what happens is the cholesterol deposits, you try to remove the cholesterol, you generate a lot of scar tissue, which then inhibits blood flow. And then you're in trouble if you can't figure out how to unblock the blood flow. So that's the main motivator and we'll see another main motivator is related to young children dying of heart attacks, which is what got Brown and Goldstein into the area of cholesterol homeostasis. So there have been three Nobel Prizes given for work on cholesterol over the years. This is a classic paper. Some of the classic papers are actually quite interesting to read, and often the original papers get things wrong. So it was mostly right, but not completely right. But anyhow, I think if you put it into the context, 1928, how would you do experiments like that? We had no IR. We had no MR. We had no mass spec. What did we have? We had ways of degrading things. People don't do that anymore. If you go back and you look at the discoveries before 1970 or something, these feats of pulling out the structures with the right stereochemistry is really, I think, quite astonishing. And I think what's most amazing to me is this old literature is actually much more reproducible than anything in the current literature. The current literature, we're spewing out papers, a lot of which will never get reproduced so we won't know if it's reproducible. But if you go back and you do anything, in the old days, you had to learn German because a lot of the original papers, all of the chemical papers, were in German. They did seminal experiments back in those days. And most of the time it was correct. So anyhow, these guys figured out the structure almost. And then Konrad Bloch figured out, along with Fritz Lynen, figured out how you make cholesterol by labeling experiments. Now, many of you-- how many did label chasing in an introductory biochemistry course? Any of you have problem sets with label chasing? So it's quite distinct. I taught with John Essigmann. All those problems were label chasing and I used to say, oh, no. Who wants to do label chasing? But the fact is now if you read any of the current papers in the literature, everybody is label chasing. And now we have much better ways of actually chasing labels using mass spec methods. So you can hardly pick up a journal nowadays without thinking about label chasing. So these guys who were way ahead of their time, but it was much harder in those days. And then here are Brown and Goldstein. They won the Nobel Prize for the discovery of low-density lipoprotein and may still win another Nobel Prize for the regulatory mechanisms that we'll talk about at the end of lecture 5. So the first thing I want to talk about in lecture 3 is focused on-- let's see. What do I want to do? Is focused on the properties of cholesterol. So we want to look at the properties of cholesterol. Then we're going to ask the question, how does cholesterol get from the diet to the bloodstream? And then we're going to ask the question, how does cholesterol get from the bloodstream into the tissues where it's essential for membrane controlling membrane fluidity? So what do we know about cholesterol itself? If you look at the structure, what do we have? We have a grease ball and a little hydrophilic head. And so this cholesterol moiety, if you look at the structure up there, is really pretty rigid. And it, in fact, rigidifies. So this is rigid-- and in fact rigidifies membranes. And so you can see this if you go back and look at this. Hard to see these little things, but those are cholesterols stuck within the phospholipid bilayers. And this is key since this is something that I think a lot of people are spending a lot more time on and we're getting much better at this now. People have stayed away from membranes because it's so-- and membrane proteins because it's so hard to work with and they stick to everything. How do you control all of this? And Brown and Goldstein really did some of the classic experiments that taught us how to deal with these type of really hydrophobic molecules. And so cholesterol is pretty important. 10% of the membranes actually have-- of the lipids in the membranes are from cholesterol. So if you look at this, you would think it wouldn't be very soluble. And in fact, the solubility of this-- solubility is about five micromolar. So it isn't very soluble, but in fact, as an adult, we have 35 to 50 grams-- we each have 35 to 50 grams of cholesterol. And we know that per day 1 gram is derived from synthesis in the liver, so the predominant organ where cholesterol is made, like we just were describing, is the liver. But we also have-- and I don't know how good these numbers are. I got them out of some book. So I'm not an expert in this, but anyhow, these are all rough numbers and you'll see these in other nutrient uptake systems. You want to have some vague idea of the contributions to the two distinct processes. We get from the diet, say, 200 to 300 milligrams from the diet. So then if you think about this, cholesterol we're going to see is transported in the blood, and we'll see how that happens. Whoops. Transported in blood. And we know something about the amounts. And if you do a calculation, that says that you would have five millimolar cholesterol. So that's impossible. The number is squishy, but it's impossible. So you'd have this insoluble mess. So the question is, how do you deal with it? And so that's what we need to think about. So how does cholesterol move-- I think [INAUDIBLE]. So how does cholesterol go from the blood to tissues, given the solubility problems? So here is again the structure of cholesterol. Again, it's pretty rigid and it inserts itself into membranes. Where do you get cholesterol from? You all know you get cholesterol from beef and chicken and eggs. I guess there aren't very many-- do any of you eat at McDonald's? Or is that a passe thing? I love McDonald's anyhow. That was my favorite when I was in Wisconsin. There was only one restaurant near where the biochemistry department was and I went there every day, and my favorite thing was like two of those things slathered in cheese with French fries. Anyhow, fortunately, I have very low cholesterol. But anyhow, you get that our diet is a major source of cholesterol and what you eat can, in fact, be problematic and part of it really sort of depends on how lucky you are genetically, right? That's sort of the key thing. So what do we do? We have this insoluble molecule and the question is, how do how are we going to get this insoluble molecule into the tissues where it's needed to control the fluidity of the membranes? That's the issue. So the second thing I'm just going to introduce you to, and this is taken from Voet and Voet. So many of you may have read that if you had 705 or something, but in 507 we don't cover this reaction. So I'm going to spend a few minutes going over it. So what has to happen is cholesterol is found in lipoprotein particles. And we know a lot about the composition of these lipoprotein particles, which-- this is taken from Voet Voet. And what you can see is-- and I think, again, the relative amounts isn't all that important. But you can see you have-- and we're going to be focused on low-density lipoprotein, which is the major deliverer of cholesterol to the tissues. And why is that true? So if we look at free cholesterol, we see we see 7% to 10%. I'm going to tell you about the structure of the lipoproteins in a minute. But most of the cholesterol is actually esterified with fatty acids, and you can see that the cholesterol esters are 35% to 40%. So if you look at the total amount of cholesterol in the LDL particles compared to all the other particles, it's much higher. So what do you see and what do you have to worry about if you're getting this from the diet? So what would you expect to see? You would expect to see proteins. You would see phospholipids. So this is a phospholipid. You would expect to see triacylglycerol. Everybody know what triacylglycerol is? We expect to see fatty acids. We expect to see cholesterol. And if you look over here, what people have done, have isolated different particles. How do they isolate the particles? The particles are isolated based on density differences. And if you look at all of these different compositions, they vary between very low-density lipoprotein, intermediate-density lipoprotein, high-density lipoprotein. They have different amounts of these different species. And in fact, most of them have very hydrophobic stuff in the outside and more hydrophilic stuff-- on the inside and more hydrophilic filled stuff on the exterior, which is more dense. And then that tells you something about how these things sediment by a subterfugation method and a density gradient. So these lipoprotein particles-- and we're going to see this kind of method in next week's recitations-- are separated by the centrifugation due to density differences. So let's just briefly look at LDL. That's what we're going to be dealing with today. And it's important because LDL is what we're going to try to take into the cell, and the composition of the LDL is key to thinking about how to studying that process in the classic Brown and Goldstein experiments. So if we look at the cartoon of LDL that you see up there, what you see is a particle. They're sort of circular. The LDL particles, which is what we're going to be focusing on, low-density lipoprotein particles, have only a single protein and this protein is called the ApoB. It's a huge protein and it covers about 50% of the surface of the particle itself. Again, these things change in size. We'll see when we actually look at the transport process. What do we know? It's on the exterior. On the exterior what you see all over the exterior, these things that are phospholipids. So the phosphate is on the outside, the fatty acids are on the inside. What else do you see on the exterior? You see a lot of cholesterol molecules, which I'm indicating like this. And then it turns out that the predominant-- and that maybe covers, I don't know, 20 Angstroms, but the particles are 200 Angstroms, 220 Angstroms. So what's in the center? And what's in the center, so this is the interior. You basically have triacylglycerols and then you have cholesterol. And remember, cholesterol has one lone 3 prime hydroxyl group, and this is a esterified with a fatty acid. And so this is also in the interior. So that's the composition of the LDL particles. And the question is, again, where did they come from starting with stuff we get from the diet? So that's what we're going to be focused on and that's what we're going to try-- the cholesterol is stuck on the surface and in the interior. Yeah? AUDIENCE: I don't understand. Aren't there different splice variants ApoB and which one is the one that's involved now? JOANNE STUBBE: There could be. There could be We're not talking about this in detail at all. I don't know how many splice variances there are. And I don't really know that much about all of these different proteins. You'd have to go read about them in detail. So I'm giving you sort of a cartoon general overview of what you need to think about. There are splice variance of almost any protein. And in humans, you have in the PCSK that we talked about, there were nine isozymes. So isozymes and eukaryotic systems are something you don't have to worry about a lot for this lecture and for what I want to say. You don't have to worry about that. And if you want to read about it, go for it. So we have LDL particles and they are distinct from all these other particles which have different densities, different proteins. And there are two cartoons I want to use. This cartoon was taken from Voet and Voet. I think it was the third issue. The one from the fourth issue, which I'll show you in a minute, I think, is much better. So I'm going to change the handout. But I just want to very briefly walk you through this. This is a really complicated process and, from my reading, is really not completely understood. But you have diet. And what do you have in your diet? Triacylglycerols, phospholipids, proteins. They get taken-- in the diet, they get into the intestine and somehow in that process they need to get packaged into one of these lipoprotein particles. And the lipoprotein particle that really is the predominant one that comes out of the intestine are these things called chylomicrons. And you can see they have a lot of proteins. They have a lot of triacylglycerol. They have a lot of phospholipids. Anyhow, the composition varies and the sizes also vary. So these chylomicrons come from the diet. So how do these lipoproteins deliver LDL to the extrahepatic tissues? That's what we're really after. And so these chylomicrons-- let me just show you the next slide for a second. I think this is probably a better one, anyhow. So these chylomicrons, somehow they have to package all this stuff into these lipoproteins. You know how that happens? I don't really know very much about it. Maybe somebody does. I don't know that much about it. So anyhow, it gets packaged into these little particles and then it goes into the intestinal lymph, which then goes into the bloodstream and then it needs to start circulating. So everything comes from the diet comes from these chylomicron particles. So what happens when you go adjacent to adipose tissue or muscles? So what you need, if you're going to be involved with fat metabolism or you need energy to run down the street, you need fatty acids. So where do the fatty acids come from? They come from the triacylglycerol. So what you have are lipases and all of these chylomicrons in the lipases. Then when you get near the tissue-- let's see. Here we get near the tissue, the muscle or the adipose tissue, a li-- does everybody know what a lipase is or do you want me to write that reaction on the board? Does everybody know what a lipase is? No? OK. So a lipase-- so here is your triacylglycerol with different fatty acids. So this is a TAG. This is glycerol. It's stereo-- this is a chiral center. And so what happens then is lipase is simply an esterase that hydrolyzes the bond. So I'm not going to draw the reaction out, but a lipase catalyzes, release-- this is a fatty acid. And it actually cuts off two of them, and so most of the time you have monoacylated fatty acids. But again, from what we're talking about, this is not really important because really what we want to do is get to the low-density lipoproteins. So what you see when you start doing this is that if you drop off triacylglycerols or monoacylglycerols here, and you drop off something else to the other tissues along the way, the sizes of your particles change size. So they call these things, then, the remnants from your starting material. And it turns out there is a receptor that takes up chylomicron remnants into the liver. So the central player in all of this is the liver. So you got to take stuff-- we get it from the diet, but we got to get it into the liver and that's where everything is controlled. And that's predominantly where cholesterol is actually biosynthesized. So if you start-- you bring in-- what are you bringing in? You're bringing in cholesterol because you've dropped off triacylglycerols and lipids and phospholipids to the tissues. So what's predominantly left? What's predominantly left is cholesterol. So from these chylomicrons you drop off fatty acids and monoacylglycerols to adipocytes or muscle, and then what you have left is cholesterol. You have a lot of stuff left, but cholesterol is a major thing, which is then going to go to the liver. And so then once this gets into the liver, the liver has all this machinery to repackage things and they can make very low-density lipoproteins again. So you can go back and look at this. It's very complicated. They then in the bloodstream can drop off stuff along the way to tissues as well. And then they change into intermediate-density lipoproteins, which then can change into low-density lipoproteins. And it's these low-density lipoproteins that are going to then deliver cholesterol, that has more cholesterol than any of these other particles, either two extrahepatic tissues or back into the liver. So you have a complicated set of transport systems that we're not going to spend any time on, but it's all related to the fact that cholesterol is basically not a happy camper in water. And so we've got to figure out how to move cholesterol around. So that just summarizes-- what I didn't show you over here, you all have heard about high-density cholesterol, low-density cholesterol. And high-density cholesterol is distinct from all these other lipoprotein particles. And it sort of scavenges excess cholesterol from these extrahepatic cells and returns it to the liver. But unlike the look the receptor-mediated endocytosis we're going to talk about with the LDL receptor, this receptor is completely distinct. I'm not going to talk about it, but the mechanism is distinct from these other receptors that people have also studied in some detail. So the other thing that I wanted to briefly say is that in addition to cholesterol what you see-- and I'm not going to spend much time on this either, but I think it's worth mentioning-- is when you get cholesterol back into the liver, what can you do with that excess cholesterol? If you have too much of it, how do you control the levels? That's the key thing we're going to be trying to focus on. What have we learned, at least to some level, in control of cholesterol levels? But it turns out in the liver-- so the key organ in all of this is the liver-- cholesterol can be metabolized to form molecules that have four rings just like cholesterol that are called bile acids. And these are multiple steps. I'm not going to draw out the steps, but if you look at a bile acid-- and I have cartoons of bile acids over here. So here's cholesterol and if you look at this-- it's hard to see it, but if you look at it, it really sort of it looks a lot like cholesterol. The only differences are you add additional hydroxyl groups. So in cholesterol we have a 3-alpha hydroxyl group. In the bile acid you have two additional hydroxyl groups put on again by cythochrome P450s. So you have a hydroxyl group at C7. You have a hydroxyl group at 12 alpha, simply means the stereochemistry. So the stereo chemistry of the hydroxyl group is on the same face. So that's what I mean here. So what you have then is hydrophobic and hydrophilic. And in addition, if you look at the very end, it turns out you have molecules glycine or taurine, which are on the handout, which has a negative charge. And again, it's on the same face. So we have a bunch of hydroxyls, something charge, and they act as emulsifying agents, and that's all you really need to know. So these become emulsifying and they really play sort of key role in also helping to take things back into the cell. And this is a really complicated process. And in fact, I think it was 15 years ago, something, people used to try to remove bile acids as a mechanism of controlling cholesterol levels. And what you did was actually-- boy, I'm way over again. What you actually did was eat-- have any of you ever worked with Dowex in an exchange? You used to eat the resin you have in the lab called Dowex because it would bind the negatively charged materials. And so, really, it was very hard for people to stomach this, but that was before we had really sort of wonder drugs-- Dowex, eating Dowex in these little grainy resins. You should go look in the lab if you're doing a UROP. That's how we used to treat high levels of cholesterol. So anyhow, bile acids also play a key role. We're not talking about this in detail. So the next time we're going to come back and we now sort of see what the properties of cholesterol are, that they're in lipoproteins. And we want to focus on the key experiments that showed how LDL is taken into cells.
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MIT_508J_Biological_Chemistry_II_Spring_2016
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R3_PreSteady_State_and_SteadyState_Kinetic_Methods_Applied_to_Translation.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT Open CourseWare at ocw.mit.edu. JOANNE STUBBE: We talked last time about kinetics, steady-state kinetics, pre-steady-state kinetics, how you design the experiments, what kinds of information you can get out of each experimental design. And we introduced all of that material. And today, what I want to do is come back to the model. You saw it at the very beginning, and you've seen it in a lecture. And specifically, where did this model come from? That's what we're going to focus on. OK, and so in order to be able to understand this model, you have to design assays. And you're going to see over and over again over the course of the semester figuring out how to design an assay, in this case, isn't so hard, but in many cases is really tough. And that's the key to being able to get kinetic information is designing the assay. So if you look here, today, we're going to be looking at GTP is hydrolyzed. So you need to think about, as a chemist, how you could study that reaction. How would you look at starting material? How would you look at product as a function of time, which is what we were talking about last time? And we're going to talk about that first. We're going to talk about use of radioisotopes first. And we've already been talking about radioisotopes in class the last couple of lectures. So we decided to focus most of our energy now on radioisotopes. And then the second kind of probe you're going to see is a fluorescent probe. We're going to use fluorescent probes over and over again. And the details of the fluorescent probes and how they work isn't going to come in until the last recitation, which is recitation 13. So from the point of view of thinking about Rodnina's paper, what you need to think about is, if you have a probe, and you stick it in a different environment, it changes. And you can watch it change, OK, without looking at the details. But that's something you do need to think about, but we're not going to talk about that. OK, so we have a way of monitoring potentially GTPase. And we'll talk about that today. What other reaction can we monitor in here? We can monitor formation of the polypeptide chain. And so that's the other thing. And both of these chemical transformations use radioactivity. OK, so that's where we're going to focus on it initially. And then hopefully-- how many of you went back and reread the paper for this week from last week? Did any of you go back and reread it? OK, so I think it's good. I just think, you know, every time I read a paper-- I read a paper. Sometimes, I've read it 10, 15 times over the course of my career. And as I learn more and think about things differently, I keep seeing new things. And this paper is just packed full of information. So I could say you could read it another 10 times, and you'd still keep learning stuff out of reading it. And in the very beginning, that's what we're trying to teach you to do. What do you look at in the paper to learn how to critically evaluate what's being presented in the model, which is maybe what you're going to build your research program on? Somebody else's data, is it correct? Is it not correct? OK, so we're going to use radioactivity. I'm going to start there. And then to look at these first few steps, which are binding steps, that's where we're going to look at the fluorescent probe. And there were three different kinds of experiments that were described in this paper-- looking at the rates of the reactions as a function of the concentration of the ribosome-- you need to think about why they looked at a concentration dependence-- measuring fluorescence changes, and then they used non-hydrolyzable GTP analog. Why did they use that? Do you remember what the non-hydrolyzable GDP analog was? So where's the n? AUDIENCE: It's between beta and gamma. JOANNE STUBBE: So it's non-hydrolyzable. It is hydrolyzable, but not under the experimental conditions. So what does it do? Why would you want to use something like that to get information about the first few steps? AUDIENCE: It's along the reaction continuum. JOANNE STUBBE: Yeah, so you don't let the reaction continue. So what that does, if it's working correctly, is it puts a block here. And then you can potentially monitor what's going on here. And from the data that you looked at, it's not really so clear what was going on there unless you went back and read the preceding paper. So there had been a decade worth of experiments on this system before this paper came out summarizing the conclusions about what they are thinking about fidelity. OK, so what we're going to do is talk about radioactivity. And our objective is simply-- and we'll come back to this at the very end-- is to use all this experimental data, the concentration dependence, the radioactive isotope experiments, the stop flow fluorescence experiments, and try to come up with a model that can explain all of the data. OK, so you make some measurement. What you're measuring is some k apparent. And that's usually a first-order rate constant because it's happening on the enzyme. OK, so you measure these numbers. Well, what do they mean? You don't know what they mean. And why don't you know what they mean? Because the kinetic mechanism is so complicated. You saw that with the steady-state analysis of km and kcat last time. So in the end, though, if you come up with a model, and it can explain all the data because you've done many, many experiments, it can be quite informative about the question we're focused on is specificity. How do you distinguish between phenylalanine and leucine and proofreading? How do you decide whether you're going to form the right peptide bond or the incorrectly charged tRNA is going to dissociate? OK, so that's what you want to come out with. You want to look at the ratio of these rate constants and the ratio k3 to k minus 2. And when you look at the experimental data, which we'll look at the end today, it should make sense to you in terms of this model. OK, but let's put it this way. In most cases, you don't come out with a unique model. It's a working hypothesis that people for the next 15 years, if it's an interesting problem, will take pot shots at to try to understand in more detail what's really going on. OK, so what I want to do is talk about two methods, but the focus probably won't get very far in terms of the second one. But today, we're going to look at radioisotopes and how you use that to do the assay for GDP hydrolysis and peptide bond formation. OK, so what is an isotope? OK, so how many of you guys have actually worked with radioisotopes? Any of you? No, OK, so you know, maybe they don't use this anymore. Biochemists for the decades have used isotopes. Every paper I read has isotopes in it. But you know, I'm old school. So maybe people don't use it. But I think the power of it is its sensitivity. I'm going to show you that today. And the other power of it is that you have no perturbation of your system and there are almost no probes like that. You're sticking on green fluorescent protein. Well, what does it do to the whole rest of the protein? You have to perturb to see, but radioisotopes have minimal perturbation. So it's still a very important probe, but it probably depends on what kinds of questions you're focused on. So what is an isotope? So an isotope is atoms with the same number of protons and a different number of neutrons. That's called the mass number. So what you have here for carbon, which is one of the common isotopes you guys will be using if you do any kind of biochemistry, we have C-12, C-13, and C-14. OK, and so this is the atomic number, which is the number of protons. OK, so the only difference between these guys is a neutron or two neutrons. OK, so there's minimal difference. And so what are the isotopes that you see used in biology? So we've already seen many of these in this paper, but we've also talked about some of them in class today and in the preceding class. So we're going to be using over the course of the semester isotopes of hydrogen. Why? Because if you look at your metabolic pathways, you're always cleaving carbon-hydrogen bonds. OK, so this isotope becomes incredibly important. C-12, C-13, anybody know where you use C-13? AUDIENCE: In NMR. JOANNE STUBBE: NMR, so if you're working for Mei Hong, you might be doing isotopic labeling using C-13. If you're doing any kind of metabolic label chasing, you're going to see the radioisotope is, which is what we're talking about, is C-14. Working So you see often, all the time, you see nitrogen and oxygen. And oxygen has three isotopes. Nitrogen has two. None of them are radioactive. OK, so you're never going to be using the methods we're describing today. But frequently, in NMR again, you might replace N-14 with an N-15. And today, we will see that we're using isotopes of phosphorus. What about phosphorus-31? Where do you see that? Have you thought about this? Maybe you have, and maybe you haven't. Phosphorus-31 versus phosphorus-32, what's the normal abundance isotope of phosphorus? 31, so phosphorus-31 has a nuclear spin of a 1/2. So you frequently use that as well in NMR. And P-32 is used-- it's radioactive and is used in today's experiments. OK, so this is something that in the back of your mind you should think about. What are stable versus unstable isotopes? And what we're talking about today is unstable isotopes. So what I want to do is we're not going to go into this in a lot of detail, but I want to describe the things I think you need to think about if you're ever going to use radioactivity and how you make measurements, quantitative measurements. And so we're going to be looking at a radioisotope. And what do we know about radioisotopes? They're unstable. OK, and depending on which atoms they are, they have different stabilities. And they decay spontaneously into some new configuration. They have a nuclear decay spontaneously into a new state. And during this process, during this decay, they emit ionizing radiation. They emit energy. So during this process-- so this is the whole thing you need to remember. They emit energy. And the energy is in the form of ionizing radiation. It could be alpha, beta, or gamma radiation. And so what we're going to be looking for is trying to detect that energy that's actually released. OK, so before I go on, we've already seen all of these isotopes used already in class, even though we've only gone through seven lectures. And when we look at the LDL receptor and cholesterol homeostasis, there aren't very many LDL receptors. You something highly sensitive, which is something that you need to think about. And I-125 which is a gamma emitter, is what you end up using. We'll come back to that in recitation, I don't know, 8 I think it is. OK, so they're unstable. And they spontaneously decay into a new configuration. And they release energy. And what we want to do is detect the energy. The ones that most of you will be focused on, if you use radioactivity in your experiments, will probably be all beta emitters. And all of these guys over here are beta emitters. And you've already seen all of these radioisotopes. OK, so what do we know about these isotopes? There's two things you need to think about. So this is the properties. And one of them is the energy of the beta particle or gamma particle released. And if you look over here, what do you see? Tritium has 18.6 with this kind of unit. The unit might not mean very much to you. All I want you to do is look at the relative energies. Versus phosphorus, 1,710, so it has much more energy released. And what does that mean? If you've never worked with radioactivity, you might not-- a lot of chemists are petrified of radioactivity. I mean you could eat most tritium and C-14. Don't tell anybody I said that. But you could eat almost all tritium or C-14-labeled molecules you end up buying. They don't really do anything to you because the energy is low. And if you wear plastic gloves or something, that protects you from any kind of energy released. P-32, on the other hand, which isn't used as frequently, does anybody know where that used to be use all the time? I've used tons of P-32 in my lifetime. Where do you think that was used initially? AUDIENCE: With DNA. JOANNE STUBBE: In what? AUDIENCE: DNA. JOANNE STUBBE: DNA sequencing, yeah. So DNA sequencing, which you guys don't do, you send it out to have somebody do it for you. We used to run these huge gels. And we used to have to run many, many sequencing gels to sequence something that was 500 base pairs long. And P-32 was the method of detection. So what you do there is it's still not that bad, but you have to have a safety shield. So if any of you use radioactivity at MIT, they have radiation safety, and you go. Even if there's somebody else in the lab using it, and you're not, you should go just read the handouts that they give you to be aware of what's going on with radioactivity. I would say the biggest issue is that, if somebody spills it and doesn't clean it up, then it can contaminate everybody's experiments in the lab. That I would say is one of the biggest issues with radioactivity. OK, so iodine, again, is a gamma emitter. So that's in a category by itself. So the other thing that I think people don't really have very much feeling for is the half-life of decay. And if you look at that, look at how many years for your C-14 to decay by 50% from whatever the number is, forever. You don't have to worry about it. You can sit it in your-- you can leave it in your refrigerator, and it's good for your lifetime anyhow. On the other hand, P-32, for example, has a half-life of 14 days. So what does that mean? It's spontaneously decaying continuously. And if you have it for 14 days, you start out with some number. We'll define what that number is. And then 14 days later, you only have half as much. OK, so you need to know something about the half-life. And the only one you need to ever think about is P-32, which they needed to think about in the experiments that are described in the Rodnina paper. So you have the energy released, and the energies are distinct. And so the question then is what we really want to do is think about quantitation. So that's going to be the key thing is we want to be able to quantitate radioactivity. And to do that, we need a method of detection. And there are a number of methods of detection. The one that-- I guess, again, I'm not sure. I think I'm the only one in the chemistry department that has a way of detecting radioactivity using an instrument called a scintillation counter. So this is sort of a very oversimplified view of what's going on in your scintillation counter. And so people come from all-- actually, they come from lots of places on campus to use it. So again, I think that's common. I don't know how many people are using radioactivity. But you have radiolabeled molecule here. It would be leucine. And you have tritiated leucine. OK, or it would be P-32-labelled GTP. Those are the two molecules that are used in these experiments. You put it into a little vessel with some kind of fluid. And the fluid that you use, whether it's organic, water, aqueous, or mixtures, depends on the molecules you're dealing with. OK, and the energy gets transferred in some way to the solvent in your solution. And then you put in a small molecule called the scintillant, which can remove the energy from the solvent and absorb that energy in some way. And again, it depends on-- the standard one we use in my lab is POPOP. You can look up scintillants in Google, and you can find out what the structures of these things are. And then these things decay. And when they decay, the energy is related to the detection method using a photomultiplier tube. So it gives you a quantitative measure of how much radioactivity you have over here and how much you get out on this side. Now, if you look at this process, it's complicated because you have energy transfer. So what can happen during this energy transfer depending on the energy of your radioisotope? Anybody got any ideas? What might you have to worry about? AUDIENCE: You're looking at efficient transfer. JOANNE STUBBE: Yeah, so the efficiency of the transfer. And if you have something in a solution, often, you're doing crude cell extracts. OK, so you have a lot of things in there that can also absorb the energy. So at any stage along the way, you can get quenching. OK, and if you get quenching, that reduces the amount you detect over here. OK, so that's going to throw your numbers off. So where is quenching a problem? Quenching is a problem-- we just looked at all these energies. OK, tritium has the lowest energy. OK, P-32 has a much higher energy. So if you look at it, and you have to figure this out for every system you work on. I've worked on tritium-labeled molecules where you couldn't be quenched by 90%. So if you have some measure-- we'll call it decompositions per minute, 10,000-- if it's quenched by 90%, you've lost a lot of your sensitivity. So you have to figure out a way to determine whether you'll get quenching or not. Otherwise, your numbers are completely off. So why do you want to quantitate your radioactivity? Where would you be using radioactivity? And why would this quenching make a difference? Yeah? AUDIENCE: This is just a question. JOANNE STUBBE: Yeah, sure. AUDIENCE: What's kind of like the nature of the solvent to the fluorescence involved. Is that like a [INAUDIBLE] kind of idea? JOANNE STUBBE: So it's just some kind of energy transfer. Yeah, so I mean it depends on what the molecules are. An it depends on what the solvent is. AUDIENCE: OK. JOANNE STUBBE: OK, so every single one of these systems, you need to go in and look at the details of what's going on. And so when you do this, people have worked out these conditions so that when you're measuring-- and so this is an important question you're asking. How do you know that what you're measuring really is related to what's way over here? So that's the absolutely the right question to ask. And so when you start, for example, the first thing you do is every scintillation counter comes with a standard. OK, and so the instrument is calibrated. And if you care about radioactivity, you have somebody come in, and they calibrate the instrument twice a year. OK, so all of this stuff is really important. And the question of sensitivity is important. We're going to see what you're measuring is something called decompositions per minute. OK, that's the readout you get from the instrument. And so you might be getting 100,000 of these things. But in fact, you might be getting five. OK, is five real? Five can be real if you count it so you get a statistical distribution to make sure the five is real, that it's not five plus or minus five. OK, so radioactivity is incredibly sensitive. And you can extend the sensitivity by just counting your material in this instrument for a very long period of time. OK, so where would you want-- where have you already seen that you would use radioisotopes? What did you see today in class, for example, in ribo-x? You looked at an experiment with ribo-x today in class. AUDIENCE: Oh, the cysteine incorporation. JOANNE STUBBE: Yeah, with cysteine incorporation. What were you looking at? AUDIENCE: It limited the radioactive system. JOANNE STUBBE: Right, but what we are using to look at this? So we're talking about the detection method. So I'm going to describe another detection method. Would you be looking at this by scintillation counting? No, so you need another method. AUDIENCE: So like one of those phosphorimagers. JOANNE STUBBE: Right, so I'm going to show you. That's the next thing. So what you could have, for example, is a TLC plate. Or you could have-- they were using a gel, an agarose gel probably. So you need a way of detect the radioactivity that's going to be distinct from scintillation counters where you use little vials and scintillation fluid. And you have a completely different method of detection. And these methods keep changing. And so I don't update them anymore. I'm not sure what the current technology is. It's all secret anyhow. So they tell you sort of something about what it is, but they don't tell you any of the details because it's all proprietary. OK, so in that case, what were we looking at? We were just looking for incorporation. We were doing some labeling experiment in the cell. OK, so we were chasing a label. So that you're going to see a lot. That's how all the metabolic pathways were figured out. The advent of C-14 as an isotopic label revolutionized our understanding of glycolysis, fatty acid biosynthesis, et cetera. And today, what are we using radioisotopes today to do? We're using it to do what? We're looking at GTP. We want to look at GTP going to GDP plus Pi. OK, so what are we using this for? To get information for our model. What do we do as a function of time? Why do we want to use gamma P-32-labeled ATP? And how do we use this in analysis? Did you even know that we're using gamma P-32-labeled GDP? How many knew that? Anybody read that in the methods section? You, over there, did you read that in the methods section? What's your name? AUDIENCE: Mathis. JOANNE STUBBE: Matt? AUDIENCE: Mathis. JOANNE STUBBE: Matt, did you read that in the methods section or not? AUDIENCE: No. JOANNE STUBBE: No, did anybody read it in the methods section? So I mean that's what-- again, this is what this recitation is all about is looking at the details of what's going on. And I think when you first start doing something like you don't know what details to look for. Some of you might have read it, but it didn't mean anything. So it went in one ear and out the other. Yeah? AUDIENCE: Wouldn't you have to use gamma labeled GTP though? I mean the hydrolysis gives you GDP and phosphate. So that's the only-- I mean, if you labeled another one, it doesn't give you as much information. JOANNE STUBBE: OK, so if you labeled-- say you labeled the base. So let's just call this base. Say you put a tritium in the base, OK, versus-- hopefully, you all know this, but this is the gamma position versus a label here. Why are you putting the label here? What's going on in this reaction? Actually, this was interesting because this is the second recitation where I don't think anybody understood what was going on in this reaction, which is rather disturbing. What's going on in this reaction? So we're going GTP. So this is G. So you have a nucleoside and three phosphates, TP. And what are you producing out the other side? GDP. So what's happening during this reaction? Yeah? AUDIENCE: [INAUDIBLE] JOANNE STUBBE: Yeah, so you're hydrolyzing it. So in some way, that's what all of these GTPases are about. You're going to see these GTPases not only in translation. You're going to see it in three of the sections that I talk about. GTPases are everywhere. OK, so what you're looking at is then some way you have hydrolysis of the gamma phosphate. OK, so why are you labeling the gamma phosphate? You could have labeled actually the alpha or the beta. AUDIENCE: You wouldn't be watching the reacting. JOANNE STUBBE: Yeah, you want to watch your reaction. So if you have an isotope here, which we're going to watch it using some method, scintillation counting or phosphorimaging, and where does the label end up? The label ends up here. OK, well, if you put the label in alpha or beta, could you follow the reaction? OK, well, you're shaking your head no. Why couldn't you follow the reaction? AUDIENCE: Because it would stick in the GDP [INAUDIBLE]---- JOANNE STUBBE: So it would be in GDP. AUDIENCE: --there's no GTP in GDP. JOANNE STUBBE: Yeah, is there a difference chemically between GDP and GTP? Again, this is what I'm finding. You need to think about the structures of everything you're working with. We're chemists. OK, what is the difference between the diphosphate and the triphosphate? AUDIENCE: It's harder to hydrolyze the next phosphate off. JOANNE STUBBE: Well, it's not. You know, all of these things-- without an enzyme, all of these things are hard to hydrolyze. Why? Because you've got negative charges all over the place, and a nucleophile can't get into the active site. So they're all hard to hydrolyze. So that's not-- you have to think about that, but that's not what I'm looking for. Yeah? AUDIENCE: So if you run a gel or something, they should come out-- GDP and GTP are going to come out in the same-ish area, whereas, obviously, phosphate-- JOANNE STUBBE: OK, so that's what you need to think about. But what can you take advantage of as a chemist where they don't come out in the same-ish area? AUDIENCE: I mean, if you label them, the gamma phosphate, then the label won't come out nearly anywhere. JOANNE STUBBE: So that's absolutely true. But what is it about this molecule? Because I've been sloppy. What is it about this molecule that allows the distinction between your starting materials and products? This is what developing an assay is all about. How are you going to monitor this reaction? So in this paper, one of the graphs looked at Pi production. We're going to look at this if we get this far. OK, so how would you distinguish between these things as a chemist? You have no idea. You, you haven't any idea, not good. OK, what about you? This isn't a hard question. Look at the structures. And as a chemist, how would you distinguish your starting material from your products? That's the question. And that is the question in any assay you have to develop. That's what you've got to figure out. You've got to figure out a way to distinguish the starting materials from the product. Now, if we have a base here, and if this is G, we have a base here. What do we know about guanine? What's its absorption look like? What's its absorption spectrum look like? AUDIENCE: 210 [INAUDIBLE]. JOANNE STUBBE: How much? AUDIENCE: Isn't it like 210 nanometers. JOANNE STUBBE: I can't hear you. You need to-- don't mumble. Look at me in the face and tell me. You know, don't be shy. I mean, we all ask questions. [INAUDIBLE] We're here to learn. Right? Yeah? AUDIENCE: It absorbs in the UV. I think it's 210 nanometers. JOANNE STUBBE: OK, so it's not 210. So you guys need to go think about amino acids and nucleic acid. It absorbs at 260. OK, so I mean, potentially, you could sit at this absorption at 260. But what does GTP look like? GDP look like? It has the same base. So you're not going to see any change. So that's useless because you need to be able to monitor a change during the reaction. OK, so what else about this molecule will easily let you, as a chemist, determine substrate from product? AUDIENCE: The charge. JOANNE STUBBE: Yeah, the charge, yes. AUDIENCE: Just do anything with the charge. JOANNE STUBBE: So here we have all of these negative-- every oxygen is negatively charged. Here we only have two phosphates. Every oxygen is negatively charged. Phosphate-- all right, let me ask this question. We'll see how much we need to be thinking about here. So we have-- what is the charged state of phosphate? Can anybody tell me? AUDIENCE: Minus 3. JOANNE STUBBE: Pardon me? AUDIENCE: Minus 3. It depends on the pH of your solution. JOANNE STUBBE: Yeah, well, we're at neutral pH. So you look at all the buffer. You know what the buffers are. They've described the buffer in their reaction. So you're at neutral pH. What is the charge? AUDIENCE: Minus 2. JOANNE STUBBE: Yeah, so it's the pKa of the first proton loss is at 1.6. And the pKa of the second proton loss is about 6.8. So you'll have a mixture between 1 and 2. So this is incredibly different from this. And that makes it-- how do you separate things? By an anion exchange column, which separates things based on charge, some kind of a TLC system, which can separate things based on charge. And so that's what you have to do in your overall assay. OK, so the second place where you're going to use radioactivity is an assay. OK, and in the paper you read, not only did they use it for GTP, they had to use it to monitor peptide bond formation. Can anybody tell me how they did that? So what are we looking at if we go back to the original? What's the product of the reaction of the EF-Tu reaction with the ribosome? What's the product you get out? AUDIENCE: [INAUDIBLE] on EF-Tu and also label the hydrogen on leucine. JOANNE STUBBE: OK, so you're labeling the hydrogen on leucine. OK, but then what are you looking at in your assay? We're developing an assay. Here we're developing an assay where GTP is going to GDP plus Pi. What are we looking at in the case of the leucine in this experiment? AUDIENCE: The leucine is incorporated into the peptide. And you have the [INAUDIBLE]. JOANNE STUBBE: OK, so but where is the dipeptide? So that's correct, yeah. AUDIENCE: It will be in a P [INAUDIBLE] on the ribosome. JOANNE STUBBE: Yeah, but what's it attached to? Is it a dipeptide? AUDIENCE: Yeah, it's attached to the less phenylalanine. JOANNE STUBBE: Yeah, and what is that attached to? AUDIENCE: Another tRNA. JOANNE STUBBE: What's the phenylalanine attached to? If you look over here, what is everything attached to? AUDIENCE: Another tRNA. JOANNE STUBBE: It's attached to a tRNA. So could you separate a tRNA with one versus two amino acids chemically? Is that easy? AUDIENCE: No. JOANNE STUBBE: Now you have charges. Right? You have huge numbers of charges on your RNA. But they're the same on all the tRNAs. So you have one amino acid, which has a carboxylate end and a second amino acid, which has the same charge. Do you think that's going to be easy to separate? No. So does anybody know what they did to make this assay work? AUDIENCE: Put the label on leucine so the leucine is incorporated. Then you're still different [INAUDIBLE].. You can have basic number. Then after the conversion, you have a signal. JOANNE STUBBE: OK, so after conversion, you have a signal. But then the question is how do you detect this. So you have-- I mean, I guess what they could have done-- so we started out with a leucine that's labeled. And so what you're saying is that you have a way of detecting your leucine on the tRNA. So this is all attached through an ester linkage. So this is attached to the tRNA. So what you would be after is separating an amino acid from a tRNA. So that's possible. You could potentially do that. But what do you think about the ester linkage? This is all the thought process that goes into an assay and making an assay robust. Do you think that ester linkage is stable? You're going to have to chromatograph it someway to separate your starting material from product. So the answer is it's not very stable. And if you don't know, you've got to figure that out. So what they do is they quench the reaction with hydroxide. OK, and why did they quench the reaction with hydroxide? So this is a rapid chemical quench like we talked about last time. Why did they do that? AUDIENCE: To hydrolyze the ester. JOANNE STUBBE: Exactly, so then what do you have? You have, you know, your dipeptide here. Or you could hydrolyze before. And then you would have no label at all. And so then you can monitor dipeptide formation. So if you looked at the details of the graph that they presented, they weren't looking at tRNA charged with a dipeptide. They were looking at the peptide. And so that should have been a clue. Immediately, you go back to understand what's going on in the assay. So you have assays. This is pretty important. And where's another place where you might want to use radio label, where you need a sensitive assay? We're going to see radioactivity is incredibly sensitive. I'm not getting very far. But what other kind of an experiment might you think about if you have some kind of a mammalian cell, and you have receptors on the cell, for example? And you don't have very many receptors on the cell. You have, you know, sub-nanomolar number of receptors. Where else might you want to use radioactivity? And we're going to see this in the cholesterol section again. Anybody got any idea? So you have some receptor on a cell. And you want to count the number of receptors. We need a quantitative way of looking at that. AUDIENCE: We need to measure uptake. JOANNE STUBBE: So uptake is another place. You absolutely would want to use it to measure uptake. It's frequently used also to measure binding. OK, so you have to figure out a way to prevent-- on the cell, you can prevent uptake by just cooling down the lipids. And then you're measuring binding. OK, and that's exactly what they did in the LDL where they count the number of LDL receptors. So the other place where you're going to see this used over and over again is some kind of binding assay. And there are many ways to measure binding. You're going to have a whole recitation on this. Most of them aren't as sensitive as the radioactive methodology. OK, so let's move on after that long digression. OK, so what you need then is a quantitative way to measure radioactivity. OK, oh, the other thing I wanted to just point out, as you pointed out before, there's another way of detecting radioactivity using a phosphorimager. And you can read about this in detail. So what you do is you have your gel or a TLC plate. You have an image plate on top of it that somehow collects all the energy emitted from your radioactive decay. And then you quantitatively release that energy in a way that allows you to quantitate the amount of radioactivity you have on your spot on the gel or your spot on the TLC plate. And for example, tritium, with the lowest energy, you might have to put a plate onto your gel for a month and a half. That's how insensitive it is. You don't have enough energy to collect enough data to give you some kind of an answer. So you need to think about the energy, and you need to think about the method of detection. Tritium is the cheapest. It's the easiest to get your hands on. s it's the least sensitive because of the low energy that's released. OK, so the other thing that I think is amazing about the phosphorimager is, if you look at the linearity of detection, it's linear over five or six orders of magnitude, whereas, in the old days, you used to use some kind of film on top. And the film would absorb the radioactive decay and make a spot. And that was linear over a period of over one order of magnitude. So you had nonlinearity. So that was really hard to do quantitation. So phosphorimager have revolutionized what you can do in terms of analyzing TLC or gels like Liz talked about today in class. OK, so what we need then is a quantitative way of actually measuring radioactivity. And what is the standard for radioactivity we use? And so the quantitation and the standard is called a curie. It also could be called becquerel after the discovery of radioactivity. And there's a relationship between the two. And what we know, the standard of radioactivity with the Curie is defined as the substance that decays at 3.7 times 10 to the 10th disintegrations per second. So one curie equals 3.7 times 10 to the 10th disintegrations per second. Or the number that you often see is 2.2 times 10 to the 12th disintegrations per minute. So this is often what you see on the bottles when you actually buy radiolabeled material. OK, so again, what you see is you're counting. Efficiency, as I've already described, varies with the energy that's released. And you have to think about quenching. That was just repeating what I've already told you. OK, and so then what do you do? So when you purchase radioactivity, how does it come? OK, so you guys are used to purchasing something from Sigma or Aldrich or wherever you get it. You look at it, and you can see something in the bottle. When you purchase radioactivity, you can't see anything. Why? Because there's no material, almost no material in your bottle. It's all radioactivity. So if you put it in a scintillation counter, you would have, you know, 10 to the ninth decompositions per minute, OK, but no material. So you can't work with it because you can't weigh it. You can't do anything with it. OK, you have-- I don't know-- a picomole of material. It depends on the material that you buy. So the question is what do you do with this material when you get it. Well, you want to be able to use it. And in our case, how are we using it? We're going to buy GTP that's gamma P-32-labeled. To be able to use this, we need to measure something. So what is the first thing we do? Has anybody got a guess? You can't use what you buy because what you buy is you'll get a little vial like this. And that's what you see, or you might be able to see some red material that's decomposed material actually. Yeah? AUDIENCE: You need like a kinase that'll exchange the phosphate with the radioactive phosphate. JOANNE STUBBE: No, I mean, you could do that if you wanted to convert it into something else. But we want the gamma P-32-labeled ATP. That's what we want to use in our assay. So what do you do to make this usable? AUDIENCE: You add some buffer. JOANNE STUBBE: Do what? AUDIENCE: You add some buffer to the [INAUDIBLE].. JOANNE STUBBE: You add some buffer. OK, does that change the amount of material? No, so we probably do have some buffer, OK, because we want to be able to transfer it into something so we can do our assays. So go ahead. AUDIENCE: Yeah, then we're going to transfer it when you have a specimen that you are going to take some buffer [INAUDIBLE]. JOANNE STUBBE: OK, so you can, but you have no material in there. So if you had a substrate that was 10 to the minus 12th molar in solution, would the enzyme ever turn it over? Probably not, because it could never find it. OK, so that's not going to work. So what is the-- go ahead. What would you do? AUDIENCE: Like would it matter if like the radiolabeled phosphorus were just like a fraction of regular phosphorus? Like could you add some like unlabeled phosphorus? JOANNE STUBBE: Exactly, and so this is the key point. The first thing you do is you take unlabeled material, and you add it into the radiolabeled material. And how much you add depends on what you're using it for. So if you're going to use assays, and you don't need a very sensitive method, you can add much more. If you're going to look at a binding consent, you know you're pushing a lower limit of detection because you have some estimate of the number of receptors. Then you would add much less. So what you're going to do then-- the first thing you do when you get radioactivity is you add unlabeled material. And I think this is something, if you didn't get anything out else out of today's discussion, I think most people won't get this. When you work with radioactivity, most of material, one molecule, only one molecule in 10 to the sixth to 10 to the ninth is radioactive. All the rest are non-radioactive. OK, so this is just telling you about the sensitivity of the method. Somehow using a scintillation counter or using these phosphorimagers, you can quantitate the amount of radioactivity you have present. So when you're dealing with radio label, most of it is unlabeled. OK, so what does that tell you then? So again, the amount of stuff is tiny. When you add cold material, what does that allow you to do? What that allows you to do is measure. And this is the key take-home message. Now you can measure the specific activity of your material. OK, so you bought radiolabel. Let's say tritium. And then you added protonated material. And the specific activity is the amount of radioactivity per the amount of material that you have present, the number of moles of material. So it's in decompositions per minute per micromolar, decompositions per nanomole. And again, you have to change everything to accommodate quenching effects. So what you measure from a scintillation counter is counts per minute, which is just decomposition per minute times quenching. So if there's 50%, you see half as much as you should be seeing. So specific activity is given in counts per minute per amount, which is usually in micromoles or nanomoles. So if you know you have 1,000 counts per minute per nanomole, and you count 100 counts, how many nanomoles do you have? So you're given your specific activity. You do an experiment. You have 1,000 counts per minute per nanomole. And when you count this-- whoops, when you count this, you end up with 100 counts. What amount of material do you have? AUDIENCE: 21 moles. JOANNE STUBBE: Yeah, so that's it. So that's the quantitative relationship you need to remember to do all these assays that are actually in the paper that was described. So let me just give you two examples of this. We're already late. But so this is tritium. OK, does anybody see anything weird with tritiated cytidine. So this was taken off of Google from Sigma. You can buy this from Sigma now. Do you think that's reasonable that we CT3 in our methyl group? So T is for tritium. What did I just tell you about our material? How much material has got a label in it? How much-- AUDIENCE: One in 10 the the fourth. JOANNE STUBBE: Yeah, so we don't have very much that's labeled. Say we had 100% labeled. Do you think that would be an issue? Say we had a million to-- 10 to the sixth to 10 to the ninth more tritium. What do you think that might do in terms of energy? Yeah? AUDIENCE: You said tritium is much weaker. We're talking about phosphorus here. So that's like a huge signal, whereas-- JOANNE STUBBE: So even with tritium, OK, you still get enough energy. If you tried to put that much tritium in your molecule, within as fast as you could isolate the material, it would be completely decomposed. So there are ways to put tritium into the molecule, but the decay would completely destroy your molecule because you have so much radioactivity. So this, which is on the web, is completely incorrect. So what you have is one molecule in 10 to the sixth that actually has tritium labeled. And how much you have, you don't know. What you need to do is add cold material, and then you need to figure out a way to quantitate the amount of material, leucine or GTP. And then you count that amount of material. And that gives you the specific activity. So let me just say one more thing. Those of you who have to go, I'm sorry I'm late. You can go So where do you get radiolabeled material from? Do you think this is easy? I mean you could buy leucine. I just showed you we could find that on the web. You can buy a gamma P-32-labeled GTP as well. Most things you can't buy. OK, so this is what distinguishes a chemist from a biologist in many cases because I could make things radiolabeled decades ago. Doing a 15-step synthesis, I was able to make molecules that allowed me to study something that nobody else could study. So the question is you need to make your label and put it in a specific position. And so what do you start with? You start with something that's easy to work with. And you try to put the label in at the very end of your synthesis. And one of the things that you often start with is sodium borotritiride. Why would you start with sodium borotritiride? What can sodium borohydride do? This is frequently used. We'll see this used later on. Anybody remember what sodium borohydride does? Yeah? AUDIENCE: It's reductive. JOANNE STUBBE: Yeah, it's a reductant. So you can reduce a ketone or an aldehyde to an alcohol. OK, so that's frequently used to put in tritium. So what are the issues with sodium borohydride? Again, this is something that you need to think about the chemistry. What are the issues with sodium borohydride? If you're going to put your label in, OK, I just told you. How much of your sodium borohydride is labeled? What do you have mostly in there? Do you have NaBT4? No, so what do we know about tritium versus hydrogen? I guess this might depend on how much organic chemistry you've had. Tritium versus hydrogen, what's the difference? Two neutrons, OK, but it's huge in terms of weight, OK, because neutrons are the same weight as the protons. So what you see is an isotope effect on the reaction. So when you use sodium borotritiride, the activity is never the same as what you got out of a bottle. You have an isotope effect. The other thing is, if any of you have ever worked with this, and you're doing this in aqueous solution, what does sodium borotritiride do? Anybody got any ideas? In water, at pH 7. AUDIENCE: Proton exchange. JOANNE STUBBE: Proton exchange. AUDIENCE: You get hydrogen gas. JOANNE STUBBE: You get hydrogen gas. You get hydrogen gas. The whole little flask would hit you in the face with the hydrogen coming off when you're in-- and what would you get? You'd get a face full of tritium, tritiated hydrogen. OK, tritiated hydrogen is not so bad because it's not very soluble. So it goes into your system and gets washed out. If you were producing tritiated water, that's bad. So that's the other place where you do this. You can get very hot labeled tritiated water. And that you have to be really careful of because, if you breathe that in, it gets mixed with all the unlabeled materials. And it takes forever to get rid of it. So I think we're not even going to get to-- I'd let you go through all of this. But what I want you to do now is go back and think about what this data means. At least you now know what the assays are. And think about the axes. And think about, you know, cognate versus near-cognate. Why do we see a lag here? What happens at 100%? You're using up all the GTP. What does that mean? That's what I want you to think about. If you come over here, and you're looking at dipeptide, not tRNA. We're looking at a dipeptide. You need to look at the axes. They're completely different. One is micromole. One is nanomole. So we're trying to get you to actually look at the primary data, which you may or may not-- how many saw this difference when you read the paper? OK, so to me, this is what we're trying to get you to do on the first test. I can tell you a lot of people have trouble looking at this. That's what we're trying to get you to do. That's why we're going through this in so much detail. And then it becomes second nature. You just start reading it. You look at the details. And you make a judgment. If you don't understand what's going on, you go look it up, or you go talk to somebody about what the issues are with the method. Here again, the lag phases are not all that different. But here, if you take the differences in amounts into account, you're only getting 1%, 1% to 2% the amount of leucine incorporated into the peptide as you would with two phenylalanines. And that's because it's a near-cognate. And what's happening? You know, you're having discrimination between peptide bond formation and dissociation. So that's the proofreading part of the overall mechanism. So I think thinking about these two slides really tells you quite a bit about whether you believe the model that Rodnina-- whether the model is reasonable given the data you actually see. All right, I'm sorry. I'm way over. So I'm going to stop here.
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MIT_508J_Biological_Chemistry_II_Spring_2016
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25_Cholesterol_Homeostasis_5_Metal_Ion_Homeostasis_1.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JOANNE STUBBE: In the last lecture, we were focused on the proteasome. And we were focusing on how you targeted the protein of interest for degradation by attachment of ubiquitin. So here's the model that I presented at the very beginning. And right now, we're focused on how do you attach ubiquitins to a protein of interest that's going to be degraded. OK, and so here's the protein of interest the last time we talked about all the linkages. We're going to be isopeptides where you have lysine on the surface, or a lysine that's accessible, that can then get attached to ubiquitin, which at a C terminal end has glycine. So that's glycine 76-- that's the linkage common to everything. And then this ubiquitin, and you can see from the structure, has a number of lysines attached depending on what the function is of the ubiquitin. It can be attached almost anywhere. For the proteasome, we're focused on lysine 48. OK, which again, makes an isopeptide linkage here. And you need multiple ubiquitins to be able to get your protein of interest degraded. OK, so what I want to do now is talk about the equipment that's required to attach this ubiquitin molecule through the glycine to the protein of interest. And so what we're focused on is the three enzymes-- E1, E2, E3. So we want to look at the attachment. And we're going to look at E1, E2, E3, and what their function is in general. And so E1-- and it turns out in human systems there are only two of these proteins-- by, again, by homology. And this is going to be what they call the activating enzyme. OK, and so E1 is going to be-- so since we have thousands of proteins that are going to be degraded and only one of these E1s, it's going to play a role in many, many reactions. It's sort of the lynchpin. Over there there's a cartoon of what I'm going to write on the board. But E1-- and when we look at the chemistry, a key player in the chemistry are cysteines and covalent catalysis by forming thioesters. OK, you've now seen this many times in the polyketide synthases. You've seen it in fatty acids synthases. You've seen it in cysteine proteases. This is the motif that nature uses over and over again. And what she does is takes ATP and then she takes ubiquitin. And I'm just going to put G 76 at the C terminal end. So this is the C terminal carboxylate. And what she does then is uses ATP to activate the carboxylate so that it can be attached to the cysteine, which then forms the thioester. OK, and sort of the strategy is the same. I'm not going to write out the details of this strategy. But there are two ways ATP can be used. What does ATP use to activate this into a good dehydrating agent? What does nature do? What are the two options? AUDIENCE: [INAUDIBLE] JOANNE STUBBE: So adenylate or phosphorylate, alpha or gamma, you see this over, and over, and over again. You have seen this used many, many times in the first half of this course. We have talked about the tRNA synthetases, the activating domains of the non-ribosomal peptide synthetases. And so what you have done in this reaction is activated the carboxylate. And just remember this so I don't have to keep writing this down. The G is there. It's always attached to the C-terminus of ubiquitin. And now what happens-- so this is activated. And so you have a base. You never do any chemistry with a thiol. And so one then forms that a thioester by nucleophilic attack on the carbonyl. So what one ends up with then in the first step is ubiquitin attached covalently to E1. OK, so let me put that up. Again, all of this is written down in your hand outs. And so what we have now is E1, where we have S attached to ubiquitin. OK, so that's the activating step. The second step is-- involves an E2. And that's called the ubiquitin conjugating enzyme. So this is going to react with E2, the ubiquitin conjugating enzyme. What we see in humans again, that there are about 40 of these proteins. And we still have thousands of proteins that are going to be targeted for degradation. So that would imply that these E2s can be used in multiple processes and during the targeting process for degradation. So we're going to see that E2-- and so there are 40 of these. And we'll see that E2 also has a sulfhydryl group that is going to play a key role in this reaction. So what we're going to do now is a simple thiotransesterification. So we're going to transfer the ubiquitin to E2 by thiotransesterification. So again, you have base. And you liberate E1. And now what you have is E2 with the attached ubiquitin. OK, so there were 40 of these things. And what does that have to tell you-- what does that tell you about E1 and E2? I haven't given you any information about structure, but there are 40 E2's, so they obviously all have different structures. But these things have got to be really flexible. And so we have a few structures, but I don't think it even comes close to allowing us to understand really sort of the specificity of these processes. And this is a major focus of many people now. They've been discovered for a long time, but people are still messing around. And I think part of the complexity relates to the flexibility, which makes them harder to crystallize. OK, and so E3 are the ubiquitin ligases. And the latest paper I read-- they keep finding new ones-- but there are greater than 600 of these guys. OK, so even there, since more proteins are targeted for degradation than 600, E3s are going to be used multiple times. And E3s are able to form protein complexes. And again, I'm giving you sort of a generic overview. But you'll see in the next module, one of the key proteins involved in iron homeostasis gets targeted for degradation by a ubiquitin ligase that exists in a complicated protein complex. So that and then adds to the complexity you need to target all of your proteins specifically for degradation in some fashion. OK, so what does E3 do? So we have an E2. And we have an E2 with ubiquitin attached. And E2 needs to interact with E3. And this, again, is a complex. It is not a single, necessarily a single polypeptide. And E3 interacts-- ultimately what we want to do is attach ubiquitin into our protein of interest. So E3 interacts with the protein of interest. And as I alluded to before, but I'm not going to talk about in any detail, what is the basis of the interaction of a protein? It's going around. It's doing its function. How do you target it for degradation? In general, you target for degradation by the N-terminal modification or by post-translational modification. And every protein is different. So you might phosphorylate it. You might hydroxylate it. In fact, I told you and Lizzie told you in the section on ClipX and ClipP, that there was this thing called the N-end rule. OK, so one of the things you can do is you can attach different amino acids to the N-terminus. And in fact, tRNAs are actually used to do that. And we're not going to talk about the details of that. But these things, you know, you have so many proteins that are floating around. How you're going to subtly control when you're going to degrade it is not trivial. And that's a focus of many people's attention. So the model here is that-- remember, everything we're doing is isopeptide linkages. OK so, this needs to be set up. This complex needs to be set up so the lysine to which the ubiquitin is going to be attached needs to be adjacent to the ubiquitin. So you have-- a lysine needs to be activated for nucleophilic attack. And now you've attached your ubiquitin. So this is a direct transfer. Again, you're forming the isopeptides. And remember that we said in the very beginning, you don't just have one ubiquitin. You have many ubiquitins. Somebody asked me after class, how do you get the many ubiquitins attached? It turns out that people have started to study this in some detail. And in many cases, the E2s and E3s interact in a processive way to attach multiple ubiquitins. I gave you a reference if anybody wants to read about this. It was published a couple of years ago about how you control polyubiquitination. And one of the handouts, they had an E4. There could be E4s that also act process to attach ubiquitin. So you can attach more than one to give you, basically, the protein of interest with the ubiquitins actually attached. OK, so that is the machinery. We know what a bit about this machinery, but that's all I'm going to actually say in this class. But every system-- there are a lot of people studying this. And there are very few of these that are understood in really sort of molecular detail. So here again is, again, that you have these kinds of isopeptide linkages. Here is that ubiquitin with the carboxylate. And here is the ubiquitin with a lysine 48 which you can attach additional ubiquitins to. And here is another cartoon of this overall process. So what I didn't tell you in general is that E3s come in flavors. So they have little domains in them. They have HECT-- is that what it's called? I can't remember the names of these domains. I haven't-- yeah, HECT, H-E-C-T domains, HECT domains, RING domains, U domains, all of which are distinct and play a role in the details of how this process actually works. So again, this is something I don't expect you to remember the details from, but this is what you can imagine happening. So in the cartoon over here, this is what I just described, that we have an E2 that has-- this little ball is ubiquitin. Here is our protein of interest, S. And so what happens is, E2 is attaching ubiquitin to the protein of interest. OK, so that's one possibility. And in this case, the transfer is directly from E2. And that's what the ones that have been studied, the RING-finger-containing domains, do. Alternatively, you can imagine that E2 could transfer ubiquitin to E3. And once it attaches it to E3, E3 could attach it to the protein of interest. So all of those are possible. And there has been one case where that's been studied, where E3 attaches to ubiquitin. So I think you're going to find actually many variations of this theme. This is an old paper, I think. And I think the more people study it, probably the more complex it will end up getting. And so that's basically sort of the machinery-- with the major consideration, which I think is actually quite interesting from a biochemical point of view, is the N-end rule, how do you modify the N to target it for degradation. Does it have a half-life of two minutes? Does it have a half-life of two hours? And what governs all of that? And you can imagine that post-translational modification also governs the half-life-- so both of those possible. And those of you interested in polyubiquitination can look at that reference. And in fact, that paper uses methodologies we've talked about in class and recitation too. They use rapid chemical quench technology to measure the rate constants for putting on multiple ubiquitins. So this rapid chemical quenched technology continues to appear over and over again when you want to look at more details about how these systems actually work. OK, so that is allowing us to get to the stage where the ubiquitin is attached to the protein of interest. So and that is via the chamber of doom, the 20s proteosome. But now what we also would like to look at a little bit-- and this is a very active area of research-- is the lid. OK, and you saw ClipX in addition to ClipP before. And so I just want to spend a minute on the 19s lid of the proteosome. And this lid has proteins coming and going. And when you isolate it, you probably lose proteins that are loosely bound. So this is, again, a complex-- you can tell that from this cartoon over here-- a machine of 15 to 20 proteins. OK, and if you look at this machine, there has been a lot of people-- actually, one of Bob Sauer's students at Berkeley has spent a lot of time studying this human counterpart and has done a lot of really beautiful cryoEM on this. So again, this methodology we've been talking about has been used. Why are they using cryoEM? Because you can imagine, this is really hard to get a picture of because it's moving around a lot. So if you look over here, what you will see is that you have a species called Rp2-- Rpt. And there are six of these. So they're all slightly different. And this is part of an AAA ATPase system. So you have the Rpt equivalent 1 through 6. And this is an ATPase. And it sits on top of the proteosome. OK, so its function is exactly like what you guys learned about with ClipX and ClipP. What does it do? It's going to pull to try to unfold the protein. And it's going to use ATP hydrolysis to then try to thread the protein into the chamber of doom. So the model, which hasn't been anywhere near as well-studied-- the best-studied system is the one that Liz talked about. That's why we chose to look at this. It's sort of doing the same thing. It's just there is orders of level more complexity associated with this. You can imagine how complicated this is in terms of thinking about-- from the Saunders single-molecule talk, you can imagine this is even more complicated. So what do you have here that's also similar to the ClipX-ClipP system? Here is a hexamer. And remember that we looked at beta and alpha, they were sevenmers. So again, just like with ClipX and ClipP, you have a mismatch. So we have a hexamer-heptamer mismatch just like you did before. And why nature has chosen to do this, I don't know. But remember, even the beta-- the alpha subunits are inactive. The beta subunits are-- only three out of the seven are active. So it's just really quite complex. Now, what are the other things that could be involved-- these other proteins could be involved in? Well now, what's distinct from the two you had in the ClipX-P system, you had something that recognized the ssRNA tag. So you had adapter proteins. So here what we need is something that recognizes the ubiquitins. And these could be-- and in the handouts that I've given you, they tell you which one of these is which. I'm not going to talk about that detail. But you have Rpn proteins that recognize ubiquitins. OK, and I have another-- people are starting to get cryoEM pictures of all of this. This is a paper in 2012. Here is the protein of interest. Here is the AAA-type ATPase system that needs to unfold the protein of interest and thread it into the chamber of doom. And you have binding sites for the polyubiquitin tail. OK, so that's one thing you need to do with your lid proteins. A second thing you need to do is that nature recycles the ubiquitins. So what you have is enzymes that are called deubiquitin enzymes. I think that's whatever they label down here. R11 in this molecule, Up6 are deubiquitin-- sorry. [INAUDIBLE] DUBs-- OK, yeah, so they are. Both of these guys that are involved in clipping off the ubiquitins and recycling. So you have another set of proteins, deubiquitinating enzymes. And you have an isopeptide linkage, remember. And what kind of an enzyme might you expect a deubiquitinating enzyme to be? What kind of activity would it have? You want to cut these things off. What are you-- AUDIENCE: Protease. JOANNE STUBBE: --going to do? Protease, OK. And it turns out almost all of them, there-- again, we're identifying them continually. They're not so sequence-identifiable by looking at bioinformatics. The ones that have been looked at or all cysteine proteases. So the ones that have been studied in detail are cysteine proteases. But remember, they're recognizing isopeptide linkages, not peptide linkages. And as in the case of cysteine proteases, what do they involve? They involve covalent catalysis. So again, here is another example of stuff you learned in the first part of the course that you're going to-- you see over, and over, and over again in nature. And hopefully, this is now becoming second nature to you guys, that these kinds of processes actually happen. OK, so this is the lid. I'm not going to say anymore about that. You see the equipment. You see how complicated it is. And every system you study in biology, and if you care about the regulation, you're probably kind of have to think about degradation. And you're going to have to individually look at the proteins of interest and figure out what the E2s and the E3s are and what the signals are that control this overall process. So this was just taken out of some recent review, but it just gives you an idea of where-- you see this is a couple of years old now-- but, where you see this kind of machinery. We're going to see it in the next section. We're going to see a key player in sensing iron is degraded by ubiquitination. In addition, you can imagine progression through the cell cycle, apoptosis, immune surveillance, they're all regulated by protein-mediated degradation. So this is a fundamental mechanism of regulation. And so having, I think, a cartoon overview that I've given you in class is really important to have in the back of your mind when you're thinking about the system that you might be working on. And this was a paper that was very recently published. And so we've been focusing on cholesterol homeostasis. And remember, when I introduced this topic, we were talking about Insig and HMG-CoA Reductase. And HMG-CoA Reductase is targeted for degradation by Insig. That's why we made this digression. And if you go back now and look at what people have pulled out of the literature-- we're going to look today very briefly at gp78 in your problem set due this week. We'll see that gp78, which people thought was the whole story, is not the whole story, that there is another E3. Hopefully you will get that out of the data that I've given you in problem set three. And there is yet another system involved in cholesterol degradation of HMG-CoA Reductase, but it's not limited to HMG-CoA Reductase. One also has degradation of the transcription factors SREBP. We've talked about those. They use different targeting enzymes. And furthermore, a lot of people have been studying the enzymes involved in cholesterol efflux. And again, these enzymes here are also targeted for degradation. So the timing of all this and what's recognized is central to think-- people thinking about regulation, not only in systems in general, but cholesterol, specifically. OK, so that's a summary of everything we've said. And finally, what I want to do now is just come back to where we started in this section to finish up. And where we started was, we were looking at the second mechanism of regulation and the key role of Insig, that you've already seen, plays a key role in SREBP control, keeping it in the endoplasmic reticulation. So now we're coming back and looking at HMG Reductase and the role of Insig in targeting its degradation. And so we've seen these players now over, and over, and over again. So I'm not going to keep drawing the structures out on the board. But remember, if you have high cholesterol, what do you want to do with HMG-CoA Reductase, if you have high cholesterol? Do what? AUDIENCE: Inhibit it. JOANNE STUBBE: Yeah, you want to inhibit it. And so the way you inhibit it is you target it to remain in the ER. And so the question then is then, how does Insig and HMGR in the presence of cholesterol-- and it turns out, the signal is not cholesterol itself, but the signal is lanosterol. And we talked about that very briefly a couple of times. Where do you see lenosterol? If you go back to the biosynthetic pathway, it's sort of in the middle. So you have acetyl-CoA. You have lanosterol. And you have another 19 steps before you get to cholesterol. And somehow, this senses lanosterol. And people are trying to understand the details of that. How do you really know that's true? That's not such an easy thing, as we've talked about in recitation. So what we want to do then is retain HMG-CoA Reductase in the ER. So these are both ER-bound. And in the presence of lanosterol, we want to target HMG-CoA Reductase for degradation. That's the goal. The question is, is how did people go about studying that? OK, and so it turns out that they have discovered three proteins, at least in one of these systems. And the protein that I'm going to talk about for a very brief period of time is gp78. That was glycoprotein78, tells you something about its molecular weight. Again, I don't expect you to remember the details. But gp78 interacts with Insig. OK and if you go back and you look at the little cartoons I've given you, Insig, again, has lots of transmembrane helices and is stuck in the ER. So what do we know about gp78? And again, you see these cartoons that Liz has used and I've been using, since we really know nothing about the detailed structures of these systems. What we know is, at the N-terminus, we have an Insig binding site. And so people had to study that. And how did they study that? Probably by mechanisms similar to what you had to-- what you thought about looking at problem set seven. It turns out that gp78 is a ubiquitin ligase, so it's an E3. So this is an E3 ubiquitin ligase. So this is-- gp78 is an E3 ubiquitin ligase. It has a RING domain. Remember we said there were little domains that alter the way you stick the ubiquitin on. Again, we don't know the details about this. It has another little domain called Ubc7. We're really into acronym worlds. But what you need to know is that this is an E2-conjugating enzyme. So what you have now is an E3, they can bind an E2. That's the cartoons we just went through over here, E3 binding to E2. E3 is the gp78. E2 is this little protein domain. And I think what's really interesting about this protein is it has another little domain called VPC. And this is an ATPase. And if you think about this, if you want to target something for degradation, where is the proteosome located that we've been talking about? Where is it located in the cell? Actually, there are multiple proteosomes, but the ones we've been focused on, where is it located? AUDIENCE: Cytosol. JOANNE STUBBE: Yeah, cytosol, so this is a membrane protein. So how do you get this membrane protein into the proteosome? OK, that's not trivial. And this protein, this VPC domain, uses energy somehow to pull this out of the membrane so it can get degraded in the proteosome. So the VPC domain, well, pulls HMGR out of membrane. And so it gets degraded in the cytosol by the proteosome, complicated, actually quite interesting-- yeah? AUDIENCE: Is it at all understood how that pulling out happens? JOANNE STUBBE: I don't-- you know, maybe, I don't know how. I haven't found anything, but I haven't looked through the literature of any of this, the details. My guess is the answer is no, but you can go look it up. And one of the questions you can ask is how frequent does that happen? How often do you want to degrade-- do you have this domain, and how often is that domain used? And what are the characteristics of that domain? Probably a lot more is known. I don't really know off the top of my head. So this is a cartoon model. And so I'm not going to draw the model out. So I'll say the model, you can just see your PowerPoint. OK, and so this is the same kind of cartoon we've been using over and over again. So Insig is the center guy. Insig interrupts with SCAP and cholesterol to keep SREBP in the ER membrane so you don't activate transcription of cholesterol biosynthesis or the LDL receptor. We spent a lot of time on this. So here, Insig is here again. And it interacts with gp78, which interacts with these other two proteins, the E2 and whatever this protein is that helps extract it from the membrane. A key player in all of this is lanosterol. You have lanosterol in the membranes. So you could do, potentially, a similar study that we talked about in recitation this past week to look at do you see a switch with lanosterol, what are the lanosterol concentrations? What are the concentrations of lanosterol? And this is a cartoon showing this. I have no idea about the details of this cartoon, but what you're going to do then is attach the ubiquitin using this E2-E3 machinery onto HMG-CoA Reductase. And remember, that protein-- we've looked at that now a number of times-- has a steroid-- sterol-sensor domain, which is lanosterol. And it also has a cytoplasmic face. That's the HMG-CoA Reductase. You can cut this off. It's also active. And we've talked about that a number of times. And so what they have here is just a cartoon of attaching ubiquitin, which then, in the end, magic, you end up with degradation of your membrane-bound system. So this is a major mechanism of regulation involving cholesterol homeostasis. But what you see when you look at the problem set that I've given you is that it's more complicated than that. So you can knock out genes and you still get it degraded. What is the timescale? How do you do the experiments? And I think that's what people are seeing with all of these things. And in part, it becomes complicated because, if these proteins need to be modified in some way, it's not so easy to tell whether they've been modified, and what it is that is recognized by the E3 ligase. OK, so I think this is sort of an exciting and interesting area. And we need some new breakthroughs so that we can better understand how these degradation systems are integrated into regulation in general. So that's just a summary of the role of Insig, in the presence of cholesterol-- or lanosterol, in keeping the levels of cholesterol low. OK, so we finished the section on cholesterol. I think I've introduced you to a lot of different kinds of concepts. I've told you how important it is in terms of therapeutics. People are continually studying this, as you saw by the news article that Liz had given me last time. We have this PCSK9 that's in clinical trials, in addition to the statins. And I think it's going to be on people's radar screens for some time to come. So I think cholesterol is cool because of the spectacular discoveries of receptor-mediated endocytosis of transcription factors that are found in the ER as opposed to being found in the nucleus. And we've also introduced you to another generic mechanism of control, that by protein-mediated degradation. So that's the end of Module 5. And what I'm going to do now-- and we've posted this information. Again, the information will always be posted ahead of class so that you can actually have the PowerPoints out there. Some things, I'm not going to write down. In this section, there is a lot more phenomenology. What I'll try to do is give you an overview of why I've picked this phenomenology, but I'm not going to write down-- it takes a long time to write down all of the phenomenology on the blackboard. And I'm not going to do that. So integrating your notes of the things I'm going to write down with your PowerPoint, I think, is really important for you to do. And I would suggest that you bring the PowerPoint so you can see what's written down and where you might want to stick in a piece of paper where I expand on something or really tell you something in much more detail than what's written in the PowerPoint. So Module 6, so as I just told you at the very beginning, these modules are not really linked together except through thinking about homeostasis. Everything in the cell is homeostasis. In the first lecture, we're going to be talking about metals and metal homeostasis in general using the periodic table. OK, but then what I'm going to do is focus on a single metal. And the single metal I'm going to be focused on is iron. And so the reading is also posted. And there are three things for you to read. One is to think about iron in the geochemical world. You know, why is iron so important? If you look at the periodic table, why aren't we using aluminum? It's the most abundant in the earth's crust. OK well using iron and not aluminum? Well, as chemists, we ought to be able to think about that. Silicon is the other thing that's one of the most abundant things in the earth's crust. Why aren't we using silica and aluminum as life-- as the basis for life? And this article, I think it's very interesting from a chemical perspective telling you about how to think about these kinds of things. Why is it true? And I'll give you a little bit of background on that. And then you can do as much or as little thinking about it as you choose. So the first one, I'm just going to give you an overview of why metals are so darn important and try to convince you that you should all know a lot more about metals than probably most of you have thought about from an introductory course. Then in Lecture 2, we're going to talk about metal homeostasis in general. And that's going to be-- that could be applied to any of the metals I'm going to show you in the periodic table, but I'm going to focus on iron. And then in the second lecture, we're going to focus on iron homeostasis in humans. And we're going to look at iron transport from the diet, where we heard this from. How does it get taken into the cell? It can get taken into the cell-- we'll see a number of ways. But receptor-mediated endocytosis, and they told us where have we seen that? There is a protein that allows iron to be transferred around. Just like with cholesterol, you had to figure out how to keep this insoluble thing soluble with-- we're going to see there is a lot of problems with iron, so we need to figure out how to control iron's chemical reactivity. So we use a protein to do that. There is a transferrin. It's a little protein called transferrin. There is a transferrin receptor. We'll talk about that. And then there are many levels at which iron is regulated. Probably the most important regulation is a peptide hormone that I'll briefly mention, but that's not what I'm going to focus on. What I'm going to focus on is a new kind of regulation based on regulation of the translational process and proteins binding to RNA. And right now, that's a very active area of research here. It doesn't have to be a protein binding to RNA, but small molecules binding to RNA. Riboswitches are being found all over the place. And so I'm going to introduce you to translational control by proteins binding to RNA. And then the third and fourth lectures are going to be focused on more on bacteria. We know a lot about bacterial systems. Almost all bacteria require iron to survive. And Liz is the expert, so she can correct anything I say incorrectly during this lecture. Where did bacteria get their ion from? Some bacteria get their iron from rocks. How the heck do you get iron out of a rock? OK, well, bacteria have figured that out. We on the other hand are way up here. We can eat bacteria. We can eat plants. They've already figured out how to get the iron out of the rocks. And so our problem is much easier. But so bacteria are amazingly creative. And I've just chosen one of the creative ways to look at how iron is obtained. So we're going to talk a little bit about the host-pathogen battle. And I'm going to use specifically Staphylococcus aureus as an example because of the resistance problems we currently have in the clinic. You can get an iron in many forms. We're going to focus on getting iron in the form of heme, which is a major source of iron for this organism. OK, so that's where we're going. Will we get finished in four lectures? Probably not. Anyhow, so what I'm going to do today is the first five or six slides of PowerPoint. And it's more phenomenology. And then we'll get into it, the more details, in the next lecture. So here is the bottom bottle that-- do any of you take Flintstone vitamins? Anyway, I'm not supposed to digress. I can't swallow vitamins, though I like them because they taste good. AUDIENCE: [INAUDIBLE] JOANNE STUBBE: Huh? AUDIENCE: When we were little. JOANNE STUBBE: Do you remember-- does anybody remember who this guy is? No, OK-- AUDIENCE: [INAUDIBLE] JOANNE STUBBE: Oh yeah, all right, so [INAUDIBLE] Fred. OK, well you know, I always have this generation-- I'm much older than you. So anyhow, I mean, what you learned about in the introductory course 5.07 is, you learned a lot about the vitamin bottle, really, how the vitamins that you have, vitamin A, vitamin C, vitamin, all the vitamin Bs, et cetera, what they do is greatly expand the repertoire of reactions that enzymes can catalyze in all your metabolic pathways. What you don't learn about in most introductory courses is the minerals. OK, so they're on the bottle too, but you sort of ignore all of that stuff. And you know, you need iron. You need copper. You need calcium. You need zinc, et cetera. And so what I want to do is to try to give you very briefly an overview of why these metals are so important. And again, the focus is going to be on iron. OK so here is our periodic table. And these are the metals that are found inside of us-- yeah, I guess maybe. We don't have tungsten. Liz, do we have tungsten? We don't have tungsten. I don't think we have tungsten in us. So these are found in bacteria and us. And so if you look at this, all of these guys over here, where have we seen magnesium before? I've been talking about that over and over again. Magnesium is bound to all nucleotides. We're going to see this again and again. We're going to talk about-- a little bit about the proper use of magnesium which makes it function in that capacity to neutralize the charge on nucleotides. Sodium, and potassium, and iron, conduction-- calcium is involved in signaling. But what we're going to be focusing on are the transition metals. OK, and specifically within the transition metals, what we're going to be focusing on is iron. And this is-- it's hard to measure the concentrations in their localizations within the cell. But you can measure the total concentration by just taking your cell and then submitting it to some kind of mass spec analysis. We can see iron versus manganese. And we're going to, again, be focused on iron, which accounts for about 8%. And it's been estimated in this article that approximately 50% of all the proteins have some kind of metal bound. OK, it might involved in catalysis. It might not. In fact, the metals more likely are not involved in catalysis. And we'll look at that distribution. So we'll come back to this a couple of times, but we're going to be focusing over here. And what are the properties of metals that make them so special to increase the repertoire of reactions that can be catalyzed inside our bodies? OK, so these guys are unique from a lot of the reactions you've already studied in your introductory biochemistry course. And so what I want to do is sort of just give you a general overview of where you see metals involved in catalysis. And then we're going to focus on iron only. OK, so where do we see catalysis? We see iron transport. We need to get iron, potassium, and sodium in the right places, or we're in trouble. Signaling-- signal transduction, we use calcium all the time. There is huge numbers of people studying calcium signaling. Where have you seen oxygen transport? In us-- we're in serious trouble if oxygen can't be carried by our hemoglobin to our tissues. And I'll show you a little bit about that. So oxygen transport is really important. Of central importance is electron transfer and proton-coupled electron transfer. Where have you seen that? You've seen that in the respiratory chain. If you go back and you look at complex I, complex II, complex III, you see all these metals in there. What are they doing? They're doing electron transfer reactions. We'll talk a little bit, but not much, about that. So not only is electron transfer involved in respiration. Electron transfer plays a central role in all of the environmental chemistry. And so while, in many introductory classes, they don't talk about this-- we talk about humans, because most people are more interested in disease-- the coolest chemistry, without question, hands down, is absolutely associated with the bacteria and the Archae. OK, they do, like, amazing things. How do you take nitrogen and do an eight-electron reduction to ammonia? How do we do that as chemists? 200 atmospheres pressure in a 400 degrees. This is an incredibly important reaction. Where does all the nitrogen from our amino acids come from? What about our nucleic acids? And we skip all this stuff. This is like-- I mean, this, to me, is sort of, like, amazing. Another thing we skip all the time is where does oxygen, how does oxygen-- how does light take water and make oxygen gas? Without that, we'd be in serious trouble. The bacteria would definitely be taking over the world. And this, I'll show you, is sort of an amazing reaction-- nucleotide reduction. We may never get there, but the last module is, I'm going to show you, you're making deoxynucleotides. The enzymes can use manganese, iron, cobalt, and iron sulfur. So they use a wide range of metals to make the building blocks required for DNA. OK, signaling, we've all-- I just talked about calcium as a signaling agent. But now it's becoming clear, because of studies from the lipid group, and studies from Chris Chang who is a former lipid group member, signaling of metals is much more common than we thought. And people are proposing that, not only is zinc a signaling agent, but also copper. And there is a lot of problems in nerve cells with oxidative damage which we're going to come back to. So thinking about the levels and sensing of these levels I think is going to be a future area that's going to be very exciting to study. You have to regulate these metals. Transcriptional, translational levels, we're going to talk about. And they're involved in many kinds of catalysis. So let me just close by showing you one last slide, oxygen carriers. You've seen this before. That's hemoglobin. You've all studied, hopefully, hemoglobin and the cooperative binding of oxygen, how it binds, how it's released-- sort of an amazing machine. That's not the only way that organisms reversibly bind oxygen. This guy, the horseshoe crab, it uses copper. This guy-- these are worms. These are found in-- they're found in the sea, right? So you go to Woods Hole and they'll extract these worms for you. Anyhow, what do they have? They have a diiron cluster. And the strategies of both-- they all have to reversibly bind oxygen. And they've all adapted to their environments to be able to do this in an efficient way. So what I'll do next time is come back and-- let me just do one more thing. Anyhow, this is-- think about this. Put it on under pillow. Think about how it works. Look at this. This is the cofactor of nitrogenase. Not only does it have iron and molybdenum, but look at that guy in the center-- carbon, carbon 4 minus. Think about that. We'll come back next time.
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MIT_508J_Biological_Chemistry_II_Spring_2016
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29_Metal_Ion_Homeostasis_5.txt
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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. JOANNE STUBBE: So what I want to do today is continue where we left off to try to get further in into this module on regulation of iron right now in terms of humans. And we're talking about the fact that regulation occurs at the translational level. And so I'm going to introduce to you the model. And I introduced you, last time, to two key players that we'll look at in a little more detail-- proteins and little pieces of RNA. And what happens is the proteins bind to the little pieces of RNA and prevent the translation of the messenger RNA into protein, or prevent degradation of the messenger RNA, allowing the translation to proceed. So that's really the take-home message. And I'll show you what the model is. So the last time, we were talking about one of the protein players, IRP1 and IRP2. And I told you that IRP1 was a cytosolic aconitase and that you had seen the aconitase reaction, which I drew in the board last time, which is conversion of citrate into isocitrate. If you look at the model up there, citrate to isocitrate, you're simply losing a molecule of water. And then over here, you're generating isocitrate. And the chemistry is facilitated by the presence of a single unique iron in the form of iron 4 sulfur cluster, which was the first example of these kinds of clusters doing chemistry in addition to electron transfer reactions that you've been exposed to before. OK. So the question then is, what is the signal? And so we're going to see that the signal is going to be related to-- let me just get myself organized here. So the signal-- the question is, what's recognized under low iron and high iron conditions? So that's what we'll be talking about, and how does this switch work to allow translation or not allow translation to occur. And the other player that we need to be introduced to before we look at how this signal works is the iron responsive element. And this is a piece of RNA-- and I'll show you that on the next slide-- within the messenger RNA. OK. So you have a structure like this, and there are so many base pairs, in this little stem loop, that's part of the messenger RNA. And you have a three nucleotide sequence, and this is a bulge sequence. And what we're going to see is that there are many of these structures. People have now done a much more extensive-- this was the model that came forth a long time ago when it was discovered. It was discovered a long time ago, but people have since done a lot of bioinformatics analysis to try to define really what do we know about this little sequence here. Is it three nucleotides? Is it more? People now think it's a little bit more, but it-- there's variability in that you need a bulge. And so what's going to happen is our iron responsive protein is going to interact with this bulge, and that's going to be what's related to depending on the location of this bulge within the message. So we'll see this bulge can be at the 3 prime end or the 5 prime end. And this location and its interaction with this protein is going to regulate the translational process. So that's what I'm going to be presenting to you. So what do I want to say? What I want to say then is for the iron response protein 1, it has, as we saw with the mitochondrial aconitase of 4 iron 4 sulfur clusters. So that could be a switch. We're doing iron sensing. What's going to cause us-- what is the sensor of iron that allows us to translate or not translate all of the proteins-- transfer and receptor, transfer and ferroportin, all of the things we were introduced to in the previous lecture. So this protein has a 4 iron 4 sulfur cluster. And when it loads the cluster, as with mitochondrial aconitase, the protein is active. So it's found in the cytosol as opposed to the mitochondria. And it can convert citrate to isocitrate. OK. So the question is, what is the switch that allows this IRP1 to interact with this little piece of RNA-- the stem loop piece of RNA? And so the switch is that you have to lose this cluster. And what you generate then is apoIRP1. And apoIRP1-- so somehow the cluster magically disappears. And when it disappears it can bind to the IRE. So in the apo form-- that means no metal-- it binds to IRE. Whereas in 4 iron 4 sulfur loaded form, it does not bind. So what that would imply-- if you think about it sort of superficially-- if you have low iron and there's no iron sulfur cluster, the apo form is going to bind. OK. I'm going to show you the model in a minute. OK. So the switch really is related to-- in this case, the sensor is related to-- the fact that we have a 4 iron 4 sulfur cluster. So we also have-- I told you before-- in addition to an IRP1, we have an IRP2. And IRP2 also looks, structurally, like a cytosolic aconitase, but it has no aconitase activity. OK. So we have the second protein, IRP2. It's also a cytosolic aconitase lookalike, but it has no activity. And why does it have no activity? You've seen, over here, the iron sulfur cluster is required to do the dehydration reaction. So it's required for activity in the mitochondrial enzyme. This has no iron sulfur cluster. So it has no iron sulfur cluster. And what it has in addition, even though it looks like IRP2-- the structurally homologous-- it has a 73 amino acid insert. So this is a distinction between the two. OK. But now, this raises the question here-- at least superficially, you can understand that you might be able to sense iron, because you have a cluster, and you can go to no cluster. OK. And you can go back and forth. And so, remember, in a couple of lectures ago I told you about biosynthetic pathways, and I showed you a picture of iron sulfur cluster assembly-- very complicated. At the end of the notes in this part of the lecture, you'll see what the model is. I'm not going to go through that. But how you assemble and disassemble, even though this model has been around for a long time, is only recently beginning to be understood. It's not trivial, because there are 10 steps to assemble a 4 iron 4 sulfur cluster. OK. But here, we don't even have any iron. So how is the IRP2, which binds to the same IREs-- and again, in vivo we don't really know all of this. People are trying to sort that out as what the what the functions of the different proteins actually are. But how does it sense? And so I just told you that the apo form of the IRP1 binds. That's also true of the IRP2. And in fact, it can only be apo, because it can't bind a cluster. So the active form, the binding form, the apoIRP2 binds the IRE. And then the question is, what is the switch? And so what we'll see is that the switch relates to the fact that IRP2 gets degraded. So when IRP2 is degraded, it can't bind. And that's how you turn the thing off. So then that takes you back a step further-- how do you target IRP2 for degradation? And this goes back to one of the reasons that I'm going to spend some time talking about degradation in mammalian systems. And so it turns out-- how does this relate then to iron sensing? And what I'm going to show you is that you have an E3 ligase. I'll show you this in cartoon form. And I'll just say, see PowerPoint. We're not going to-- it's not really completely understood, so I'm not going to talk about it in detail. But what it has attached to it is an FXBL5 domain that looks like a protein we've seen before earlier that has an iron in it. So many of you probably don't remember hemerythrin, but that's the little enzyme in worms that reversibly binds oxygen. So that was incredible. It's structurally homologous to that little protein. So this is-- again, the details are not known. But it can bind iron and it can sense oxygen. So if you're at low iron, there's no iron bound to this little domain. And so there's a consequence. I'll show you what that is. But if it's high iron, it has a different consequence. So the sensing is back a step. Its back a step into-- remember, I told you E3 ubiquitin ligases are multienzyme complexes. So this is part-- you'll see in a minute-- of the multienzyme complex. And so under certain sets of conditions when you have high iron, what happens is this is targeted for degradation. I'll show you what the model is for how this works. So the models are the same. That is, apo in both cases bind. In one case, the iron sensor is directly related to the IRP itself, because it has an iron sulfur cluster. And in the second case, it's indirectly related to an iron cluster that's associated with the E3 ubiquitin ligase. So another point I think I want to raise-- and this will get us into the next module, which I'm not going to spend very much time on-- the reason that we have iron module juxtaposed to the reactive oxygen species is they're really intimately linked. We've talked about how iron 2 can generate hydroxide radical or hydrogen peroxide. These are iron sulfur clusters are also oxygen-sensitive. This is oxygen-dependent. So again, what you're seeing is not only do we have iron sensors, but we will see that iron sensing and oxygen sensing are linked. And I would say-- I was trying to make up your exam, and I was trying to put in a linkage so you would all of a sudden see this, and the more I read, the more confused I got. So the fact is, there are many, many papers published on this now, and the linkages-- the proteins involved-- do many things. And so sorting this out into a very simple model is really still tough. But what I what I believe right now is both iron and oxygen sensing are linked through this type of a model. So let me now just show you a little bit about IRP1. We know a lot about IRP1, because we have structures. So this is the structure actually of the cytosolic aconitase, and this is with the 4 iron 4 sulfur cluster bound. So what happens when you get to the apo form? What happens in the apo form, you now have a little piece of RNA bound. And this little piece of RNA always has a bulge with a cysteine in it. And it always has some kind of a loop. And we'll see in a second that the sequence of that loop can be variable. But here you can see that. So these little balls here that are iron sulfur clusters are the cytidine bulge in this loop. So you see the thing changes confirmation and as binding to an IRE. So that is the switch. And then the question is, how does that work at level of controlling translation, which I'm going to show you in a second. OK. So where do we see these iron-responsive elements in our messenger RNA? So messenger RNA-- go back and look at all of the players I introduced you to the last time. We have a transferrin receptor that's involved in uptake. We have DMT1. That's a dimetal transporter involved in iron uptake. So intuitively, you should ask the question, if you are at low iron, do you want to take up more iron? So you want to turn on the transferrin receptor. You want to turn on the DMT1 protein. So I think most of it makes intuitive sense. The linkage to oxygen, I think, is less intuitive. If you have a lot of iron, what do you want to do? You want to store the iron. So you in some way want to make more of the ferritin. And then the other thing, this HIF 2 alpha is a transcription factor hypoxia-- inducible transcription factor-- that's linked to many, many things-- a huge number of people are working on this now-- one of which is this linkage to iron. But it senses anaerobiasis. And so you can see, it's also linked by one of these little elements. And the next slide just shows you a more recent one where people started doing a lot of bioinformatics on this. The previous slide was from a few years ago. And again, the details, what you need to see is you, in all cases, have stem loops, a little bulge of a cytosine, and then you have some kind of a loop at the top of the stem loop. And if you look down here and you go through-- so these little stem loops are going to be either at the 3 prime or the 5 prime end of your messenger RNA. And so, for example, one of the things you see is aminolevulinic acid. Does anybody know what pathway that's involved in? AUDIENCE: Heme synthesis. JOANNE STUBBE: Yeah. So it's a rate-limiting step in heme synthesis. So there would be a place that would make sense. Remember, I told you all the iron is in heme and hemoglobin. OK. So almost all of these stem loops that you'll see, if you go back and you look through your notes, will make sense in terms of the big picture of how you want to control the levels of these proteins to deal with high iron or to deal with low iron. OK. So that's iron 2. And so here's the picture of IRP2. And this is the model for how IRP2 works. And so here's the case when you have high iron. And when you have iron, this part here, the Fbox, Skp1, and Cul are all part of the SCF E3 ubiquitin ligase. And I don't expect you to remember the names, but remember I told you the E3 ubiquitin ligase is the one that does what? It attaches ubiquitin on to the proteins, targeting it for degradation. So this little part is the ubiquitin ligase. Here's your E2. Remember, you always need an E2 and an E3. And somehow the E2 is attaching this onto the IRP2, which is targeting it for degradation by the proteasome. OK. So this is exactly like the model we put forth a couple of lectures ago. So again, this is the part that's most interesting. If you go back and you look at hemerythrin, which irreversibly binds oxygen with two irons, you have a diirons site. And earlier in your notes, I showed you what that site looks like. This site is intact because the protein is folded. Under conditions of very low iron, what happens is this becomes unfolded. And then this part of the protein gets targeted for degradation by another E3 ligase-- not this one. And then you've lost your sensor. So the IRP2 remains stable. So you might think this is complicated-- and maybe I didn't spend enough time going through this-- but you should go back, and you should look at the explanation again. So the two key switches are here and here. This one is the more complicated switch. Everybody thought everything was understood once they found this little iron binding protein that models hemerythrin, but nothing could be farther from the truth. We still really don't understand, overall, how this fits into the big picture. But it's not an accident that iron and oxygen are required to fold this into this little bundle that looks like hemerythrin. So those are the switches. And so now, what I want to do is put forth a model. So let me see. So what is the model for how you want to turn these things off and on? OK. So we have two things-- we have IREs-- Iron-Responsive Elements-- that can bind in front of the message to be transcribed or at the end. So what we're going to look at two sets of conditions-- one is under conditions we have low iron, and one is under conditions where you have high iron. How do you sense those conditions? So that's the question. So again, we're sensing low versus high iron. So let's look at low iron first. So what we're going to see is we're going to have a stem loop. So here's my messenger RNA, and here's the 3 prime end, and here's the 5 prime end. And here, you initiate translation. And so if it binds-- if this protein IRP2 or IRP1, both in the apo form-- IRE-- and they do both bind. People are trying to sort all of this out. So this is IRP1 or IRP2. What happens to the translation? What happens to the translation is it's inhibited. So if you have this little stem loop in the front of your message, you inhibit translation. And so what I'm showing you now-- and then I will give you some examples-- is the key to thinking about this. And most of it actually is intuitive once you remember what all the factors are that are involved in iron homeostasis that we've already gone over. So binding inhibits translation. OK. So then we have the second case at low iron. And again, we have a 5 prime end, and then we have a 3 prime end, again, of our messenger RNA. And here is the initiation of translation. And in this case as well, you have the same sort of structures of stem loops. They are similar but distinct. And what can happen with the apo form of IRP1 or IRP2? Again, it can bind to these stem loops. So you can have-- this chalk is not working-- your proteins, they're all bound-- they may or may not be all bound. I don't think we know that much. But what you see is the number of stem loops at the 3 prime end is variable. It depends on the message. So number of stem loops is variable. So what does this binding do? What this binding does is it prevents the messenger RNA from being degraded. So it basically stabilizes the messenger RNA. So this model binding prevents messenger RNA degradation. So it stabilizes the messenger RNA. So that's the model. OK. So now, let's just look at a couple of examples. And then what you can do later on is go back and think about this more of what's going on. So again, this is the model. And let me just make sure I go through the ones I want to do. So we're still at low iron. And we'll do two at low iron. And then we'll look at the consequences of what happens at high iron, and does it make intuitive sense based on what we think the function of these proteins are that are going to be translated? So let's look at low iron. So we're at low iron. And let's look at ferritin. OK. So what is ferritin? It's the iron storage protein. So under low iron, do we want to store iron? No. So if you have the choice of these two modes of regulation, what would you choose? Where would you put your iron-responsive element? At the 5 prime end the 3 prime end? So we're at low iron. We don't want to store iron. So we don't want storage. So what would we do? If these are the two choices-- and these are the two choices from experimental data. There are many other variations on this you could have imagined, but this is the model that everybody agrees on at this stage. So where would you put your stem loop? Yeah. You'd put it at the 5 prime end. And why would you put it at the 5 prime end? Because it prevents conversion of your message from ferritin into the protein. So you have less of the ferritin. So what you see now is you have-- again, so this is a stem loop at the 5 prime end prevents translation and have lower concentration of ferritin. So that's exactly what you would expect. Some of the others are less intuitive, but we've seen ferroportin. Remember, ferroportin is the iron 2 transporter in many cells which allows the iron to come from the inside to, get picked up by transferrin, and redistributed to the tissue. So it, in conjunction with the hepcidin peptide hormone we briefly talked about plays a really important role actually in controlling where the iron ends up going. And in fact, what you would like to be able to do-- say you had not very much iron, where would you want to put your iron? Would you want to put your iron in some metabolic pathway that's not so important, or would you want to put your iron in a metabolic pathway that's very important? You would want to put it into the pathway where you really need it to survive. And so this is a subtle tuning on all of this. And so an example of how this can be tuned if you look at an iron-responsive element binding protein is succinate dehydrogenase. Any of you ever heard of succinate dehydrogenase? And where have you heard of it? You have heard of it, you just probably don't remember it. [INTERPOSING VOICES] JOANNE STUBBE: Yeah. So it's in the TCA cycle. So it converts succinate, which is a hydrocarbon, into an olefin, an alpha beta unsat-- into fumarate. So remember the TCA cycle, you can tune it down or you can tune it up. So if you really were desperate for iron, you would probably tune down the TCA cycle. So in fact, if you look, you'll see a stem loop in front of succinate dehydrogenase which prevents its translation and tunes down the pathway. So there's a subtle example of how nature has-- at least is the way we rationalize the experimental observations of what nature has done. Now ferroportin, which is the way I started on this, sets priorities. And it does this in conjunction with hepcidin, which we already talked about. Remember, hepcidin can target ferroportins for degradation. And this allows the iron to be distributed in defined ways within the cell. And in fact, what you want to do, in this case, is have the stem loop at the 5 prime end so that you don't export the iron inside the cell to the outside. So that's what it does. And some of these, as I'm saying, are easier to rationalize than others. The ferritin one is really easy to rationalize. The ferroportin is easy to rationalize based on what I just told you. But what you see also is that in some of these systems-- I don't know how much you guys thought about RNA, but you know messenger RNA can be spliced. In different cells it's spliced differently. You've also seen that cell types, in terms of iron homeostasis, the enterocyte, the macrophage system in the spleen, red blood cells are much more important, it might be, if you're in some other tissue, the splicing site is different, and you don't have a stem loop. So you can alter the regulation by alternate splicing systems. So these are these two are at the 5 prime end. What about the transferrin receptor? So let me put this down here. What about the transferrin receptor? What would you expect at low iron-- we're still at low iron-- the regulation to be from the transferrin receptor? What do you want to do at low iron? So this is another example, low iron. Let's look at the transferrin receptor. What does the transferrin receptor do? Hopefully you know this. Yeah. AUDIENCE: [INAUDIBLE] JOANNE STUBBE: You need to speak louder, I can't hear anything you said. You just went like this. That didn't mean anything to me. AUDIENCE: It helps to intake iron. JOANNE STUBBE: Yeah. So it helps to intake iron. So if you have low iron, what do you want to do? AUDIENCE: You want to increase-- JOANNE STUBBE: Yeah. So you want to increase that. So where would you put the stem loop? AUDIENCE: 3 prime end. JOANNE STUBBE: Yeah. So you put it at the 3 prime end, because that stabilizes the messenger RNA of the transferrin. So here, at low iron, you want to increase iron uptake. And that means that if you have the 3 prime end, you're going to stabilize the message. So you can go through each one of the proteins that we discussed in the last lecture. And before you look at it, try to rationalize under different sets of conditions. This is low iron. What would you expect to happen at high iron? Here, let's just look at this one so I don't have to draw this again. But what would happen to the transfer receptor at high iron? Do you want to take more iron into the cell? No, you don't. So what you want to do is get rid of the transferrin receptor. So now what do you do? At high iron, if you're IRP1, you switch to pick up the iron sulfur cluster. It no longer binds. And so now what happens? So this is all bound. So it's stabilized and bound. In this case now, messenger RNA is degraded. So the big players in iron homeostasis, I think, are easy to rationalize. If once you know-- this might not be so rational why you would stabilize messenger RNA or whatever, but this is the way nature designed this. Once you remember this-- and remember, the switches are just apo binding, and somehow they sense iron and they no longer bind, whatever the details are-- you should be able to understand in different kinds of cell types how you might regulate the iron at the translational level. So I think that is all I wanted to say. This is just a summary of what we've done in the human part. And we're thinking about-- I gave you a big picture of what happens in humans. This is the summary of that big picture with all these factors that are regulated at the translational level by the iron-responsive binding proteins. And so you can go back and look at this cartoon. Whether you want to store it, whether you want to distribute it, whether you want to put some in the mitochondria-- all of that kind of stuff is regulated at the translational level. So in this module, the second lecture, which was longer than I wanted it to be-- but that's life-- was focused on the big picture for human and how iron is transported. And uptake, which we talked about by divalent metal irons, transporters, and by transferrin in the plus 3 state, and this question of regulation at the translational level. Now everything-- the hepcidin, we didn't touch on very much. Very complicated, but it plays a major role systemically. Whereas these others-- what we were just talking about is more specific for each cell type. And different cell types want to have regulation in different ways. So the bigger picture is the hepcidin. And it was discovered a while back, but I still would say we don't understand a lot about what's going on in terms of that hormonal regulation. So now what I want to do, as advertised, is move into bacteria, and how do bacteria do the same thing. They have the same problem. We talked about metal homeostasis-- exact same problem in human and bacteria. But in the end, the bacteria want to survive and we want to survive, so we have the battle between the bacteria and us for iron. So what I want to do is introduce you to the bacteria-- generically, how they take up metal to use for the same things that we use it for-- a little less complicated, maybe, than humans. And then what we're going to do is I'll introduce you to this war between bacteria and humans. And then we're going to focus on one bacteria that's a major issue nowadays-- Staphylococcus aureus-- because of resistance problems. This is a problem that Liz's lab has worked on. And specifically, I'm going to give you one example of how Staphylococcus aureus gets iron out of our hemoglobin. That's going to be the example. And the system that you'll see, it's amazingly cool. But you'll see, there's still many things we don't really understand in a lot of detail. So what I want to do now is introduce you-- how am I doing timewise? OK. So what I really would like to do is draw this out on the board, because it's complicated. And I know what happens if you use PowerPoint, you go through it at 100 miles an hour. But I'm going to be using more PowerPoint to get through something. So anyhow, this is an overview of where we're going if you forget. So what I wanted to do, at least a little bit, we're going to be focusing on gram-positive and gram-negative systems. And I want to tell you what is the difference. You all know or have heard about gram-positive and gram-negative bacteria. They use different strategies. They use the same strategies, but they use distinct strategies because of their structures. And so what I want to do is give you an overview, and then we'll focus specifically on Staph aureus. So in gram-positive, here we have our plasma membrane. So this is the plasma membrane. And this big guy here is PG-- the peptidoglycan. And we'll see, in Staph aureus, the peptidoglycan is going to play a key role. So you need to understand the structure of the peptidoglycan. So I am going to spend a little bit of time describing to you the structure. It's also the major target of many antibacterial agents that are currently used. Why? Because it's unique to bacteria. So you have only this plasma membrane. There's no outer membrane. That's going to be distinct from gram-negative. And so the question is, how do you get iron from the outside to the inside? And so one of the ways you can take in iron is-- you've already seen this, and you've talked about it in detail in the first half of the course-- siderophores. And we've already talked about the fact that we have many, many different kinds of siderophores. And somehow these siderophores-- and we'll look at a few structures-- can get to the outside of a cell. They pick up iron in the plus 3 state, and then they need to bring it back to the plasma membrane. And then somehow it needs to get transported across the plasma membrane. This is a transporter. Most of them are called ABC transporters and they require ATP. And when they get across, they take the siderophore with the iron into the cell. So that looks simple enough. We'll see that the strategy of gram-negative bacteria is going to be distinct, because it has an outer membrane. So how would you get the iron out of the siderophore? And so I'm going to push this up, and I'll come back down again. So how do you get the iron out of the siderophore? And of course, what that depends on is the reduction potential of the iron. So we will see with enterobactin, in which you already looked at, the reduction potential under neutral conditions is minus 750 millivolts. Minus means that it's really hard to reduce. It wants to be oxidized. It's outside the realm of what you can do inside the cell. So And we want to reduce iron 3 to iron 2. Why? Because we increase the exchangeability of our ligands. That's why that was introduced before. So you could reduce this, potentially. And so I'll just put a question mark there. And so then what you have is a siderophore. So let me just write this down so you don't forget. So this is the siderophore. And then you have your iron. So what happens to the siderophore with no iron? It can now get recycled back to pick up more iron. So let me just put this here. This is recycled. And we're going to be focusing on here taking up iron from siderophores, but we'll see that you can take up iron from hemes. And you have the same issue. You're going to use the same strategy. You'll bring it into the cell. You've got to get the iron out of the heme, and you have to recycle it. So if you can't reduce it, what do you do? Does anybody remember what you do with enterobactin? Anybody remember the KD? We talked about this last time, but it bonds like a son of a gun. It's hard to reduce. You have ester linkages in enterobactin. If you go back and look at the structure, there are proteins that can hydrolize the ester linkages. So ring opens-- makes it bind less tightly, and so it can be released. So in the case of enterobactin, you have an esterase. So let me just show you that, and then we'll come back again to the gram-negative. But if you look at the siderophores, there are 500 siderophores. Here is enterobactin. here are the esters. You can hydrolyze them to release. You can't reduce, because, again, the more negative, the more it wants to be oxidized. And the range of reduction inside the cell is maybe minus 500. You can't get that much above that. But if you look down here at citrate-- remember, we were talking about citrate-- unusual in that citrate is part of this aconitase IRP1 and IRP2 system. But what's the reduction potentially are completely different. So if you had iron citrate, you could easily reduce it under physiological conditions. So the strategies you need to be able to release the iron to then use the iron for what you want to do is distinct depending on the siderophore. So if we go back, now let's just look over here and I'll draw that in parallel. So what's the difference between gram-positive and gram-negative? So let's draw that out. And then what I'm going to show you, rapidly, is, again, the strategies with heme are subtly different, but very, very similar. We have different sets of proteins. So with gram-negative we have an outer membrane. So this is gram-negative. And what we have in the outer membrane are proteins. It has a lot of proteins. And it has a big protein with a ball in it. And these proteins-- it has 27 beta strands, and these are beta barrels. So there are many, many of these proteins. In fact if any of you heard Dan Cohn's talk this past semester, he's figured out how do these things get made down here and get inserted in the outer membrane. It's an interesting problem. So these are beta barrel proteins, and they have 27 strands. We then have a peptidoglycan. But the peptidoglycan is distinct. It's much smaller. It doesn't take up anywhere near as much space. And then you have your plasma membrane. And then in the plasma membrane-- so this is a plasma membrane-- you still need to do the same thing. You need to get your siderophore into the cell. So what do you have here? You still have transporters. And those transporters are going to allow your siderophore to go into the cell, just like we saw with the gram-positive. So over here then, we have a siderophore-- again, the same types of siderophores. So somehow it needs to get inside the cell. And we have many of these beta barrels, and a lot of them are specific for a given siderophores. There are many, many of these things. We'll look at E. coli. There are 10 different ways to get iron from the environment into the cell. That tells you how important all of this is. And it turns out that you also have, inside the cell, a periplasmic binding protein that can pick up the iron when it gets transferred across here. So you have a periplasmic binding protein. And one of the questions is, how does the siderophore get transferred? And to do that in gram-negative bacteria, you need a machine. And that machine is composed of three proteins. It's called the tan protein. If you look over there in pink, you have tanB. It's exbB. And this should be not C, but exbD. So there are three proteins required. And they somehow can use the proton motor force from the inner plasma membrane to allow transport across the outer membrane. So in all of these, one has tanB. So this is tanB. And tanB can recognize part of the beta barrel. So it interacts with the beta barrel protein. And this is exbD and exbB. And again, you generate a proton motor force which allows the siderophore to get into the cell. It then gets transferred to a periplasmic binding protein. And then what does it have to do? So from here, it has to go through our transporter. So let me put this up here, just like we just did before. So your siderophore-- so you can't see the bottom of my transporter-- comes through So this is the plasma membrane. And what do you have? You have the same problem. You have to get the iron out of the siderophore. And so the problem is exactly the same and gram-negative and gram-positive. So you somehow have to get it in. It's more complicated to get it in with gram-negative because of the different constructions of the peptidoglycans and the outer membrane. The other thing I wanted to say about the other outer membrane is-- which I don't know if you guys know, but I think it's incredibly important and is a major issue in a human disease-- the fact that, in addition to this outer membrane in these beta barrels, the whole outer surface is covered with sort of amazing molecules called lipopolysaccharides. So the whole outer surface is covered with LPS-- I'm not going to write it out-- lipopolysaccharides. Which, actually, one of my best friends elucidated the whole pathway for how that works. It's a beautiful, beautiful set of biochemical studies to figure out how this thing is made. It's got lipids. It's got all these sugars. It's got all this stuff hanging off of it. And this thing is really important in human health. If you read about infections, they're always talking about lipopolysaccharides. So what I'm going to do next time, just by way of showing you to introduce you to this-- you can see here in the next cartoon we have the same problem when we want to take up hemes as opposed to siderophores. And we're going to focus on hemes. So this is a cartoon, very similar to the one you just saw. And there are a couple of proteins on the outside that you need to think about. How are you going to get the heme across the peptidoglycan or into the cells? So the model is very similar. You should look at that. And then we'll see this is what your problem set is going to be on. This is for Staph aureus. And we'll see that if you get a heme, there's going to be bucket brigade that can transfer the heme through proteins covalently bound to the peptidoglycan into the cell. It's sort an amazing system, and that's what we're going to talk about for probably the first half of the next lecture. So you're going to have to read on that on your own to solve the. problem.
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